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Visualizing the Bezier curve: an approach provided by Geogebra and

the CAS Maple

Francisco Regis Vieira alves – fregis@ifce.edu.br

Abstract: Admittedly, in Algebraic Geometry we observe an abstract and formalist character.

In a specific way, the classical problem relatively to find a parameterization to an algebraic

curve is traditionally discussed in the literature. However, a computational approach permits

obtain an approximation, when we take a set of control points. In this paper, we discuss aspects

and application of the Bezier curve. This approach is described by P. Bézier and P. Casteljau.

We will show some preliminary examples and, in these cases, we can obtain a graphical and

dynamical interpretation by the Dynamic System Geogebra – DSG. On the other hand, we will

describe a way that permits to visualize a Bézier curves with a larger quantity of control points.

For some situations, we obtain the final curve supported by the CAS Maple.

Key-words: Bezier curve, Visualization, Geogebra, CAS Maple.

1. Introduction

There are many problems in Algebraic Geometry linked to the question to find a

specific parameterization for an algebraic plane curve described by the

condition ( , ) 0f X Y = . Moreover, if we have a rational parameterization, there is

possible to indicate its description in terms of Cartesian coordinates. Unfortunately, in

some cases, we must find a curve that passes through some specific points.

In this way, Vainsencher (2009, p. 116) explains that “the Bezier´s curves can be used

with certain computational and aesthetic advantages. Are provided that now, we are

content with a rational curve that ‘visually adjust’ the graphic distribution of points”.

On the other hand, frequently we can obtain a parameterization for a particular algebraic

curve. In fact, lets consider two curves:

(i) 6 2 3 5 0x x y y− − = (ii) ( )( ) ( sin( )), (1 cos( ))t a t t a tβ = − − (YATES, 1947, p. 64)

In the first case, we can take 6 2 3 5 0x x y y− − = and given t IR∈ , we will find the

intersections points of C with the line y tx= . For this, we compute 6 2 3 5 6 2 3 5 6 3 5 5 50 ( ) ( ) 0 0x x y y x x tx tx x t x t x− − = ∴ − − = ↔ − − = . So, if we consider

5 3 5 3 50 ( ) 0x x x t t x t t≠ ∴ − − = ↔ = + . Finally, we still compute 4 6y tx t t= = + . So, we

obtain that 3 5 4 6( ) ( , )t t t t tα = + + . In the figure 1, with the DSG´s help, we can easily

describe its trace. However, we perceive some limitations, when we have considered the

Cartesians coordinates (on the left side). In fact, we can perceive a strange behavior near

at the origin. On the right side, however, we can obtain with DSG, the graphical

behavior from the parameterization form. While in the second case, we have considered

the Cycloid that represents a path of a point of a circle rolling upon a fixed line

(YATES, 1947, p. 64). Correspondingly also the first item, we can compare the both

representation related to the same mathematical object (see fig. 1).

In the next section, we will discuss some specific properties related to this object and,

specially, a recursive analytical method for obtain a Bezier curve of degree n.

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Figure 1. Visualizing a graphical behavior of a algebraic curve.

2. Some considerations about the Bézier curve

When we considering 1 1 1 2 2 2 3 3 3( , ), ( , ), ( , ), , ( , )d d dP x y P x y P x y P x y= = = =… we seek a

rational curve which passes through of these points. Moreover, we need that the

tangents at these points containing the segments 1 2 2 3 3 4 1, , , , d dP P P P P P P P−… . On

the other hand, en virtue to understanding this basic construction, we need to talk some

information about the Bernstein’s polynomials. Indeed, we know the Bernstein’s

polynomials defined by ,

!( ) (1 ) (1 )

!( )!

i n i i n i

i n

n nB t t t t t

i i n i

− − = − = −

− , with 0 i n≤ ≤ .

(*).

So, a Bézier curve ( )r t of degree n is represented by ,

0

( ) ( )n

i i n

i

r t b B t=

=∑ under

condition0 1t≤ ≤ . We can conclude that a Bézier curve is a polynomial, the degree of which is one less than the number of control points.

A parameterization is obtained from a recursive way (VAINSENCHER, 2009, p. 116).

Indeed, we can start with the polygonal determined by the 1d − segments. We write 1

1 1 2

1

2 2 3

1

1 1

( ) (1 )

( ) (1 )

( ) (1 )d d d

t t P tP

t t P tP

t t P tP

σ

σ

σ − −

= − +

= − + = − +

�

. In the next step we must substitute each consecutives

pairs of polygonal by an interpolation. In our case, we take a parabola in the following

way:

2 1 1

1 1 2

2 1 1

2 2 3

2 1 1

2 2 1

( ) (1 ) ( ) ( )

( ) (1 ) ( ) ( )

( ) (1 ) ( ) ( )d d d

t t t t t

t t t t t

t t t t t

σ σ σ

σ σ σ

σ σ σ− − −

= − +

= − + = − +

�

. Lets consider the set of the points:

1 2 3 4( 1,1), (0, 0), ( 1.2, 1.2), (2, 1.5)P P P P= − = = − − = − . Easily, we find:

3

1

1 1 2

1

2 2 3

1

3 3 4

( ) (1 ) (1 ) ( 1,1) (0,0) ( 1,1 )

( ) (1 ) (1 )(0,0) ( 1.2, 1.2) ( 1.2 , 1.2 )

( ) (1 ) (1 )( 1.2, 1.2) (2, 1.5) (3.2 1.2, 0.3 1.2)

t t P tP t t t t

t t P tP t t t t

t t P tP t t t t

σ

σ

σ

= − + = − ⋅ − + ⋅ = − −

= − + = − + − − = − −

= − + = − − − + − = − − −

.

In the next step, we compute: 2 2 2

1

2 2 2

2

( ) (1 )( 1,1 ) ( 1.2 , 1.2 ) ( 2.2 2 1, 0.2 2 1)

( ) (1 )( 1.2 , 1.2 ) (3.2 1.2, 0.3 1.2) (4.4 2.4 ,0.9 2.4 )

t t t t t t t t t t t

t t t t t t t t t t t

σ

σ

= − − − + − − = − + − − − +

= − − − + − − − = − −. Finally, we

find the following parameterized cubic curve described by 3 2 2 3 2 3 2

1 1 1( ) (1 ) ( ) ( ) (6.6 6.6 3 1,1.1 0.6 3 1)t t t t t t t t t t tσ σ σ= − + = − + − − − + .

Figure 2. Visualizing a Bézier curve with four control points

We still take another the set of points: 1 2 3 4( 1, 0), (0, 0), (1, 2), (2, 0)Q Q Q Q= − = = = .

Following a similar procedure, we establish that:

1

1

1

2

1

3

( ) (1 )( 1,0) (0,0) ( 1,0)

( ) (1 )(0,0) (1,2) ( ,2 )

( ) (1 )(1,2) (2,0) (1 ,2 2 )

t t t t

t t t t t

t t t t t

σ

σ

σ

= − − + = −

= − + =

= − + = + −

.

We continue by the calculations

2 2

1

2 2

2

( ) (1 )( 1,0) ( , 2 ) ( , 2 )

( ) (1 )( , 2 ) (1 , 2 2 ) (2 , 4 4 )

t t t t t t t t

t t t t t t t t t t

σ

σ

= − − + =

= − + + − = − and,

finally, we obtain a Bezier cubic 3 2 2 2 2 3

1 ( ) (1 )( ,2 ) (2 , 4 4 ) ( ,6 6 )t t t t t t t t t t t tσ = − + − = + − .

In the figure 3, we show the dynamic construction associated to which of these

recursive parameterization for four control points. Moreover, we perceive in color red,

only the variation correspondingly to the restriction 0 1t≤ ≤ . In these cases, is easy to

obtain the final form of a Bezier cubic curve 3 ( )tα . On the other hand, incidentally, we

have found here only two Bezier cubic. Degree 3 or cubic are usually used in Computer

graphics (WATT, 2000).

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Figure 3. Visualizing a Bézier curve with four control points

Figure 4. Farin (2002, p. 47 – 84) comments the Casteljau algorithm of a Bezier curve of degree six

and the Degree elevation process

We show an arduous work en virtue to describe and visualize a Bézier curve with nine

control points. On the left side, we have used the Casteljau´s algorithmic (FARIN, 2002,

p. 45) formally described in the following way

( )1 1

1

0

( ) (1 ) ( ) ( )1 e 0 i

( )

r r r

i i i

i i

P t t P t tP tr n n r

P t P

− −+

= − +≤ ≤ ≤ ≤ −

= en virtue to obtain the

recursive curves. And on the right side, we obtain the final trajectory related a (red)

curve from the set of control points{ }0 1 2 3 4 5 6 7 8 9, , , , , , , , ,P P P P P P P P P P . We have

considered nine arbitrary points. From the specialized literature, we know that a Bezier

curve is contained within the convex hull of its control points (FARIN, 2002, p. 60).

(see figure 6). Farin (2002, p. 16) explains that “the convex combination points is

always inside those points, which is an observation that leads to the definition of the

convex hull of a point set: this is the set that is formed by all convex combination of a

point set”. In the figure 7, with the CAS Maple, we can understanding that is impossible

to make this process with a larger control points, without the actual technology.

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Figure 5. Visualizing a Bézier curve with nine control points with DSG

Figure 6. Visualizing the convex hull related to a Bezier curve

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Formally, a convex hull of the set of points 1 2( , ,..., )nX x x x= is defined to be the set

0 0 1 1

0

( ) | 1, 0n

n n i i

i

CH X a x a x a x a a=

= + + = ≥

∑� . From this formal definition, we just

enunciate the theorem (Convex hull property): Every point of a Bézier curve lies inside

the convex hull of its defining control points. In other words, for all [0,1]t ∈ ,

0 1 2( ) ( , , , , )nB t CH P P P P∈ … . A visualization provided by technology, we can acquire an

intuitive meaning for this definition and the formal theorem.

Figure 7. Analytical syntax required in the CAS Maple

3. Visualizing the degree elevation of Bézier curves

In this section, we will take a set of points { }0 1 2 1, , , , ,n n

P P P P P−… we apply the

Casteljau´s algorithimic en virtue to obtain a curve of degree n. From this, we desire to

describe another set of points { }0 1 2 1, , , , ,n n

Q Q Q Q Q +… and indicate a Bézier curve

of degree n+1. Well, we will use the following model

1 1 , i=0...n+11 1

i i i

i iQ P P

n n−

= + −

+ + (WATT, 2000, 69).

For exemplify this expression, we take the following control points:

0 1 2 3 4(0,0); (0, 4); (2, 4); (2,3); (1.5,3)P P P P P= = = = = . So, en virtue the expression

(**), we compute the new control set points:

0 0

1

2

3

4

5 4

(0, 0)

1 4 16(0, 0) (0, 4) (0, )

5 5 5

2 3 6(0, 4) (2, 4) ( , 4)

5 5 5

3 2 18(2, 4) (2, 3) (2, )

5 5 5

4 1 19(2, 3) (1.5, 4) ( , 3)

5 5 10

(1.5, 3)

Q P

Q

Q

Q

Q

Q P

= =

= + =

= + =

= + =

= + =

= =

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Well, in the figure 8, on the right side, we can perceive the Bezier curve 5 ( )Q t . We still

observe that this curve is obtained from the following parameterization:

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((1 - t) (2 (1 - t) t² + t (2 (1 - t) t + 2t)) + t ((1 - t) (2 (1 - t) t + 2t) + t (2 - 2t + t (2 - 1 / 2 t)))( )

(1 - t) ((1 - t) (4 (1 - t) t + 4t) + t (4 - 4t + t (4 - t))) + t ((1 - t) (4 - 4t +P t =

t (4 - t)) + t ((1 - t) (4 - t) + 3t)))

In some cases, when we dealing with a higher degree, this software manifests several

limitations relatively the algebraic computational en virtue to express the final curve.

Figure 8. Visualizing the elevation process with the DSG

4. Visualizing the reversing degree elevation of Bézier curves We find that the degree elevation process produces redundancy. On the other hand,

when we have a Bezier curve of degree n , is possible to obtain a Bezier curve of degree

1n − ? In the figure 9, we visualize this mathematical dynamical process.

Figure 9. Visualizing the process of reversing degree with Geogebra´s help

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Let consider the control set points: 0 1 2 3(0, 0); (2, 6); (3, 0); (5, 4)Q Q Q Q= = = =

4 5 6; (7,1); (5,5); (10,6)Q Q Q= = = . We will follow the specific procedure:

( ) ( )

0 0

1 1 0

2 2 1

3 3 2

(0, 0)

6 1 6 1(2, 6) (0, 0) (2.4, 7.2)

5 5 5 5

6 2 6 2 12 36(3, 0) ( , ) (3.3, 3.6)

4 4 4 4 5 5

6 12 (5, 4) 1 (3.3, 3.

3 5

P Q

P Q P

P Q P

P Q P

= =

= − = − =

= − = − = −

= − = − −

���

�� ���

��� ��

��� ���

( ) ( )

( ) ( )

4 4 3

5 5 4

6) (6.7,11.6)

6 43 (7,1) 2 (6.7,11.6) (7.6, 20, 2)

2 2

6 56 (5, 5) 5 (7.6, 20.2) ( 8,131)

1 1

P Q P

P Q P

=

= − = − = −

= − = − − = −

��� ���

��� ���

In this case, we have used that 1 , i=0,...,n-1i i i

n iP Q P

n i n i−

= +

− −

�� ����

(FARIN, 2002, p.

85). From a geometric point of view, we recall the description of Farin (2002, p. 81)

when explains that “one way to proceed in such a situation is to increase the flexibility

of the polygon by adding another vertex to it”. Moreover, the degree elevation process

can be viewed as a process to introduce redundancy (see figure 8). On the other hand,

the degree reduction process is very inaccurate (see figure 9). Indeed, “in general, exact

degree reduction is not possible” (FARIN, 2002, p. 85). For example, if we consider a

cubic with an inflexion point, we can not obtain a quadratic curve which is a bad

approximation.

5. Final remarks In this paper, we have discussed some properties and examples related to the Bézier

curves. This topic is discussed without much detail in the context of Algebraic

Geometry (VAINSENCHER, 2009, p. 116-117). So, in some cases, we showed in this

paper that is possible to realize the algorithmic computational related to this object.

However, when we dealing with a higher degree Bezier curve, we have to consider

some problems concerning to obtain the final parameterized curve (ser figure 5).

Indeed, we showed some examples relatively the parameterization of a Bezier curves

with nine control points. Its description is made possible by the use of the CAS Maple.

Moreover, we observe in each situation, a prominent rule of the Bernstein polynomials

(*). In fact, we always describe a Bezier curve as a finite combination of Bernstein

polynomials. From this fact, we apply one of the most important results in the

approximation theory (FARIN, 2002, p. 90).

Finally, in application context, we recall the words due to Farin (2002, p. 112) when we

observe that “according to Bezier, designers a Renault were quickly getting used to

manipulating control points of a curve in order to create a particular shape. Other

designers may not like the concept of control points and would prefer to manipulate

points on the curve directly”. His words indicate the possibilities of technological

application of Bezier curves (DEKKERS, 2010). Moreover, with regard to academic

education, we pointed out the importance of the students understand this mathematical

constructive process of obtaining plane and spaces curves (see figure 10) through

approximations (ALVES, 2014).

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In fact, Farin (2002, p. 166) warns that “there is probably no “best” parameterization,

since any method can be defeated by a suitably chosen data set”. We can extract a

important pedagogical implication relatively to understanding the mathematical

knowledge supported by a constructive approach and has certain limitations.

Figure 10. Peters; Marsh, D & Jordan (2012, p. 123) discuss an computational approach of Bezier

curves in 3D

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13, nº 2, 115-134.

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