Weighted radial displacement: A geometric look at Bézier conics and quadrics

Computer Aided Geometric Design 17 (2000) 267–289www.elsevier.com/locate/comaid

Weighted radial displacement:A geometric look at Bézier conics and quadrics

Javier Sánchez-Reyesa,∗, Marco Palusznyb,1a Department of Mechanical Engineering, Polytechnic University of Catalonia,

ETSEIB, Diagonal 647, 08028, Barcelona, Spainb Laboratorio de Computación Gráfica y Geometría Aplicada,

Escuela de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela,Apartado 47809, Los Chaguaramos, Caracas 1041, Venezuela

Received January 1999; revised September 1999

Abstract

We describe a simple geometric construction for Bézier conics and quadrics, based on a toolcalled Weighted Radial Displacement (WRD). The shape of a rational Bézier curve or surface ismodified via WRD by choosing an arbitrary pointO and displacing the control points along radialdirections throughO, changing simultaneously the weights. To construct a conic throughO, take anarbitrary segment representing a curve of degreen= 1, degree raise it ton= 2 and apply a WRD.Analogously, if a degree-elevated triangle is modified using a WRD, we get a quadric throughO.Any quadratic Bézier patch on a nondegenerate quadric, which defines a stereographic projection,can be obtained through this method. We present a practical algorithm to detect such quadratic Bézierpatches lying on quadrics. Bézier patches on degenerate quadrics are also derived via a WRD. 2000Elsevier Science B.V. All rights reserved.

Keywords:Rational triangular quadratic Bézier patch; Conic; Quadric; Weighted radialdisplacement; Stereographic projection; Degree elevation; Möbius reparameterization

1. Introduction

The rational Bernstein–Bézier scheme has become a de facto standard for therepresentation of curves and surfaces in CAGD (Farin, 1997), primarily because itencompasses under a unified mathematical model both freeform and classical analytical

∗ Corresponding author. Present address: ETS de Ingenieros Industriales, Avda. Camilo Jase Cela, 3, CampusUniversitario, 13071, Cuidad Real, Spain. E-mail: [email protected] E-mail: [email protected].

0167-8396/00/$ – see front matter 2000 Elsevier Science B.V. All rights reserved.PII: S0167-8396(99)00051-5

268 J. Sánchez-Reyes, M. Paluszny / Computer Aided Geometric Design 17 (2000) 267–289

shapes. Among the latter, algebraic curves and surfaces of degreen = 2 (conics andquadrics) traditionally play an important role in architecture and mechanical engineering.Thus, the problem of representing patches on a quadric as Bézier surfaces has beenaddressed in a number of articles. The simplest representation is a quadratic triangularpatch (Lodha and Warren, 1990; Boehm and Hansford, 1991; Teller and Sequin, 1991),yet a rectangular tensor-product representation is also possible (Boehm, 1993; Dietz etal., 1993, 1995).

Another question to solve is how to detect whether a given quadratic triangular patch lieson a quadric or not. In general, such a patch lies on an implicit surface of degree 4, calledSteiner surface (Piegl, 1985; Sederberg and Anderson, 1985). Boehm and Hansford, 1991gave three conditions for a quadric patch to lie on a quadric, namely that the three boundarycurves:

(1) Meet in one pointO.(2) Have coplanar tangents atO.(3) Assume the parameter value±∞ at O.Nevertheless, Boehm and Prautzsch (1994), in Remark 6 of Chapter 23, do not mention

condition (3). Analogously, Farin listed these three conditions in his book on NURBS(Farin, 1995), but dropped the last one in the second, third and fourth edition (Farin, 1997)of his book on CAGD. This is hence a somewhat obscure point in the CAGD literature.In addition, as one of the referees pointed out, the case where a boundary is a straight lineshould also be properly considered.

The conditions above are geometric and simple to enunciate. However, as Farin (1995)points out, they cannot be checked in a numerically reliable way in many cases. That iswhy Albrecht (1998a, 1998b) recently developed an alternative method, utilizing resultsdue to Degen (1996) on Veronese surfaces in 5-dimensional projective space.

In this article, in addition to confirming that the third condition is actually unnecessary,we show that most results regarding Bézier conics and quadrics take a very simple form interms of a geometric construction, which we calledWeighted Radial Displacement(WRDfor short). Given a rational Bézier curve or surface, we modify it via WRD by choosingan arbitrary pointO, and then displacing each control point along the corresponding radialdirection throughO. Simultaneously, we change the weight associated to the control point,keeping the product weight by distance toO constant. If we take a segment, (a linearBézier curve of degree one) degree raise it up to degree two, and modify it via WRD, aconic throughO is obtained. Analogously, from a degree-elevated triangle we generate aquadric throughO. The WRD concept also yields simple conditions to solve the “quadricor not” problem, without resorting to exotic surfaces in higher-dimensional spaces.

The paper is arranged as follows. In Section 2, the chief properties of the WRDare reviewed. In Section 3, we discuss how to generate conics via WRD. The WRDconstruction also furnishes simple conditions to determine if a given point lies on the conicdefined by a Bézier curve, as shown in Section 4. The construction of Bézier patches onquadrics via WRD is carried out in Section 5. Conditions for a quadratic Bézier patchto lie on a quadric are analysed in detail in Section 6. In Section 7 we develop a specialconstruction for patches on degenerate quadrics (cones and cylinders) that is also based ona WRD. Finally, conclusions are summarized in Section 8.

J. Sánchez-Reyes, M. Paluszny / Computer Aided Geometric Design 17 (2000) 267–289269

2. Weighted Radial Displacement (WRD)

The Weighted Radial Displacement (WRD) constitutes a unified approach to NURBSshape modification, which builds on a perspective functional transformation of arbitraryorigin (Sánchez-Reyes, 1997). In this chapter we briefly review this tool, focusing ourattention on the particular Bézier case, as our final goal is the generation of Bézier conicsand quadrics.

2.1. Defining a WRD

A planar rational Bézier curveb(u), with control pointsbi = [xi yi]T and weightswi ,can be interpreted as the perspective projection onto the planez= 1 of a polynomial Béziercurve in Euclidean 3D space. Control points of this nonrational curve have homogeneouscoordinates:[

xiwi yiwi wi]T = [pi wi]T, pi =wibi . (1)

Hence, a degree-n rational Bézier curveb(u),u ∈ [0,1], can be written as:

b(u)= p(u)w(u)

, p(u)=n∑i=0

piBni (u), w(u)=n∑i=0

wiBni (u), (2)

whereBni (u) denotes the Bernstein polynomials of degreen. The denominatorw(u) willbe referred to as theweight function.

We choose now an arbitrary pointO and, without lack of generalization, we express thecurve takingO as the origin of coordinates, which greatly simplifies the resulting formulas.Note that changing the origin modifies the coordinates of pointsbi by adding a commonvector, whereas this does not apply to the homogeneous coordinatespi (1). If we lift the3D control points in thez-direction by choosing arbitrary factorsλi :[

pi wi]T→ [

pi λiwi]T, (3)

this operation movesbi along a radial direction passing throughO and, simultaneously,changes the associated weightwi :

bi→ b∗i = bi/λi ,

wi→w∗i = λiwi . (4)

We can utilize the control point as shape handle: we slide the control point along the rayObi until reaching the desired new locationb∗i , derive the factorλi and compute the newweightw∗i (4).

The operation (3) modifies the weight function, but not the numeratorp(u) ofexpression (2). Hence the new curveb∗(u) is given by:

b∗(u)= p(u)w∗(u)

= b(u)λ(u)

, λ(u)= w∗(u)w(u)

,

whereλ−1(u) denotes a functional scale factor. Therefore,b∗(u) is a perspective functionaltransformation (with centreO) of the original curveb(u), that is, a point corresponding to


a fixed parameter u moves on a radial direction throughO. Clearly, the WRD defines amappingb(u)→ b∗(u), and the inverse map is also a WRD of centreO (and factorsλ−1

i ).For simplicity, we have considered only curves. Nevertheless, these concepts extend to

surfaces in a straightforward manner, because a WRD in essence reduces to changing thelast homogeneous coordinate of the control points, as indicated by expression (3).

2.2. Properties and particular cases of the WRD

Two interesting cases arise for particular locations of the centreO:• O at infinity and converted to a direction: the pointsbi move along this direction,

the weights remain unchanged and the WRD reduces to the standard displacement ofcontrol points.• O on the curve, that is,O= b(u0) for a certain parameter valueu0: the position and

tangent direction of the curve atO do not vary.Other two remarkable particular cases are those corresponding to homogeneous linear

transformations of the whole curve:• If all pointsbi are submitted to a scaling of centreO, the whole curve is scaled.• If all points bi are projected onto a line, usingO as centre of projection, we have a

central projection of the curve onto the line. In the case of a quadratic curveb(u) andO on the conic defined byb(u), this is a stereographic projection.

A positive property of the WRD is its invariance with respect to rational linear (Möbius)reparameterizations. If a curve is first reparameterized, and second submitted to a WRDdefined by a set of parametersλi , the result is the same than if we apply first the WRD andthen reparameterize. This property is demonstrated in (Sánchez-Reyes, 1997) for curves.It also applies to triangular surfaces, because the reparameterization of such surfaces (Joeand Wang, 1994) just multiplies the original weights by certain factors, without movingthe control points.

3. WRD for constructing Bézier conics

3.1. The linear case

As an introduction, we discuss the WRD in the case of a linear curve. This investigationalso furnishes a geometric look at the Möbius reparameterization of a rational Bézier curve.

Consider a segmentc0c1 represented as a polynomial Bézier curvec(u) of degreen= 1,that is, with control pointsc0,c1, and unit weights. Choose a pointO and apply a WRD toc(u). As a result (Fig. 1), the control points move to new locationsb0,b1 along radial linesthroughO, and the original unit weights change tow0,w1, so that:

b0= c0/w0, b1= c1/w1. (5)

Observe thatc(u) andb(u) are related by a central projection (of centreO). The resultingrational curveb(u) does not trace out the segmentb0b1 in a linear fashion any more, unlessw0=w1 and in consequence the WRD degenerates to a scaling.

Given an arbitrary linear rational curveb(u), we could always find a degree-1 curvec(u)with unit weights such thatc(u),b(u) are related by a WRD: simply choose an arbitrary


Fig. 1. WRD for curves: the linear case.

pointO and take the pointsc0,c1 according to Eq. (5). Sincec(u) is an affine image of theparametric domainu ∈ [0,1], we can think of a rational linear mappingu→ b(u) as theinverse of a central projection.

This construction provides a geometric interpretation for a Möbius reparameterizationv(u) of a rational Bézier curve. In the Bernstein basis, such a rational linear functionv(u)

is given by two weightsw0,w1:

v(u)= w1u

w0(1− u)+w1u.

If the segmentsc0c1 andb0b1 (5) are affine images of the domainsu ∈ [0,1] andv ∈ [0,1],respectively, then the WRDc(u)→ b(u)maps affine images ofu to affine images ofv(u).

3.2. The canonical construction of conics

In this section we present the simplest WRD-based construction of Bézier conics, theso-calledcanonical construction.

Consider an arbitrary segmentc0c2 (an affine image of the domainu ∈ [0,1]), endowedwith a linear parameterization:

c(u)= c0(1− u)+ c2u. (6)

We apply now the following two-step procedure:(1) Degree-elevate ton = 2 the linear polynomial Bézier curvec(u) through standard

degree elevation. We obtain a quadratic Bézier representation ofc(u) with controlpoints{c0,c1= (c0+ c2)/2,c2} that still parametrizes in a linear fashionc0c2.

(2) Choose a centreO (not on the linec0c2) and apply a WRD to this quadratic curve,by selecting the weightswi . In a coordinate system with originO, the resultingcurveb(u) takes the form:

b(u)= c(u)w(u)

, w(u)=2∑i=0

wiB2i (u). (7)


(a) (b)

Fig. 2. Canonical construction for conics via WRD.

The curveb(u) is clearly a conic segment, because any rational quadratic curve is aconic. In a nondegenerate case, i.e.,b(u) is not a straight line, this conic displays thefollowing properties (Fig. 2(a)):• It passes throughO= b(±∞).• Its tangent lineLO at O is parallel to the segmentc0c2.To demonstrate the first property, observe that in the quotient (7) the numeratorc(u) (6)

is linear, whereas the denominatorw(u) is quadratic. Hence, the limit atu=±∞ vanishes.The second property is easily checked by introducing the change of variableu→ 1/s andcomputing the derivative ats = 0.

As we apply the WRD and move the control points along the radial linesOci , wegenerate a family of conics with a common pointO and tangent there. This comes as nosurprise, because a WRD with centre at a point on a curve yields a new curve that passesthroughO and keeps the tangentLO, as commented in Section 2.2.

It is worth studying the particular case where we choose a centreO at infinity, that is,Otransforms to a direction. To fix ideas, assume that this direction is they-axis (Fig. 2(b)). Toexpress the curveb(u) we use now a system of coordinates(x, y)with arbitrary origin. TheWRD degenerates to a parallel displacement of the control points along they-axis, keepingthe original unitary weights. This geometry gives rise to control pointsbi = [xi yi] withabscissasxi regularly spaced. Hence we get a parabola of vertical axis, the nonparametricquadratic Bézier curve:

y(u)=2∑i=0

yiB2i (u).

Another interesting case occurs when the WRD affects only the middle point, so thatb0 = c0, b2 = c2, and we get a curveb(u) in standard form(w0 = w2 = 1). Thenc0c2becomes a chord of the conic, and the rayOc1 a diameter, that is, it contains the centreC


Fig. 3. Bézier conic segment in standard form.

of the conic (Lee, 1987). This chord, parallel toLO, is also parallel to the shoulder tangent(Farin, 1995). Fig. 3 illustrates this geometry.

The inverse mapb(u)→ c(u) is clearly the stereographic projection of centreO fromthe conic segment onto the linec0c2. This correspondence between stereographic maps andrational quadratic Bézier curves was already explored by Teller and Sequin (1991), whoobserved that it provides a simple inversion procedure (see also (Wang and Joe, 1995)).

Lodha and Warren (1990) present a construction of Bézier conic segments, calledprojective functional representation, that is strongly connected to the WRD construction.The designer chooses a reference trianglec0c2O, identifies afocal vertexO, and selects theBézier pointsbi on the raysOc0,Oc1 andOc2. However, the curveb(u) is computed inan intricate way. The system finds the linear perspective functional transformationτ suchthat these rays are mapped to parallel lines and appliesτ to the control polygon to createthe control polygon for a nonparametric Bézier curve. Finally such a curve is mappedback to the reference triangle viaτ−1 to generateb(u). Clearly, the WRD proposed isconceptually simpler and provide us with a deeper geometric look at the parameterizationof conic segments in Bézier form.

3.3. Canonical construction associated with a given conic segment and centre

If we are given an arbitrary straight line segment (i.e., a degenerate conic), it alwaysadmits a trivial canonical construction, where only the first step (degree-elevation) iscarried out. Assume now that we are given an arbitrary segment of a nondegenerate conicΓ , of endpointsb0,b2, and a pointO onΓ . Can we generate this segment via the canonicalconstruction described in the preceding section? In this section we check that the answeris always affirmative.

Trace out the raysOb0,Ob2, draw any line parallel to the tangentLO and compute theintersection pointsc0,c2 with the rays. This is the starting segmentc0c2. First we degreeelevate it, obtaining the midpointc1. Second, we apply a WRD, by slidingc0,c2 along therays until they match the pointsb0,b2, and movingc1 until we recapture the tangent line atone of the endpoints (for instanceb0). Clearly the original conicΓ is reproduced, because


we have constructed a conic that shares withΓ three points(O,b0,b2) and two tangents(at O andb0).

As a consequence, note that the midpointc1 must lie precisely on the ray defined byOand the pointb1 where the tangents atb0,b2 meet.

3.4. Alternative representation of a linearly parameterized segment

In step (1) of the canonical construction we obtained an integral quadratic Bézierrepresentation of a linearly parameterized segment via standard degree-elevation. As analternative, we may utilize a generalized degree elevation (Denker and Herron, 1997).Given a linearly parameterized segmentd0d2, we multiply and divide its Bézier form byan arbitrary linear functionw(u) and then divide by the unity:

d(u)= [(1− u)d0+ ud2]w(u)[(1− u)+ u]w(u) , w(u)= w0(1− u)+ w2u. (8)

Thus we generate a rational quadratic Bézier curved(u) that traces out with linear precision(Farin and Jung, 1995) the segmentd0d2. Formally speaking, we have inserted abase pointu0, the root ofw(u):

u0= w0

w0− w2. (9)

In the Bernstein–Bézier form (2), the quadratic curved(u), of control pointsdi andweightswi , takes the form:

d(u)= p(u)w(u)

= w0d0(1− u)2+ w1d12(1− u)u+ w2d2u2

w0(1− u)2+ w12(1− u)u+ w2u2 . (10)

Expanding (8) and equating with expression (10), we derive the inner weight and controlpoint:

w1= 1

2(w0+ w2), d1= (1− α)d0+ αd2, α = w0

w0+ w2(11)

We have a degree of freedom in this generalized degree raising, namely the base pointu0,or the factorα determining the position ofd1. For u0 at infinity (α = 1/2) we recapturethe standard degree elevation. The apparent two degrees of freedomw0, w2 reduce to one,because multiplying and dividing by a constant the quotient (8) leaves it unchanged.

The invariant of this construction is the cross ratio cr of the four collinear points{d0,d1,d2,d(u0)}, corresponding to the parameter values{0, α,1, u0}. Introducing thevalues ofu0 (9), andα (11), we find that cr(d0,d1,d2,d(u0))=−1, using the definitionof cross-ratio given in (Boehm and Prautzsch, 1994). Therefore, these four points areharmonic.

The numerator ofd(u) in expression (8) is a certain parabolap(u):

p(u)= [(1− u)d0+ ud2]w(u), (12)

As u0 is a root ofw(u), from (12) we deduce the following properties:(1) p(u) passes through the origin of coordinatesO= p(u0).


(2) Its tangent lineLO at O meets the lined0d2 at d(u0):

d(u0)= d0(1− u0)+ d2u0. (13)

We could also expressp(u) as a Bézier curve of control pointspi :

p(u)=2∑i=0

piB2i (u).

Equating with (12) we derive the expressions forpi :

p0= w0d0, p2= w2d2, 2p1= αp0+ βp2, α = β−1= w2

w0. (14)

3.5. Extended construction of conic segments

The generalized degree-elevation discussed in the preceding Section allows a moreflexible construction of conics, which will be referred to as theextended construction.

Let us apply an arbitrary WRD of centreO (not on the line defined byd0d2) to thestraight lined(u) (8). As the WRD does not modify its numerator, the resulting conicsegmentb(u) takes the form:

b(u)= p(u)w(u)

, p(u)= [(1− u)d0+ ud2]w(u), (15)

wherew(u) indicates the new weight function. Asp(u) andb(u) are related by a WRD ofcentreO= p(u0), in a nondegenerate case the curveb(u) inherits the position and tangentdirection ofp(u) at O (Fig. 4(a)). In other words:

(1) b(u) passes throughO= b(u0);(2) Its tangentLO meets the lined0d2 at the pointd(u0).Since the centreO lies on the conicb(u), the WRDb(u)→ d(u) is again a stereographic

projection onto the lined0d2. It corresponds to a more general case, where the line of

(a) (b)

Fig. 4. Extended construction for conics via WRD.


Fig. 5. Particular case of a degenerate conicb(u) (i.e., a straight line).

projection is not necessarily parallel to the tangentLO at the centre of projectionO. Theextra degree of freedom provided by the extended degree-elevation allows us to choose theslope ofLO.

The limit case of a centreO at infinity is noteworthy. Without loss of generality, assumethatO is converted to a direction along they-axis (Fig. 4(b)). The WRD degenerates to aparallel displacement of the control points along this axis, keeping the original weightswi .This geometry yields a rational nonparametric curve:

y(u)= 1

w(u)

2∑i=0

wiyiB2i (u),

where we use a system of coordinates(x, y) with arbitrary origin to express the curveb(u) = [u,y(u)]. This is a hyperbola with a vertical asymptote cutting the lined0d2 atd(u0), since the denominatorw(u) vanishes foru= u0. Needless to say, ifα = 1/2 in (11),the extended construction reduces to the canonical one and we obtain a parabola, whichcorresponds to a limit hyperbola with a vertical asymptote lying atu0=∞.

In a degenerate case, such thatb(u) is a straight line (Fig. 5), the WRD corresponds tothe central projection described in Section 3.1. Hence, the two properties above transformto:

(1) b(u) has real (minimum) degree one and inherits the base pointu0 from d(u).(2) The lineLO defined byO andb(u0) meets the lined0d2 at d(u0).

Expression (15) yields 0/0 for the imageI = b(u0) of the base point, owing to the commonfactorw(u) in the numerator and denominator. Needless to say, to computeb(u0) we mustremove such a common factor.

In this case, whereb(u) is a straight line, the lineLO could still be considered the tangentto b(u) atu0 if we interpretb(u) as a limiting hyperbola that degenerates to its asymptotes.The first asymptote coincides with the line itself, and the second one is the lineLO. Thishyperbola, which degenerates toLO atu0, suffers a sudden change in its tangent atb(u0).


3.6. Correspondence between the canonical and extended constructions

Fig. 6. Correspondence between the standard and extended construction for conics.

Suppose we have obtained segment of a conicΓ via an extended construction withcentreO. According to Section 3.3, we could generate the same conic segment throughthe standard construction. Clearly, we get different Bézier representations of the sameconic segment, corresponding to different parametrizations. The correspondence betweensuch representations admits a geometric interpretation (Fig. 6). The segmentsc0c2 andd0d2, affine images of the parameter domains in the canonical and extended constructions,respectively, are related through a perspective projection of centreO, which, as explainedin Section 3.1, can be interpreted as a Möbius reparameterization.

4. Point classification algorithm for conics in Bézier form

4.1. Equivalent conditions for a pointO to lie on a conic

The WRD construction leads to several ways for determining whether a pointO lies ona conicΓ . Let us formalize this result.Theorem 1. Given a pointO and a Bézier curveb(u) of control pointsbi that are notaligned and weightswi , the following statements are equivalent:

(i) Expressingb(u) in a coordinate system with originO, its numeratorp(u) (2)factors as the product(12):

p(u)= [(1− u)d0+ ud2]w(u), w(u)= w0(1− u)+ w2u.

(ii) b(u) admits an extended construction with centreO.(iii) O lies on the conicΓ defined byb(u).


(iv) The coordinates[αβ] of 2p1 in the basis{p0,p2}, wherepi =wibi , are such that:

αβ = 1. (16)

Proof. It suffices to prove the equivalence between (i) and (ii), (iii), (iv):(i)⇔ (ii) From expression (15) we have(ii)⇒ (i). The implication(i)⇒ (ii) is also

clear: the construction has centreO, starting segmentd0d2, and base pointu0 indicated by (9).

(i)⇔ (iii ) The implication(i)⇒ (iii ) is straightforward, becauseu0 is a root ofw(u).On the other hand, if(iii ) is satisfied, then both components ofp(u) vanishsimultaneously for a certainu0, which implies(i).

(i)⇔ (iv) The implication(i)⇒ (iv) is clear from Eq. (14). Regarding(iv)⇒ (i), justchoose an arbitrary nonzerow0 and derive from (14) the values that allowfactoringp(u):

w2= αw0, d0= p0

w0, d2= p2

w2. 2 (17)

Remark 1. Condition(iv) provides a simple algorithm to determine whether a given pointlies on the conicΓ defined byb(u). If O has coordinates[x y] in a given coordinatesystem, then[α β] are linear functions ofx, y and, in consequence, the left side ofEq. (16) is quadratic inx, y. Hence, the equality (16) corresponds to the implicit equationf (x, y)= 0 ofΓ . However, the point here is not to derive a closed-form for the coefficientsin f (x, y)= 0, but to note that, since we employ the WRD deformation, the homogeneouspointspi =wibi are readily available and hence so are their coordinates[α β] in the basis{p0,p2}. Such coordinates play a key role in other computations, as exemplified in thefollowing remark.

Remark 2. If O lies onΓ , combining (9), (13), (14) we find a very simple expression, interms of[α β], for the direction defined by the tangent lineLO:

βp2− αp0. (18)

Remark 3. Theorem 1 also solves the inversion problem: given a quadratic curveb(u) anda pointq on b(u), find the parameter valueuq suchq = b(uq). Just choose an arbitrarypoint O on b(u) and compute the pointsd0,d2 (17). As the segmentd0d2 is an affineimage of the parametric domain[0,1], the intersection between the rayOq and the lined0d2 corresponds to the affine image ofuq.

4.2. Case of a degenerate conic

Theorem 1 does not apply in case the given curveb(u) is a straight line. However, itdoes if we assume now that the given pointO does not lie on the line and rewrite(iii ) as:(iii ): b(u) contains a base pointu0, so that it has minimum degree one.We are dealing with a particular instance of an extended construction, whereb(u) =

d(u) and only the first step (the extended degree-elevation) is performed. Therefore, theoriginal proofs for(i)⇔ (ii) and (i)⇔ (iv) hold. The implication(iii )⇒ (i) is trivial.


On the other hand, if(i) is satisfied, thenp(u) vanishes for a certainu0. As b(u) doesnot pass through the originO, then foru0 the numerator ofb(u) must vanish too, whichimplies (iii ). Finally, note that Eq. (18) still indicates the direction of the lineLO, whichwas defined asOb(u0) in a degenerate case.

4.3. Case of a point at infinity

The point classification for the case of a pointO at infinity requires a special treatment.Now one wants to determine if a given direction corresponds to an asymptote of ahyperbolab(u) (or an axis of a parabola). Without loss of generality, suppose that sucha direction is they-axis.

Condition (16) involves the coordinates[α β] of 2p1 in the basis{p0,p2}, that is, thesolutions of the linear system:

2p1= αp0+ βp2. (19)

If the control points arebi = [xi yi]T in an original system of coordinates(x, y), in asystem with origin moved toO = [0 −yO]T they transform to[xi yO + yi]T. Hence, thelinear equation (19) can be written as:

2w1

[x1

yO + y1

]= αw0

[x0

yO + y0

]+ βw2

[x2

yO+ y2

].

Dividing by yO the second line of the equation above, foryO→∞ we obtain:

2w1

[x1

1

]= αw0

[x0

1

]+ βw2

[x2

1

].

We conclude that, for a pointO at infinity, in condition(iv) of Theorem 1 we must employpointspi = wixi , wherexi denotes the projection ofbi onto the liney = 1, instead ofpi = wibi . Analogously, in condition(i) we take the projectionx(u) of b(u) ontoy = 1,instead ofb(u).

5. WRD for constructing triangular Bézier quadrics

Henceforth we adopt the following notation concerning triangular Bézier patches.The boldface symbolu designates the customary barycentric coordinatesu,v,w, whereu + v + w = 1. We denote a generic pointbijk by bi . We also use for the corners theabbreviations:

A = (n,0,0), B= (0, n,0), C= (0,0, n),and, in the quadratic case, for the inner points:

AB = (1,1,0), BC= (0,1,1), CA= (1,0,1).


5.1. The linear case

As for curves, to begin we investigate the WRD in a linear case. Take an arbitrary triangle1c of verticescA,cB,cC endowed with a linear parameterization:

c(u)= cAu+ cBv+ cCw, (20)

and interpret it as a polynomial triangular Bézier patch of degreen= 1. Choose a pointOand apply a WRD toc(u). As a result, the control pointsci move to new locationsbi alongradial lines throughO, and the original unit weights change to valueswi such that:

bi = ci/wi . (21)

Therefore,c(u) andb(u) are related by a projection of centreO (see Fig. 7). In addition,given an arbitrary rational triangular surfaceb(u) of degree one, we could always finda polynomial patchc(u) such thatc(u) andb(u) are related by a WRD: simply choosean arbitrary pointO and take the pointsci indicated by the equality (21). Asc(u) is anaffine image of the barycentric domain(u, v,w), we can always interpret a rational linearmappingu→ b(u) as the inverse of a central projection.

Similarly to the curve case, this construction leads to a geometric interpretation of aMöbius reparameterization:

v(u)= wAAu+wBBv +wCCwwAu+wBv +wCv

,

whereA,B,C denote points in parameter space with barycentric coordinates A,B,C. Ifthe triangles1c and1d are affine images of the barycentric domainsu andv, respectively,then the WRDc(u)→ b(u) maps affine images ofu to affine images ofv(u).

Fig. 7. WRD for triangular surfaces: the linear case.


5.2. The canonical construction of quadrics

In analogy to the case of conics, we adopt the termcanonical constructionto denote thesimplest WRD-based construction of quadrics. Consider an arbitrary triangle1c expressedin Bézier formc(u) (20) and apply the following two-step procedure:

(1) Degree-elevate ton = 2 the triangle1c through standard degree elevation. Weobtain three new control points, the midpoints of the edges of1c, correspondingto a quadratic Bézier representation ofc(u) that still parameterizes1c in a linearfashion.

(2) Choose a centreO (not on the plane defined by1c) and apply a WRD to thisquadratic surface. The resulting quadratic Bézier patch is:

b(u)= c(u)w(u)

,

wherew(u) denotes a quadratic weight function.The question is whetherb(u) lies on certain quadricQ, and the answer is affirmative.

The initial homogeneous triangle[c(u) 1] lies on an affine 2D plane inR4. TheWRD moves their control points along a fixed direction, as only the last homogeneouscomponent is altered. Therefore, this occurs in a 3D hyperplane inR4. In fact, it is thenonparametric construction of a paraboloid as the graph of a quadratic function (Boehmand Hansford, 1991). The surfaceb(u) stems from projecting this paraboloid onto thehyperplanew= 1, thereby lying on a quadric.

Observe that this 3D WRD reduces to a 2D WRD at each of the three boundary planes{u = 0, v = 0,w = 0}. Therefore all nondegenerate boundary conics ofb(u) display thefollowing properties (Fig. 8):

(1) They pass throughO for parameter values±∞.

Fig. 8. Canonical construction for Bézier quadrics.


(2) At O they are coplanar, since they have tangents parallel to the edges of1c.It is worth analysing the particular case of a centreO at infinity. The WRD degenerates

to a parallel displacement of the control points, keeping the original unit weights. Thisconstruction yields the graph of a nonparametric function, in other words, the resultingpatch lies on a paraboloid.

Similarly to the case of curves the WRDb(u)→ d(u) is the inverse of an stereographicprojection fromQ onto the projection planeP defined by the triangle1c. This geometryallows a simple inversion procedure (Teller and Sequin, 1991).

5.3. Conditions for a patch to admit the canonical construction

In the 2D case we have seen that, given an arbitrary pointO and conic segment, wecan always obtain via the canonical construction a Bézier curveb(u) that parametrizes theconic segment. In the 3D case the situation is different, since not every triangular patch ona quadricQ admits the canonical construction.

Proposition 1. Given a triangular portionQ of a quadricQ, the following statements areequivalent:

(i) Q can be parameterized via a canonical construction of centreO.(ii) A pointO exists, such that all nondegenerate boundaries ofQ pass throughO.

Proof. First consider the special case where all the boundaries are straight lines and,consequently,Q is planar. In this trivial case(ii) must be interpreted as “no restrictionin the position ofO”. Indeed, we can always parameterize a triangular planar patch via acanonical construction of arbitrary centreO. Simply take as starting triangle1c = Q anddo not carry out step (2) (the WRD).

In a general case, by construction,(i)⇒ (ii). If (ii) is satisfied, consider the three raysfrom O to the corner points ofQ, take a plane parallel to the planeTO tangent toQ atO and compute the three intersections between the rays andTO. These three points definethe starting triangle1c of a canonical construction. We can reproduce any given boundaryvia a WRD on the plane where it lies, starting from the corresponding side of1c, whichaffects solely the Bézier points of this particular conic. The two planar WRD actions ona vertex of1c produce a common point for two boundaries, thereby coinciding on thatvertex. Hence, the three planar WRD assemble into a three dimensional WRD. Thus weobtain a patchb(u) on a quadricQ′ with the same boundaries as the original patch. ClearlyQ′ =Q, as both quadrics share at least 4 points (O and the three corner points of the patch)and the tangent planes at them.2Remark. Lodha and Warren (1990) developed a projective functional representation forquadrics, and derived a similar theorem where(i) reads thatQ can be represented inprojective functional form. Obviously, this fact implies the equivalence of both methods,yet the WRD is again conceptually simpler.


5.4. Alternative representation of a linearly parameterized triangle

In step (1) of the canonical construction, we obtained an integral quadratic Bézierrepresentation of a linearly parameterized triangle via standard degree-elevation. As analternative, we may utilize a generalized degree elevation. Given a linearly parameterizedtriangle1d of verticesdA,dB,dC, we multiply and divide its Bézier form by an arbitrarylinear functionw(u) and then divide by the unity:

d(u)= (udA + vdB +wdC)w(u)[u+ v +w]w(u) , w(u)= wAu+ wBv + wCw. (22)

Thus we obtain a rational quadratic Bézier patchd(u) that parametrizes1d with linearprecision. We have inserted the base linew(u)= 0, which can be interpreted as a line onthe planeP defined by1d . This extended degree elevation has two degrees of freedom,corresponding to the location of the base line inP .

Note that the numerator ofd(u) (22) is a certain polynomial surfacep(u):

p(u)= (udA + vdB +wdC)w(u). (23)

Rewriting it in the Bernstein basis:

p(u)= upA + vpB +wpC+ 2uvpAB + 2vwpBC+ 2wupCA, (24)

and equating (23),(24) we get the corner points ofp(u):

pA = wAdA, pB = wBdB, pC= wCdC,

and its inner points:

2pAB = wB

wApA + wA

wBpB, 2pBC= wC

wBpB + wB

wCpC, 2pCA = wA

wCpC+ wC

wApA .

5.5. Extended construction of patches on quadrics

As for curves, the extended degree-raising leads to a more general method (extendedconstruction) for obtaining Bézier patches on quadrics. Let us apply an arbitrary WRD, ofarbitrary centreO not on the planeP defined by1c, to the patchd(u). In a coordinatesystem with originO this WRD does not modify the numerator ofd(u), and the resultingquadratic patchb(u) can be written as:

b(u)= p(u)w(u)

, p(u)= (udA + vdB +wdC)w(u),

wherew(u) denotes a quadratic weight function, andp(u) is the polynomial surface (23).Clearly, at each border we reproduce the 2D extended construction of conic segments

presented in Section 3.5. Therefore, the following conditions are satisfied (see Fig. 9):(1) The nondegenerate boundaries meet at a common point, the centreO; the straight

line boundaries contain base points.(2) The linesLO of the three boundaries are coplanar (tangent line atO for a

nondegenerate boundary, line joiningO and the imageI of the base point for astraight line): these lines meet the planeP at points on the base line, and hence lieon the planeTO defined by the base line andO.


(a)

(b)

Fig. 9. Extended construction for Bézier patches on quadrics: (a) Patch with nondegenerateboundaries; (b) Patch with one degenerate boundary (w = 0).

Next we give a simple demonstration confirming that these two conditions are sufficientfor the Bézier patchb(u) to lie on a quadric, and, in consequence, the extended constructionindeed yields a patch on a quadric. Consider the three rays fromO to the corner points of


b(u). Take an arbitrary plane parallel toTO and compute the three intersections with therays. These three points define the starting triangle1c of a canonical construction. Wecan reproduce each boundary through the 2D canonical construction and thus obtain apatch on a quadricQ with the same control pointsbi and boundaries thanb(u). We couldalways reparameterize both patches so that they had unitary weights at the corner points(Albrecht, 1995) and hence share the same weights and control points. We conclude thatb(u) coincides with the patch generated via the standard construction, up to a Möbiusreparameterization, thereby lying onQ. This reparameterization corresponds to the centralprojection that maps the triangle1c into1d (see Section 5.1).

The WRD b(u)→ d(u) is the inverse of a stereographic projection fromQ onto theprojection planeP . It corresponds to a more general case, whereP is not necessarilyparallel to the tangent planeTO at the centre of projectionO. This stereographic projectionprovides a one-to-one map betweenQ andP , except for the singular pointO, mappedto the singular orfundamental line, our base line in the domain space. As already notedby Teller and Séquin (1991), this line is the intersection of the planesP andTO. In thecanonical construction both planes are parallel and, consequently, the fundamental linelies at infinity. The extended degree elevation employed in the extended constructiongives us two extra degrees of freedom, allowing us to set explicitly the location of thefundamental line.

6. Conditions for a quadratic patch to lie on a quadric

6.1. Conditions in terms of a WRD

Thanks to our investigation on the WRD construction, we are now ready to formalize ina precise manner the conditions for a quadratic Bézier patch to lie on a quadric.

Theorem 2. Given a quadratic Bézier patchb(u), the following statements are equivalent:(i) b(u) lies on a quadric andu→ b(u) defines the inverse of a stereographic map.(ii) b(u) admits an extended WRD construction with a certain centreO.(iii) (a) The nondegenerate boundaries ofb(u) meet at a common pointO; those

degenerate have a base point.(b) The linesLO of the three boundaries(tangent line atO for a nondegenerate

boundary, line fromO to the image of the base point for a straight line) arecoplanar.

Proof. The implications(ii)⇒ (iii ) and(iii )⇒ (i) were demonstrated in Section 5.5. Itjust remains to show that(i)⇒ (ii). This is straightforward, as a projection is a particularcase of a WRD, and in consequence the inverse map is also a WRD.2Remarks. According to Niebuhr (1992), only in the case of degenerate quadricsQ (thosewith Gaussian curvatureK = 0, i.e., cones and cylinders), there exist Bézier patches onQ


that do not define a stereographic projection. From this fact and the equivalence of(i), (ii)in Theorem 2, it follows that:• The WRD construction generates all possible quadratic patches on nondegenerate

quadrics.• Theorem 2 gives conditions sufficient for a patchb(u) to lie on a quadric, and

necessary to lie on a nondegenerate quadric.

The third condition listed in the Introduction, that the boundaries assume the parametervalue±∞ at O, is fulfilled only by the canonical construction. We conclude that such acondition is not necessary. In fact, it can be always met after a certain reparametrization,if (iii ) is actually satisfied. Simply consider that in this case we can generate the patchvia either the extended construction, according to(ii), or the canonical construction(Proposition 1). These two patches are related by a Möbius reparametrization, as justifiedin Section 5.5.

6.2. A simple procedure to identify patches on quadrics

In this Section we show how to translate condition(iii ) of Theorem 2 into two practical,easy to check relationships.

First we are to compute the candidate pointO. In case the three boundaries ofb(u) arestraight lines,b(u) is planar. Therefore, if(ii) is satisfied for a certainO thenb(u) hasminimum degree one and, consequently, admits a WRD construction with arbitrary pointO. Thus, we can choose any pointO (not on the plane defined byb(u)).

In a nonplanar case, the only candidate pointO is the intersection of the boundaryplanes (u = 0, v = 0,w = 0), defined by triples of boundary control points. However,one boundary conic may degenerate into a straight line, even two boundaries in the caseof a doubly ruled quadric (one sheeted hyperboloid or hyperbolic paraboloid). Then thecorresponding boundary plane is ill defined. Nevertheless, ifb(u) indeed lies on a quadric,then any line in parameter space intersects the fundamental line, and hence its image passesthroughO. Therefore one may use, as a replacement for a degenerate boundary, such a linein the parameter space, for instance a radial line joining a vertex with the midpoint ofthe opposite edge. No more than two straight lines can pass through a regular point of anonplanar quadric. Hence, no more than two radial lines through a vertex, which includesthe boundaries, can degenerate into straight lines. In fact the candidate pointO can becalculated as the intersection of any three planes determined by boundaries or radial lines,planes defined by triples of control points. This wide choice of planes helps us computeOin a numerically reliable way.

OnceO has been computed, we must determine if the condition(iii (a)) is satisfied. Justexpressb(u) in a coordinate system with origin atO, compute the homogeneous pointspi = wibi and check condition (16) for each boundary conic. In the case ofO at infinity,for instance thez-direction, we calculate pointspi = wizi , wherezi is the projection ofbi onto the planez = 1 perpendicular to the direction defined byO. If [α β γ ] denotecoordinates in the basis{pA,pB,pC}, then those of the interior pointspAB,pBC,pCAsatisfy the relationships:

αABβAB = βBCγBC= γCAαCA = 1. (25)


If condition (iii (a)) is fulfilled, according to expression (18) the linesLO throughO incondition(iii (b)) have vectors of coordinates:

w = 0: [−αAB βAB 0] , u= 0: [ 0 −βBC γBC ] ,

v = 0: [αCA 0 −γCA ] .

These vectors are coplanar if and only if their determinant vanishes:

αABβBCγCA = βABγBCαCA. (26)

Observe the simplicity and cyclic symmetry of the equalities (25), (26) to check, whichare tantamount to(iii ) in Theorem 2.

7. Quadric patches on cylinders and cones

In this section we investigate how to utilize a WRD for constructing the aforementionedpatches on degenerate quadrics (cones and cylinders) that do not define a stereographicprojection. In essence we reinterpret in terms of a WRD, describing a constructive method,a result given in (Boehm and Hansford, 1991; Boehm, 1993).

A coneQ is defined by a conicΓ (directrix) and a pointO (vertex). Assume we havea segment ofΓ represented in Bézier form asb(u), corresponding to the boundaryv = 0of our triangular patch onQ. We apply the following WRD-based method, illustrated inFig. 10:

(1) Duplicate the Bézier curve and subdivide the overlapping copy at arbitraryparameter valueua ∈ [0,1]. This operation yields two curves, the boundariesu =0,w= 0, of a degenerate triangular patch.

(2) Stretch this patch by applying a WRD with centreO. The resulting surface lies onQ, as in a WRD all points move along rays throughO.

Fig. 10. Constructing a triangular patch on a cone via WRD.


As subdividing a Bézier curve does not modify the location of the point correspondingto∞, the three boundaries have such points on a common ruling. However, this propertyno longer holds if we reparametrize the patch after step (1) or, equivalently, if wereparameterize the boundaries after step (1).

Finally, observe that, in the particular case whereO lies at infinity, this WRDconstruction produces a cylinder.

8. Conclusions

The WRD (Weighted Radial Displacement) is a powerful constructive tool that providesan accessible and geometric look at Bézier conics and quadrics. If we take a straightsegment or a triangle and modify it via WRD of arbitrary centreO, we define a perspectivecentral projection. If we degree elevate the segment or triangle and then apply a WRD,we obtain a conic or quadric throughO. The inverse of such a WRD corresponds to astereographic projection of centreO. Any conic segment in quadratic Bézier form, or anyquadratic Bézier patch on a nondegenerate quadric admits this WRD-based construction,which in addition gives a clear geometric interpretation of the Möbius reparametrization.The WRD also leads to a method for defining Bézier patches on degenerate quadrics (i.e.,cones and cylinders), thereby providing a novel, unified treatment of quadrics.

A quadratic patchb(u) lies on a quadric if the following conditions are met:(a) The nondegenerate boundaries ofb(u)meet at a common pointO; those degenerate

have a base point.(b) The linesLO of the three boundaries are coplanar. Such lines are the tangents atO

for nondegenerate boundaries, and lines fromO to the image of the base point forstraight lines.

These two conditions translate into very simple numerical relationships to be satisfiedby the homogeneous control points of the patch. Another condition appearing in somereferences, that the boundaries take the parameter values±∞ at O, is unnecessary. Itis satisfied for a particular case of the WRD construction, and can be always met if wereparameterize the patch.

Conics and quadrics are the particular quadratic instance of a wider class of curves andsurfaces, called monoids. Most of the ideas exposed in this paper carry over to curves andsurfaces of higher degree. If we degree-elevate a curve of arbitrary degree, by inserting abase pointu0, and then apply a WRD of centreO, we get a new curve that passes throughOfor u= u0. Clearly, inserting several base points, followed by a WRD, furnishes a monoidalcurve, and a similar procedure holds for constructing monoidal surfaces. Research is underway on such monoids.

Acknowledgements

This research has been supported thePrograma de Cooperación Interuniversitaria(AECI-Intercampus). J. Sánchez-Reyes is currently a visiting professor at the School ofEngineering in Ciudad Real, University of Castilla-La Mancha.


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Weighted radial displacement: A geometric look at Bézier conics and quadrics

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