Transformasi Obyek (Kasus 2D)anny.if.its.ac.id/wp-content/uploads/Grafika08-TM4.pdf · Mampu...

Kuliah Grafika Komputer -

Teknik Informatika ITS Anny Yuniarti - 2008 1

Transformasi Obyek

(Kasus 2D)

Grafika Komputer

Semester Ganjil 2008


Teknik Informatika ITS 2 Anny Yuniarti - 2008

Kompetensi

1 Mampu membangun tool untuk mentransformasi obyek

2 Mampu memahami konsep transformasi affine yang merupakan kombinasi dari rotasi penskalaan dan translasi

3 Mampu mengimplementasikan konsep transformasi dalam sebuah program OpenGL



Referensi

Computer Graphics using OpenGL 3rd

Edition by FS Hill Jr and Stephen M Kelley

ndash Chapter 5

Computer Graphics with OpenGL 3rd Edition

by Donald Hearn and MPauline Baker ndash

Chapter 5

httpwwwcsvirginiaedu



Pokok Bahasan

1 Transformasi 2D Dasar

1 Translasi

2 Rotasi

3 Penskalaan

2 Kombinasi Transformasi

3 Transformasi Affine



Kegunaan Transformasi

Me-reposisi ataupun me-resize obyek

Pada proses viewing untuk mengubah world-coordinate menjadi display untuk suatu output device

Untuk aplikasi-aplikasi tertentu

CAD mengatur orientasi dan ukuran suatu komponen pada suatu rancangan

Animasi menggerakkan letak kamera atau obyek pada sebuah layar sesuai path tertentu



Jenis-jenis Transformasi

Transformasi Geometri

Diterapkan pada deskripsi geometri suatu obyek

Gunanya untuk mengubah posisi orientasi atau ukuran suatu obyek

Transformasi modeling Digunakan untuk membentuk sebuah layar atau membuat deskripsi

hierarki suatu obyek kompleks yang terdiri dari beberapa bagian

Contoh pesawat terdiri dari sayap ekor mesin dan lain-lain dimana masing-masing dapat dispesifikasikan dalam komponen-komponen level ke-2 dan seterusnya

Transformasi modeling mendeskripsikan bagaimana setiap komponen disusun untuk rancangan keseluruhan



2D Modeling Transformations

Scale Rotate

Translate

Scale Translate

x

y

World Coordinates

Modeling

Coordinates




x

y

World Coordinates

Modeling

Coordinates

Letrsquos look

at this in

detailhellip




x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates

Scale 3 3 Rotate -90

Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates



Transformasi Geometri 2D

Translasi 2D

xrsquo = x + tx yrsquo = y + ty

(tx ty) disebut vektor translasi atau pergeseran

Dalam bentuk matriks

Persamaan translasi 2D

Prsquo = P + T

Line polygon translasi titik endpoints render

Untuk menghapus obyek yang asli display it in background color before translating it

Circle ellipse spline curve translasi titik-titik yang mendefinisikan (misalnya koordinat titik pusat) render obyek hasil translasi di posisi yang baru

y

x

t

tT

y

xP

y

xP




Scaling 2D

xrsquo = xmiddot sx yrsquo = ymiddot sy


Uniform scaling berarti skalar pengali sama untuk semua komponen

y

x

s

s

y

x

y

x

0

0

2



Non-uniform scaling skalar yang berbeda untuk tiap komponen


X 2

Y 05




Rotasi 2D

xrsquo = r cos (Φ + )

yrsquo = r sin (Φ + )

Trig Identityhellip

xrsquo = r cos(f) cos( ) ndash r sin(f) sin( )

yrsquo = r sin(f) cos( ) + r cos(f) sin( )

Original pointhellip

x = r cos (f)

y = r sin (f)

Hasil substitusihellip

xrsquo = x cos( ) - y sin( )

yrsquo = x sin( ) + y cos( )

(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D

Shear pada arah x relatif terhadap sumbu x xrsquo = x + shx y yrsquo = y

Parameter shx dapat berupa bilangan real


Shear pada arah y relatif terhadap sumbu y xrsquo = x yrsquo = y + shy x


y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D

Pencerminan terhadap garis y = 0 (sumbu x) xrsquo = x yrsquo = -y


Pencerminan terhadap garis x = 0 (sumbu y) xrsquo = -x yrsquo = y


Pencerminan terhadap titik pusat (0 0) xrsquo = -x yrsquo = -y


y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01



Basic 2D Transformations

Translation xrsquo = x + tx

yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

Transformations can be combined

(with simple algebra)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)




xrsquo = (xsx)cos θ) - (ysy)sin(θ)

yrsquo = (xsx)sin(θ) + (ysy)cos(θ)

(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ) yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = ((xsx)cos(θ) - (ysy)sin(θ)) + tx

yrsquo = ((xsx)sin(θ) + (ysy)cos(θ)) + ty

(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





Matrix Representation

Represent 2D transformation by a matrix

Multiply matrix by column vector

apply transformation to point

yx

dcba

yx

dcba

dycxy

byaxx




Transformations combined by multiplication

yx

lk

ji

hg

fedcba

yx

Matrices are a convenient and efficient way

to represent a sequence of transformations



2x2 Matrices

What types of transformations can be

represented with a 2x2 matrix

2D Identity

yyxx

yx

yx

1001

2D Scale around (00)

ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices



2D Rotate around (00)

yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices



2D Mirror about Y axis

yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx

Only linear 2D transformations

can be represented with a 2x2 matrix

NO



Linear Transformations

Linear transformations are combinations of hellip Scale

Rotation

Shear and

Mirror

Properties of linear transformations Satisfies

Origin maps to origin

Lines map to lines

Parallel lines remain parallel

Ratios are preserved

)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x



Homogeneous Coordinates

Q How can we represent translation as a

3x3 matrix

y

x

tyy

txx




Homogeneous coordinates

represent coordinates in 2

dimensions with a 3-

vector

1

coords shomogeneou y

x

y

x

Homogeneous coordinates seem unintuitive but

they make graphics operations much easier




Q How can we represent translation as a 3x3

matrix

A Using the rightmost column

100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation

Example of translation

11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1





Add a 3rd coordinate to every 2D point

(x y w) represents a point at location (xw yw)

(x y 0) represents a point at infinity

(0 0 0) is not allowed

Convenient

coordinate system to

represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y




Basic 2D transformations as 3x3 matrices

1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale



Affine Transformations

Affine transformations are combinations of hellip

Linear transformations and

Translations

Properties of affine transformations

Origin does not necessarily map to origin

Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition

Transformations can be combined by

matrix multiplication

w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001

prsquo = T(txty) R( ) S(sxsy) p



Matrix Composition

Matrices are a convenient and efficient way to

represent a sequence of transformations

General purpose representation

Hardware matrix multiply

prsquo = (T (R (Sp) ) )

prsquo = (TRS) p



Matrix Composition

Be aware order of transformations matters

Matrix multiplication is not commutative

prsquo = T R S p

ldquoGlobalrdquo ldquoLocalrdquo



Matrix Composition

What if we want to rotate and translate

Ex Rotate line segment by 45 degrees about

endpoint a and lengthen

a a



Multiplication Order ndash Wrong

Way Our line is defined by two endpoints

Applying a rotation of 45 degrees R(45) affects both points

We could try to translate both endpoints to return endpoint a

to its original position but by how much

Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a



Multiplication Order - Correct

Isolate endpoint a from rotation effects

First translate line so a is at origin T (-3)

Then rotate line 45 degrees R(45)

Then translate back so a is where it was T(3)

a

a

a

a



Will this sequence of operations work

Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition

After correctly ordering the matrices

Multiply matrices together

What results is one matrix ndash store it (on stack)

Multiply this matrix by the vector of each vertex

All vertices easily transformed with one matrix

multiply



Reverse Rotations

Q How do you undo a rotation of R( )

A Apply the inverse of the rotationhellip R-1( ) = R(- )

How to construct R-1( ) = R(- )

Inside the rotation matrix cos( ) = cos(- )

The cosine elements of the inverse rotation matrix are unchanged

The sign of the sine elements will flip

Thereforehellip R-1( ) = R(- ) = RT( )



include ltwindowshgt

include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt

GLsizei winWidth=600 winHeight=600 set initial display window size

GLfloat xwcMin=00xwcMax=2250 set range for world coordinates

GLfloat ywcMin=00ywcMax=2250

class wcPt2D

public GLfloat xy

typedef GLfloat Matrix3x3 [3][3]

Matrix3x3 matComposite

const GLdouble pi=314159

void init(void)

glClearColor(10101000) set color of display window to white

Construct the 3 by 3 identity matrix

void matrix3x3SetIdentity (Matrix3x3 matIdent3x3)

GLint row col

for(row=0rowlt3row++)

for(col=0collt3col++)

matIdent3x3[row][col]=(row==col)

Premultiply matrix m1 times matrix m2 store result in m2

void matrix3x3PreMultiply (Matrix3x3 m1 Matrix3x3 m2)

GLint row col

Matrix3x3 matTemp


for(col=0collt3col++) matTemp[row][col]=m1[row][0]m2[0][col]+m1[row][1]m2[1][col]+m1[row][2]m2[2][col]



m2[row][col]=matTemp[row][col]

void translate2D(GLfloat tx GLfloat ty)

Matrix3x3 matTrans1

matrix3x3SetIdentity(matTrans1) initialize translation matrix to identity

matTrans1[0][2]=tx

matTrans1[1][2]=ty

matrix3x3PreMultiply(matTrans1matComposite)concatenate matTrans1 with composite matrix

void rotate2D(wcPt2D pivotPt GLfloat theta)

Matrix3x3 matRot

matrix3x3SetIdentity(matRot) initialize rotation matrix to identity

matRot[0][0]=cos(theta)

matRot[0][1]=-sin(theta)

matRot[0][2]=pivotPtx (1-cos(theta))+pivotPty sin(theta)

matRot[1][0]=sin(theta)


matRot[1][2]=pivotPty (1-cos(theta))-pivotPtx sin(theta)

matrix3x3PreMultiply(matRotmatComposite)concatenate matRot with the composite matrix

void scale2D(GLfloat sx GLfloat sy wcPt2D fixedPt)

Matrix3x3 matScale

matrix3x3SetIdentity(matScale) initialize scaling matrix to identity

matScale[0][0]=sx

matScale[0][2]=(1-sx)fixedPtx

matScale[1][1]=sy

matScale[1][2]=(1-sy)fixedPty

matrix3x3PreMultiply(matScalematComposite)concatenate matScale with the composite matrix

using the composite matrix calculate transformed coordinates

void transformVerts2D(GLint nVerts wcPt2D verts)

GLint k

GLfloat temp

for(k=0kltnVertsk++)

temp=matComposite[0][0]verts[k]x+matComposite[0][1]verts[k]y+matComposite[0][2] verts[k]y=matComposite[1][0]verts[k]x+matComposite[1][1]verts[k]y+matComposite[1][2]

verts[k]x=temp

void triangle(wcPt2D verts)

GLint k

glBegin(GL_TRIANGLES)

for(k=0klt3k++)

glVertex2f(verts[k]xverts[k]y)

glEnd()

void displayFcn(void)

GLint nVerts=3 define initial position for triangle

wcPt2D verts[3]=500250150025010001000

wcPt2D centroidPt calculate position of triangle centroid

GLint k xSum=0 ySum=0


xSum+=verts[k]x

ySum+=verts[k]y

centroidPtx=GLfloat(xSum)GLfloat(nVerts)

centroidPty=GLfloat(ySum)GLfloat(nVerts)

wcPt2D pivPtfixedPt set geometric transformation parameters

pivPt=centroidPt

fixedPt=centroidPt

GLfloat tx=00 ty=1000

GLfloat sx=05 sy=05

GLdouble theta=pi20

glClear(GL_COLOR_BUFFER_BIT) clear display window

glColor3f(000010) set initial fill color to blue

triangle(verts) display blue triangle

matrix3x3SetIdentity(matComposite)initialize composite matrix to identity

construct composite matrix for transformation sequence

scale2D(sxsyfixedPt) first transformation Scale

rotate2D(pivPttheta) second transformation Rotate

translate2D(txty) final transformation Translate

transformVerts2D(nVertsverts)apply composite matrix to triangle vertices

glColor3f(100000) set color for transformed triangle

triangle(verts) display red transformed triangle

glFlush()

void winReshapeFcn(GLint newWidth GLint newHeight)

glMatrixMode(GL_PROJECTION)

glLoadIdentity()

gluOrtho2D(xwcMinxwcMaxywcMinywcMax)

glClear(GL_COLOR_BUFFER_BIT)

void main(int argc char argv)

glutInit(ampargcargv)

glutInitDisplayMode(GLUT_SINGLE|GLUT_RGB)

glutInitWindowPosition(5050)

glutInitWindowSize(winWidthwinHeight)

glutCreateWindow(Geometric Transformation Sequence)

init()

glutDisplayFunc(displayFcn)

glutReshapeFunc(winReshapeFcn)

glutMainLoop()



Some Interesting Images






Tugas Kelas C

Tulis sebuah program animasi yang

mengimplementasikan prosedur rotasi 2D

Inputnya sebuah polygon yang dirotasi terhadap sebuah

pivot point menggunakan sudut yang kecil

Untuk meningkatkan kecepatan gunakan aproksimasi

dari fungsi sin dan cos dan untuk menghindari

akumulasi error yang berlebihan reset nilai koordinat

awal pada setiap putaran baru



Tugas Kelas X

Modifikasi contoh program sebelumnya supaya

Parameter transformasi dapat dispesifikasikan

sebagai input dari user

Dapat diaplikasikan pada semua polygon dengan

verteks-verteksnya dispesikasikan oleh input dari

user

Urutan transformasi geometrik ditentukan oleh input

dari user



Tugas Kelas A

Implementasi program untuk membuat gambar

Untuk membuatnya buat sebuah fungsi yang menggambar polygon gambar (b) (seperlima dari bintang)

Lalu transformasikan dengan rotasi untuk menggambar bintang secara keseluruhan



Tugas Kelas B


Untuk membuatnya buat sebuah fungsi yang menggambar sebuah motif (gambar kiri)

Lalu transformasikan dengan refleksi untuk mendapatkan sebuah spoke yang utuh lalu rotasi untuk menggambar snowflake secara keseluruhan



Kompetensi

1 Mampu membangun tool untuk mentransformasi obyek

2 Mampu memahami konsep transformasi affine yang merupakan kombinasi dari rotasi penskalaan dan translasi

3 Mampu mengimplementasikan konsep transformasi dalam sebuah program OpenGL



Referensi



ndash Chapter 5



Chapter 5




Pokok Bahasan


1 Translasi

2 Rotasi

3 Penskalaan
























Scale Rotate

Translate

Scale Translate

x

y

World Coordinates

Modeling

Coordinates




x

y

World Coordinates

Modeling

Coordinates

Letrsquos look

at this in

detailhellip




x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Referensi



ndash Chapter 5



Chapter 5




Pokok Bahasan


1 Translasi

2 Rotasi

3 Penskalaan
























Scale Rotate

Translate

Scale Translate

x

y

World Coordinates

Modeling

Coordinates




x

y

World Coordinates

Modeling

Coordinates

Letrsquos look

at this in

detailhellip




x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Pokok Bahasan


1 Translasi

2 Rotasi

3 Penskalaan
























Scale Rotate

Translate

Scale Translate

x

y

World Coordinates

Modeling

Coordinates




x

y

World Coordinates

Modeling

Coordinates

Letrsquos look

at this in

detailhellip




x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B

























Scale Rotate

Translate

Scale Translate

x

y

World Coordinates

Modeling

Coordinates




x

y

World Coordinates

Modeling

Coordinates

Letrsquos look

at this in

detailhellip




x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







x

y

World Coordinates

Modeling

Coordinates

Letrsquos look

at this in

detailhellip




x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







x

y

Modeling

Coordinates

Initial location

at (0 0) with

x- and y-axes

aligned




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







x

y

Modeling

Coordinates


Translate 5 3




x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







x

y

Modeling

Coordinates


Translate 5 3

World Coordinates




Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







Translasi 2D





Prsquo = P + T




y

x

t

tT

y

xP

y

xP




Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







Scaling 2D




y

x

s

s

y

x

y

x

0

0

2





X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








X 2

Y 05




Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







Rotasi 2D



Trig Identityhellip




x = r cos (f)

y = r sin (f)




(x y)

(xrsquo yrsquo)

r r

y

x

y

x

cossin

sincos




Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







Shear 2D






y

xsh

y

x x

10

1

y

x

shy

x

y 1

01




Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







Reflection 2D







y

x

y

x

10

01

y

x

y

x

10

01

y

x

y

x

10

01





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)







yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy


ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)

xrsquo = xsx

yrsquo = ysy

(xy)

(xrsquoyrsquo)






(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B









(xrsquoyrsquo)


yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy



ycos(θ)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)



(xrsquoyrsquo)





yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








yrsquo = y + ty

Scale xrsquo = x sx

yrsquo = y sy

Rotation

xrsquo = xcos(θ) -

ysin(θ)

yrsquo = xsin(θ) +

ycos(θ)









yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B










yx

dcba

yx

dcba

dycxy

byaxx





yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








yx

lk

ji

hg

fedcba

yx





2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






2x2 Matrices



2D Identity

yyxx

yx

yx

1001


ysy

xsx

y

x

y

x

s

s

y

x

y

x

0

0



2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






2x2 Matrices




yxy

yxx

cossin

sincos

y

x

y

x

cossin

sincos

2D Shear

yxshy

yshxx

y

x

y

x

sh

sh

y

x

y

x

1

1



2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






2x2 Matrices




yyxx

yx

yx

1001

2D Mirror over (00)

yyxx

yx

yx

1001



2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






2x2 Matrices



2D Translation

y

x

tyy

txx



NO





Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








Rotation

Shear and

Mirror



Lines map to lines



)()()( 22112211 pppp TsTsssT

y

x

dc

ba

y

x





3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








3x3 matrix

y

x

tyy

txx







vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B










vector

1


x

y

x







matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








matrix


100

10

01

y

x

t

t

ranslationT

y

x

tyy

txx



Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Translation


11100

10

01

1

y

x

y

x

ty

tx

y

x

t

t

y

x

tx = 2

ty = 1









Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B











Convenient


represent many

useful

transformations

1 2

1

2 (211) or (422) or (633)

x

y





1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B








1100

0cossin

0sincos

1

y

x

y

x

1100

10

01

1

y

x

t

t

y

x

y

x

1100

01

01

1

y

x

sh

sh

y

x

y

x

Translate

Rotate Shear

1100

00

00

1

y

x

s

s

y

x

y

x

Scale






Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B









Translations



Lines map to lines



w

yx

fedcba

w

yx

100



Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Matrix Composition



w

yx

sysx

tytx

w

yx

100

0000

1000cossin0sincos

100

1001




Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Matrix Composition






prsquo = (TRS) p



Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Matrix Composition



prsquo = T R S p




Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Matrix Composition




a a








Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B











Wrong Correct

T(-3) R(45) T(3) R(45)

a a

a








a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B











a

a

a

a




Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







Matrix Composition

1

1100

010

301

100

0)45cos()45sin(

0)45sin()45cos(

100

010

301

y

x

y

x

a

a

a

a



Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Matrix Composition






multiply



Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Reverse Rotations











include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B







include ltGLgluthgt

include ltstdlibhgt

include ltmathhgt




class wcPt2D

public GLfloat xy




void init(void)




GLint row col






GLint row col

Matrix3x3 matTemp







Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B





Matrix3x3 matTrans1


matTrans1[0][2]=tx

matTrans1[1][2]=ty



Matrix3x3 matRot










Matrix3x3 matScale


matScale[0][0]=sx


matScale[1][1]=sy





GLint k

GLfloat temp



verts[k]x=temp


GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B





GLint k


for(k=0klt3k++)


glEnd()



wcPt2D verts[3]=500250150025010001000




xSum+=verts[k]x

ySum+=verts[k]y




pivPt=centroidPt

fixedPt=centroidPt


GLfloat sx=05 sy=05

GLdouble theta=pi20












glFlush()



glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






glLoadIdentity()









init()



glutMainLoop()









Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B












Tugas Kelas C











Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Tugas Kelas X






user


dari user



Tugas Kelas A






Tugas Kelas B






Tugas Kelas A






Tugas Kelas B






Tugas Kelas B




Transformasi Obyek (Kasus 2D)anny.if.its.ac.id/wp-content/uploads/Grafika08-TM4.pdf · Mampu...

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