Existence conditions for Coons patches interpolating geodesic … · 2010-11-10 · a local...

Computer Aided Geometric Design 26 (2009) 599–614

Contents lists available at ScienceDirect

Computer Aided Geometric Design

www.elsevier.com/locate/cagd

Existence conditions for Coons patches interpolating geodesic boundarycurves

R.T. Farouki b, N. Szafran a, L. Biard a,∗a Laboratoire Jean Kuntzmann, Université Joseph Fourier – Grenoble, Franceb Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 19 March 2008Received in revised form 10 October 2008Accepted 1 January 2009Available online 7 January 2009

Keywords:Geodesic curvesSurface reconstructionHermite/Coons interpolationSurface smoothing

Given two pairs of regular space curves r1(u), r3(u) and r2(v), r4(v) that define acurvilinear rectangle, we consider the problem of constructing a C2 surface patch R(u, v)

for which these four boundary curves correspond to geodesics of the surface. Thepossibility of constructing such a surface patch is shown to depend on the given boundarycurves satisfying two types of consistency constraints. The first constraint is global innature, and is concerned with compatibility of the variation of the principal normalsalong the four curves with the normal to an oriented surface. The second constraint isa local differential condition, relating the curvatures and torsions of the curves meeting ateach of the four patch corners to the angle between those curves. For curves satisfyingthese constraints, the surface patch is constructed using a bicubically-blended Coonsinterpolation process.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Given four parametric space curves r1(u), r3(u) and r2(v), r4(v) specifying a curvilinear rectangle, the Coons interpola-tion procedure (Coons, 1964, 1974; Farin, 2002; Gordon, 1983) defines a surface patch R(u, v) bounded by these four curves,such that R(u,0) = r1(u), R(u,1) = r3(u) and R(0, v) = r2(v), R(1, v) = r4(v). The Coons scheme is motivated by the factthat, in numerous surface design or reconstruction contexts, only the boundary curves of a surface patch are specified, anda means of smoothly “filling in” the interior is needed. In addition to the four boundary curves, the bicubically-blendedCoons patch requires transverse derivative data along them — i.e., the four vector functions Rv (u,0), Rv(u,1) and Ru(0, v),Ru(1, v) must also be specified.

In the Coons interpolation scheme, the boundary curves and derivative data (which together specify the tangent planevariation along the patch boundary) are considered to impart sufficient information to ensure the desired surface shape. Inthis paper, the Coons patch is modified to admit a more fundamental geometrical significance to the specified boundarycurves. Namely, these curves are stipulated to be geodesics on the constructed surface.

The motivation for this study comes from the problem of constructing analytic computer representations of free-formsurfaces from positional/orientational measurements obtained with the Morphosense — a flexible ribbon-like device withembedded microsensors that, when placed on a physical surface, assumes the shape of a geodesic (Sprynski et al., 2008). Byplacing the Morphosense on the surface at regular intervals along two different directions, it is subdivided into rectangularpatches with geodesic boundary curves.

* Corresponding author.E-mail address: [email protected] (L. Biard).

0167-8396/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.cagd.2009.01.003

http://www.ScienceDirect.com/

http://www.elsevier.com/locate/cagd

mailto:[email protected]

http://dx.doi.org/10.1016/j.cagd.2009.01.003

600 R.T. Farouki et al. / Computer Aided Geometric Design 26 (2009) 599–614

The construction of surfaces that incorporate one or two given space curves as geodesics has been considered by severalauthors (Bennis et al., 1991; Sánchez-Reyes and Dorado, 2008; Tucker, 1997; Wang et al., 2004), in the context of applicationssuch as distortion-free mapping of textures onto free-form surfaces; specifying fabric shapes for garment and shoe design;and in the layout of fibers in composite material structures. However, the problem of constructing rectangular surfacepatches, when all four prescribed boundary curves are required to be geodesics of the resulting surface, does not appear tohave been previously studied.

This paper is concerned with extending prior studies that address the problem of interpolating two boundary geodesicsby a surface “strip” (Paluszny, 2008; Sprynski et al., 2008) to the case where all four boundary curves of a rectangularpatch are specified as geodesics of the constructed surface. In particular, we are concerned with identifying constraintson the boundary curves, whose satisfaction constitutes a sufficient-and-necessary condition for the existence of analyticsurfaces that interpolate those curves as geodesics. A companion paper (Farouki et al., 2008) will address the construction of(polynomial or rational) Bézier surface patches that interpolate four given Bézier curves, satisfying the existence conditions,as geodesic boundaries.

The plan for this paper is as follows. After reviewing basic concepts concerning the geometry of curves on surfaces inSection 2, the ideas of Sprynski et al. (2008) are extended in Section 3 to identify a local condition that must be satisfied bytwo intersecting space curves if those curves are to be geodesics of an analytic surface. A global compatibility condition, thatmust be satisfied by the geodesic boundary curves of a rectangular patch to ensure a continuous oriented surface normal,is then identified in Section 4.

Section 5 describes a modified Coons algorithm, for boundary data satisfying the conditions of Sections 3 and 4, toconstruct surface patches with the given boundary curves as geodesics. Finally, Section 6 presents some representativesurfaces computed using this algorithm, and Section 7 summarizes our main results and identifies key issues that warrantfurther investigation.

2. Background on curves and surfaces

In the following discussion, all curves and surfaces are considered to be regular and “sufficiently smooth.” A curve isregular if it admits a tangent line at each point, while a surface is regular if it admits a tangent plane at each point. Allsurfaces are considered to be oriented. A surface is said to be oriented if its unit normal vector (4) is continuous on eachclosed regular curve on the surface.

The inner product of two vectors u, v in R3 is denoted by 〈u,v〉. Similarly, the plane through a point p in R

3 spannedby two linearly-independent vectors u, v is denoted by [p,u,v].

For linearly-independent unit vectors u, v and a unit vector n such that n ⊥ u and n ⊥ v, we denote by (u,v)n theoriented angle between u and v in the sense of n. Precisely, the angle A = (u,v)n is defined (see Fig. 1) by

sin A = det(u,v,n), cos A = 〈u,v〉. (1)

The variable s is employed to denote arc length along a space curve. Note that the arc-length parameterization r : s �→ r(s)of a curve satisfies ‖r′(s)‖ = 1 and r′(s) ⊥ r′′(s) for all s. However, in this paper, a general parameterization r : t �→ r(t) isoften used in the surface construction problem. The parameters of functions may sometimes be omitted when no confusioncan arise.

• With each point r(s) of a curve satisfying r′′(s) = 0, we associate the Serret–Frenet frame (e(s),n(s),b(s)) where e(s) =r′(s), n(s) = r′′(s)/‖r′′(s)‖, and b(s) = e(s)×n(s) are, respectively, the unit tangent, principal normal, and binormal vectorsof the curve at the point r(s). The arc-length derivative of the Serret–Frenet frame is governed by the relations

d

ds

[ e(s)n(s)b(s)

]=

[ 0 k(s) 0−k(s) 0 τ (s)

0 −τ (s) 0

][ e(s)n(s)b(s)

], (2)

where the curvature k(s) and torsion τ (s) of the curve r(s) are defined by

k(s) = ∥∥r′′(s)∥∥ and τ (s) = det(r′(s), r′′(s), r′′′(s))

‖r′′(s)‖2. (3)

Fig. 1. Angle measurement.

R.T. Farouki et al. / Computer Aided Geometric Design 26 (2009) 599–614 601

The osculating plane at each curve point r(s) is spanned by the two vectors e(s), n(s) and does not depend on the curveparameterization. If k(s) = 0 for some s, then r′′(s) = 0 and the normal vector n(s) and osculating plane are undefinedat that point. This condition identifies an inflection of the curve.

• On a regular oriented surface (u, v) �→ R(u, v), the unit normal is defined at each point in terms of the partial deriva-tives Ru = ∂R/∂u, Rv = ∂R/∂v by

N(u, v) = Ru(u, v) × Rv(u, v)

‖Ru(u, v) × Rv(u, v)‖ . (4)

• Consider a curve r(s) = R(u(s), v(s)) on a surface R(u, v), where s denotes arc length for the space curve r(s), butnot necessarily for the plane curve defined by s �→ (u(s), v(s)). With each point r(s), we associate the Darboux frame(e(s),h(s),N(s)) — where e(s) is the unit tangent vector of the curve, N(s) is the unit normal vector of the surface atthe point R(u(s), v(s)) = r(s), and h(s) = N(s) × e(s). The arc-length derivative of the Darboux frame is given by therelations

d

ds

[ e(s)h(s)N(s)

]=

[ 0 kg(s) kn(s)−kg(s) 0 −τg(s)−kn(s) τg(s) 0

][ e(s)h(s)N(s)

], (5)

which define the normal curvature kn(s), the geodesic curvature kg(s), and the geodesic torsion τg(s) at each point of thecurve r(s) as

kn =⟨

de

ds,N

⟩, kg =

⟨de

ds,h

⟩, τg =

⟨dN

ds,h

⟩. (6)

• A regular curve r(t) on a surface R(u, v) is called a geodesic of the surface if its geodesic curvature is identically zero.From (2) and (5), this is equivalent to requiring that

N(s) = ±n(s), h(s) = ∓b(s) (7)

— i.e., the Frenet and Darboux frames agree modulo signs. Hence, we have the following useful characterizations ofgeodesic curves.

A regular curve t �→ r(t) is a geodesic on the surface R(u, v) if and only if

(D1) the geodesic curvature of r(t) is identically zero;(D2) the principal normal at each non-inflection point of r(t) is orthogonal to the surface tangent plane at the point R(u(t), v(t)) =

r(t);(D3) the osculating plane at each non-inflection point of r(t) is orthogonal to the surface tangent plane at the point R(u(t), v(t)) =

r(t).

In the case of an isolated inflection point of the curve, we may encounter cases (Do Carmo, 1976) where the osculatingplane cannot be defined by continuity, since the limit planes approaching it from the left and the right disagree. For non-isolated inflection points, the vanishing of the curvature over an interval in the parameter implies that the correspondingcurve segment degenerates to a straight line (and hence a geodesic) on the surface. For such a segment, the Darboux framecan be defined at each point, but the Frenet frame is undefined.

3. Geodesic curves crossing on a regular surface

The normal curvature kn and geodesic torsion τg at any point of a curve r(t) on a regular C2 oriented surface R(u, v)

depend only on the tangent direction of r(t) at that point. Namely, if Πp is the tangent plane at a given point p of thesurface R(u, v), and L is any line in Πp passing through p, all curves on R(u, v) that go through p, and have L as theirtangent line there, possess the same normal curvature kn and geodesic torsion τg . This property allows us to define thenormal curvature Kn(L) and geodesic torsion T g(L) at each point p of a surface R(u, v), in the direction of any line L in thetangent plane Πp .

At a fixed surface point p, the normal curvature Kn(L) exhibits two extrema K1, K2 (the principal curvatures), correspond-ing to two orthogonal directions L1, L2 (the principal directions). Consider an orthonormal basis (e1,e2) for Πp aligned withthe principal directions L1, L2 such that e1 × e2 defines the unit surface normal N at p. Then each line L in the tangentplane Πp corresponds to an angle α (with 0 � α < π ), such that the normal curvature of the surface at point p can beregarded as a π -periodic function of the angle α. It can be shown (Do Carmo, 1976) that

Kn(α) = K1 cos2 α + K2 sin2 α. (8)

Similarly, the geodesic torsion of the surface at point p can be regarded as a π -periodic function T g(α) of the angle α,given (Do Carmo, 1976) by

T g(α) = (K1 − K2) sinα cosα. (9)


Remark. A point p at which K1 = K2 is called an umbilic of the surface. At such a point, the normal curvature Kn(α) definedby (8) is the same in all directions α (so the surface shape is locally like a sphere), while the geodesic torsion (9) vanishesidentically.

For subsequent use, we now derive Proposition 3, which identifies a necessary condition for two intersecting space curvesto be geodesics on a regular surface. First, notice that we have only three degrees of freedom in the curvature distribution,namely the two principal curvatures K1, K2 and the orientation of the principal directions. Thus, the normal curvature andgeodesic torsion are locally determined in any direction from relations (8) and (9). So, the normal curvatures and geodesictorsions along two directions in a point of a regular surface must satisfy one compatibility equation, which is established inthe following lemma.

Lemma 1. For two directions in the surface tangent plane Πp specified by the angles α1 , α2 the normal curvature and the geodesictorsion satisfy the relation

sin(α2 − α1)[

T g(α1) + T g(α2)] = cos(α2 − α1)

[Kn(α1) − Kn(α2)

]. (10)

Proof. By direct computation, we have

sin(α2 − α1)[

T g(α1) + T g(α2)]

= (K1 − K2)(sinα2 cosα1 − sinα1 cosα2)(sinα1 cosα1 + sinα2 cosα2)

= (K1 − K2)[sinα1 sinα2

(cos2 α1 − cos2 α2

) + cosα1 cosα2(sin2 α2 − sin2 α1

)]= 1

2(K1 − K2)

[sinα1 sinα2(cos 2α1 − cos 2α2) + cosα1 cosα2(cos 2α1 − cos 2α2)

]= 1

2(K1 − K2)

[cos(α2 − α1)(cos 2α1 − cos 2α2)

]= 1

2cos(α2 − α1)

[K1

(2 cos2 α1 − 1 − 2 cos2 α2 + 1

) − K2(1 − 2 sin2 α1 − 1 + 2 sin2 α2

)]= cos(α2 − α1)

[Kn(α1) − Kn(α2)

]. �

We now consider in greater detail the relations (7) that characterize a geodesic curve on a surface.

Lemma 2. Consider a regular curve r(s) on a surface R(u, v) with notations as introduced in Section 2. Then, setting N(s) = σn(s)with σ = ±1, we have

h(s) = −σb(s), kg(s) = 0, τg(s) = −τ (s), kn(s) = σk(s). (11)

Proof. Setting N(s) = σn(s) so that h(s) = −σb(s), a direct calculation using relations (2) gives

d

ds

[ e(s)h(s)N(s)

]= d

ds

[ e(s)−σb(s)σn(s)

]=

[ 0 k(s) 00 στ(s) 0

−σk(s) 0 στ(s)

][ e(s)n(s)b(s)

]

=[ 0 0 σk(s)

0 0 τ (s)−σk(s) −τ (s) 0

][ e(s)h(s)N(s)

], (12)

which corresponds to the stated results by comparison with relations (5). �Proposition 3. Consider two geodesics r1(s) and r2(s) parameterized by arc length on the surface R(u, v), with principal normal n1(s)and n2(s), crossing at the point p = r1(s1) = r2(s2). Assuming that p is not an inflection on either of the curves r1(s) and r2(s), letα = (r′

1(s1), r′2(s2))Np be the oriented angle between them at p, in the sense of the surface normal Np at that point. Also, let ki(s) and

τi(s) be the curvature and torsion of ri(s) for i = 1,2. Then, for the values σ1, σ2 ∈ {−1,+1} such that Np = σ1n1(s1) = σ2n2(s2),we have[

τ1(s1) + τ2(s2)]

sinα = [σ2k2(s2) − σ1k1(s1)

]cosα. (13)

Proof. Consider first any geodesic r(s), free of inflections, parameterized by arc length on the surface R(u, v). Its geodesiccurvature satisfies kg(s) ≡ 0, and its principal normal n(s) agrees (modulo sign) with the surface normal N(s) (here N(s) isthe restriction of the unit surface normal to the curve r(s)). Setting N(s) = σn(s) with σ = ±1, Lemma 2 gives

τg(s) = −τ (s) and kn(s) = σk(s). (14)

Hence, if the tangents to the two geodesics r1(s) and r2(s) at the point p are identified by angles α1 and α2 in the plane Πp ,the result follows directly from Lemma 1, with α = α2 − α1. �


Fig. 2. Geodesics on a cylinder.

Proposition 4 (Consistency with surface orientation). The geodesic crossing relation (13) in Proposition 3 does not depend on thesurface orientation.

Proof. Consider a new parameterization of the surface in Proposition 3, that induces at each point a surface normal oppositeto the original normal. Then the angle α and the values σ1, σ2 will change sign, but the curvatures and the torsions, whichare intrinsic to the curves r1(s) and r2(s), remain unchanged. Hence, we obtain[

τ1(s1) + τ2(s2)]

sin(−α) = [(−σ2)k2(s2) − (−σ1)k1(s1)

]cos(−α), (15)

which is identical to (13). �Corollary 5. If two space curves intersecting at a point p do not satisfy the relation (13), no regular C2 oriented surface can interpolatethese curves in such a manner that they are geodesics of the surface.

Example. Consider the cylinder x2 + y2 = R2, and for a1,a2 ∈ R (with a1 = a2) the geodesic curves (circular helices) definedby ri(t) = (R cos t, R sin t,ait) on it, crossing at p = r1(0) = r2(0) = (R,0,0) with the derivatives r′

i(0) = (0, R,ai) at thatpoint. Let αi be the angle that r′

i(0) makes with the horizontal plane. Note that, for each circular helix, the principal normaln(t) = (− cos t,− sin t,0) points to the inside of the cylinder. From Proposition 4, we can choose an orientation of thecylinder such that σ1 = σ2 = 1, so that positive angles αi must be counted clockwise in Fig. 2. Thus, the positive angle αbetween the two vectors r′

i(0) is α = α1 − α2, and we have

tanα = tan(α1 − α2) = a1/R − a2/R

1 + (a1/R)(a2/R)= R(a1 − a2)

R2 + a1a2.

For each helical curve ri(t), the curvature ki(t) and torsion τi(t) are constant, ki = R/(a2i + R2) and τi = ai/(a2

i + R2), and adirect computation gives

k2(0) − k1(0)

τ1(0) + τ2(0)= R(a1 − a2)

R2 + a1a2= tanα = sinα

cosα,

which is relation (13). As another simple example, one can easily check that this relation is satisfied by great-circle geodesicscrossing on a sphere.

Proposition 6. (See Struik, 1976.) Through each point of a smooth surface, there exists a geodesic in every direction, and a geodesic isuniquely determined by an initial point and tangent direction at that point.

Henceforth we shall consider only distinct crossing geodesics, with a non-zero angle of intersection. For such curves, thetangent plane Πp of the surface R(u, v) at the point p is spanned by the two derivatives r′

1(s1), r′2(s2).

4. Geodesic interpolation problem

We now consider the problem of defining a smooth rectangular surface patch, bounded by four given parametric curves,in such a manner that these border curves are geodesics of the surface. This problem, introduced in Section 4.1, is calledthe “geodesic interpolation problem.” After introducing notations in Section 4.2 and identifying a local corner compatibilitycondition on the curve principal normals in Section 4.3, we introduce in Section 4.4 the global normal compatibility con-straint, which ensures that the surface normal vector varies continuously around the patch boundary. Finally, Section 4.5specifies the local differential constraints that must be satisfied by the border curves meeting at each patch corner, in orderto ensure the existence of a surface for which these curves are geodesics. The details of the surface construction procedurewill be presented in Section 5.


Fig. 3. Surface patch boundary curves.

4.1. Specification of the problem

Consider, as illustrated in Fig. 3, four regular space curves r1(u), r3(u) and r2(v), r4(v) with u, v ∈ [0,1] such that

r1(0) = r2(0) = p00,

r1(1) = r4(0) = p10,

r2(1) = r3(0) = p01,

r3(1) = r4(1) = p11. (16)

As noted earlier, these curves are assumed to be “sufficiently smooth.” Also, consistent with the last remark in Section 3,their derivatives at each corner pi j , i, j = 0,1, are assumed to be linearly independent, in order to define the tangentplane Πi j of the interpolating surface R(u, v) (see below) at pi j .

Our goal is to construct a surface that interpolates these four curves in such a manner that they are geodesics onthe constructed surface. More precisely, if R(u, v) for (u, v) ∈ [0,1]2 denotes the interpolating surface, it must exhibit thefollowing three properties.

• South-North interpolating property: R(u,0) = r1(u) and R(u,1) = r3(u) for u ∈ [0,1];• West-East interpolating property: R(0, v) = r2(v) and R(1, v) = r4(v) for v ∈ [0,1];• Geodesic property: The principal normal at each non-inflectional point of the curves r1(u), r3(u) and r2(v), r4(v)

is normal to the surface R(u, v).

Note that each of the geodesic curves r1(u), r3(u) and r2(v), r4(v) is also an isoparametric curve of the surface R(u, v).

4.2. Notations

Let t be either of the parametric variables u or v . For i = 1,2,3,4 we denote by (ei(t),ni(t),bi(t)) and ki(t), τi(t) theSerret–Frenet frame and the curvature and torsion of the curves ri(t) at each non-inflection point. Namely,

ei(t) = r′i(t)

||r′i(t)||

, bi(t) = r′i(t) × r′′

i (t)

||r′i(t) × r′′

i (t)|| , ni(t) = bi(t) × ei(t), (17)

ki(t) = ‖r′i(t) × r′′

i (t)‖‖r′

i(t)‖3, τi(t) = det(r′

i(t), r′′i (t), r′′′

i (t))

‖r′i(t) × r′′

i (t)‖2. (18)

At inflection points, the curvature is zero and the torsion is undefined. Note that, whereas the principal normal vector isintrinsic to the curve, the tangent and binormal vectors depend on the sense of the parameterization.

4.3. Condition (C1): osculating constraints at corners

From the geodesic definitions in Section 2 we see that, at each corner pi j , the principal normals of the boundary curvesthat meet there must agree modulo sign. Hence, the boundary curves must satisfy the following constraints:

corner p00: n1(0) = ±n2(0), corner p10: n1(1) = ±n4(0),

corner p01: n2(1) = ±n3(0), corner p11: n3(1) = ±n4(1). (19)

These conditions imply that, at each corner pi j , the boundary curves meeting there have osculating planes orthogonal tothe surface tangent plane Πi j .


4.4. Condition (C2): global normal orientation constraint

Let r(t) for t ∈ [0,4] denote the “concatenation” of the four boundary curves r2(t), r3(t), r4(t), r1(t), defined as follows:

r(t) = r2(t), t ∈ [0,1], r(t) = r3(t − 1), t ∈ [1,2],r(t) = r4(3 − t), t ∈ [2,3], r(t) = r1(4 − t), t ∈ [3,4]. (20)

Consider the principal normal n(t) of the concatenated curve r(t), defined on the interval t ∈ [0,4]: n(t) is simply theconcatenation of the principal normals ni(t) of the four boundary curves ri(t), according to the parameterization (20) usedfor the definition of r(t).

Let N(u, v) be the unit normal (4) to the interpolating surface R(u, v) and let N(t) for t ∈ [0,4] be its restriction to theboundary r(t) of the surface patch. The curve r(t) : R �→ R

3 and the vectors n(t) : R �→ S 2 and N(t) : R �→ S2 (where S2 isthe unit sphere) are regarded as periodic functions, of period 4. Since we desire a regular oriented surface, N(t) must be acontinuous vector function, with N(4) = N(0). Furthermore, since the curves ri(t) are specified to be geodesics on R(u, v)

we must have N(t) = ±n(t) for t ∈ [0,4].Now the vector function n(t) depends only on the prescribed boundary curves ri(t). If these curves are regular, n(t) can

be discontinuous — i.e., exhibit a sudden reversal — only at their inflection points, or at the parameter values t = 0,1,2,3,4identifying the patch corners, where the curves ri(t) meet.

Hence, the existence of a regular oriented surface R(u, v) that interpolates the concatenated boundary r(t), with the four individualcurves ri(t) as geodesics, is contingent on the existence of a continuous unit vector function N(t) such that N(t) = ±n(t) for all t ∈ R.Specifically, a regular oriented interpolating surface can exist only if the unit vector function n(t) exhibits an even number of reversalson the interval t ∈ [0,4).

Remark. We specifically exclude the case of “pathological” inflections — i.e., points where k = 0, and the left and rightlimits n− and n+ of the principal normal are not parallel or anti-parallel (see Section 2), since no solution can be found ifsuch points are present.

Fig. 4 shows an example in which the variation of n(t) is consistent with the existence of an oriented surface patch witha normal N(t) that is continuous around the patch boundary, while Fig. 5 shows a case for which the variation of n(t) isincompatible with such a surface. In the Fig. 4 example, the normal n(t) exhibits two reversals — one at corner p00, and theother at an inflection I in the middle of the curve r2(t). In this case, a globally continuous solution for N(t) can be specifiedby taking N(t) = −n(t) for t ∈ (0,0.5) and N(t) = n(t) otherwise. In the example of Fig. 5, there is only one reversal of n(t),at the corner p00. Consequently, there is no continuous solution for N(t).

4.5. Condition (C3): geodesic crossing constraints at corners

By Corollary 5, the boundary curves ri(t) must satisfy the crossing constraint (13) at each corner pi j , with crossingangle Aij defined (see Fig. 6) by:

A00 = (e1(0),e2(0)

)N(0)

: sin A00 = det(e1(0),e2(0),N(0)

), cos A00 = ⟨

e1(0),e2(0)⟩,

A01 = (e3(0),e2(1)

)N(1)

: sin A01 = det(e3(0),e2(1),N(1)

), cos A01 = ⟨

e3(0),e2(1)⟩,

A11 = (e3(1),e4(1)

)N(2)

: sin A11 = det(e3(1),e4(1),N(2)

), cos A11 = ⟨

e3(1),e4(1)⟩,

A10 = (e1(1),e4(0)

)N(3)

: sin A10 = det(e1(1),e4(0),N(3)

), cos A10 = ⟨

e1(1),e4(0)⟩. (21)

Fig. 4. Left: patch boundaries that satisfy the global normal orientation constraint. Right: continuous surface normal N(t) along the patch boundary.


Fig. 5. A set of surface patch boundary curves that are inconsistent with the global normal orientation constraint.

Fig. 6. Geodesic crossing constraints at the four corners of the surface patch. At each corner pi j the surface normal N is directed toward the observer.

Hence, considering scalars σiL, σiR ∈ {−1,+1} defined by

N(1) = σ2R n2(1) = σ3Ln3(0), N(2) = σ3R n3(1) = σ4R n4(1),

N(0) = σ1Ln1(0) = σ2Ln2(0), N(3) = σ1R n1(1) = σ4Ln4(0), (22)

the geodesic crossing constraints at the four patch corners become

p00:[σ1Lk1(0) − σ2Lk2(0)

]cos A00 + [

τ1(0) + τ2(0)]

sin A00 = 0,

p01:[σ3Lk3(0) − σ2Rk2(1)

]cos A01 + [

τ3(0) + τ2(1)]

sin A01 = 0,

p11:[σ3Rk3(1) − σ4Rk4(1)

]cos A11 + [

τ3(1) + τ4(1)]

sin A11 = 0,

p10:[σ1Rk1(1) − σ4Lk4(0)

]cos A10 + [

τ1(1) + τ4(0)]

sin A10 = 0. (23)

We consider the construction of surface patches with given geodesic boundary curves, knowing a priori that these curvessatisfy the above constraints. This stipulation can be met by, for example, selecting the boundary curves as known geodesicson simple analytic surfaces. For general free-form boundary curves, their construction so as to satisfy the system of con-straints (23) — or the modification of initial boundary curves so as to satisfy them — is a substantive problem in its ownright, which shall be addressed separately in another paper.


Fig. 7. Coons interpolation of four geodesic boundary curves.

5. Main steps of the construction

The algorithm comprises the following steps (see Fig. 7) and is based on Coons interpolation. The approach is similar tothat used in Sprynski et al. (2008).

• Step 1. Tangent plane. Along each of the four boundary curves ri(t), the tangent plane Πi(t) of the surface R(u, v) isdetermined by the surface normal vector N(t) along the patch boundary. Specifically,

Πi(t) = [ri(t),ei(t),hi(t)

].

• Step 2. Departure/arrival vectors for Hermite interpolation. A unit vector Ti(t) in the tangent plane Πi(t) at each pointof ri(t), and another unit vector Ti+2(t) in the tangent plane Πi+2(t) at each point of ri+2(t), must be specified for i = 1,2.These vectors specify the departure/arrival directions between the corresponding points on opposite sides ri(t), ri+2(t) ofthe patch boundary.

Consider the situation shown in Fig. 8. A natural choice is to assume Ti(t) = bi(t) for i = 1,3,4 and Ti(t) = −bi(t) fori = 2. However, we shall see in step 4 that the vectors Ti must satisfy one more constraint at the four patch corners. So


Fig. 8. Vectors along patch boundary.

we adopt a more general form for the departure/arrival vectors, based on the Darboux frame, yielding a larger family ofinterpolating surfaces. Namely, we set

Ti(t) = cosαi(t)ei(t) + sinαi(t)hi(t), i = 1,2,3,4. (24)

Here, the angle functions αi(t) specify the inclination of the vectors Ti relative to the boundary curve tangents ei . Note thatwe specify unit departure/arrival vectors Ti , since we shall subsequently introduce scalar magnitude functions to control the“parametric speed” of the isoparametric curves on the Coons interpolating surface patch. These functions can be used tomanipulate the surface shape for optimum smoothness.

• Step 3. Surface construction by bicubic Coons interpolation. Let d1(u), d3(u) and d2(v), d4(v) be scalar functions, used tomodulate the magnitude of the vectors T1(u), T3(u) and T2(v), T4(v). Consider the three surfaces R13(u, v), R24(u, v), andR0(u, v) defined by

R13(u, v) = [H0(v) H1(v) H2(v) H3(v)

]⎡⎢⎢⎣

r1(u)

d1(u)T1(u)

d3(u)T3(u)

r3(u)

⎤⎥⎥⎦ ,

R24(u, v) = [H0(u) H1(u) H2(u) H3(u)

]⎡⎢⎢⎣

r2(v)

d2(v)T2(v)

d4(v)T4(v)

r4(v)

⎤⎥⎥⎦ ,

R0(u, v) = [H0(u) H1(u) H2(u) H3(u)

]⎡⎢⎢⎣

p00 r′2(0) r′

2(1) p01

r′1(0) Ruv(0,0) Ruv(0,1) r′

3(0)

r′1(1) Ruv(1,0) Ruv(1,1) r′

3(1)

p10 r′4(0) r′

4(1) p11

⎤⎥⎥⎦

⎡⎢⎣

H0(v)

H1(v)

H2(v)

H3(v)

⎤⎥⎦ .

In the above expressions for R13(u, v), R24(u, v), R0(u, v), the functions

H0(t) = 2t3 − 3t2 + 1, H1(t) = t3 − 2t2 + t,

H2(t) = t3 − t2, H3(t) = −2t3 + 3t2,

are the cubic Hermite polynomials (Farin, 2002). The twist vectors Ruv(i, j) at the four patch corners i, j = 0,1 will bedetermined in step 5 below.

The desired interpolating surface is then defined (Farin, 2002) by

R(u, v) = R13(u, v) + R24(u, v) − R0(u, v).

• Step 4. Interpolating properties of tangent vectors at corners. Along the boundary v = 0, the equation of the surface R(u, v)

reduces to

R(u,0) = r1(u) + [H0(u) H1(u) H2(u) H3(u)

]⎡⎢⎣

r2(0) − p00d2(0)T2(0) − r′

1(0)

d4(0)T4(0) − r′1(1)

r4(0) − p10

⎤⎥⎦ (25)

and the partial derivatives Ru = ∂R/∂u, Rv = ∂R/∂v along this curve are

Ru(u,0) = r′1(u) + [

H ′0(u) H ′

1(u) H ′2(u) H ′

3(u)]⎡⎢⎣

r2(0) − p00d2(0)T2(0) − r′

1(0)

d4(0)T4(0) − r′1(1)

⎤⎥⎦ , (26)

r4(0) − p10


Rv(u,0) = d1(u)T1(u) + [H0(u), H1(u), H2(u), H3(u)

]⎡⎢⎢⎢⎣

r′2(0) − r′

2(0)

ddv d2(v)T2(v)|v=0 − Ruv(0,0)

ddv d4(v)T4(v)|v=0 − Ruv(1,0)

r′4(0) − r′

4(0)

⎤⎥⎥⎥⎦ . (27)

From expressions (25) and (26), we deduce that the interpolation constraints R(u,0) = r1(u) and Ru(u,0) = r′1(u) are satis-

fied if and only if d2(0)T2(0) = r′1(0) and d4(0)T4(0) = r′

1(1). Hence, for the four boundary curves, we obtain the followingnatural constraints:

d1(0)T1(0) = r′2(0) and d1(1)T1(1) = r′

4(0), (28)

d2(0)T2(0) = r′1(0) and d2(1)T2(1) = r′

3(0), (29)

d3(0)T3(0) = r′2(1) and d3(1)T3(1) = r′

4(1), (30)

d4(0)T4(0) = r′1(1) and d4(1)T4(1) = r′

3(1). (31)

In other words, the products di(t)Ti(t) of the magnitude functions with the corresponding departure/arrival vectors mustcoincide with the derivatives of the two adjacent curves at the extremities of each boundary curve ri(t).

To study these conditions in greater detail we consider, for example, the first condition in (28). Invoking the defini-tion (24) of vectors Ti(t), and noting that the magnitude functions di(t) are required to be positive, we have

d1(0)T1(0) = r′2(0) ⇐⇒

{d1(0) = ‖r′

2(0)‖cosα1(0)e1(0) + sinα1(0)h1(0) = e2(0)

⇐⇒ d1(0) = ∥∥r′2(0)

∥∥ and α1(0) = A00.

The relations (28)–(31) thus lead us to introduce the following scalar coefficients diL , diR , αiL , αiR for i = 1, . . . ,4 (seeFig. 9):

d1L := d1(0) = ∥∥r′2(0)

∥∥, d1R := d1(1) = ∥∥r′4(0)

∥∥,

d2L := d2(0) = ∥∥r′1(0)

∥∥, d2R := d2(1) = ∥∥r′3(0)

∥∥,

d3L := d3(0) = ∥∥r′2(1)

∥∥, d3R := d3(1) = ∥∥r′4(1)

∥∥,

d4L := d4(0) = ∥∥r′1(1)

∥∥, d4R := d4(1) = ∥∥r′3(1)

∥∥, (32)

α1L := α1(0) = +A00, α1R := α1(1) = +A10,

α2L := α2(0) = −A00, α2R := α2(1) = −A01,

α3L := α3(0) = +A01, α3R := α3(1) = +A11,

α4L := α4(0) = −A10, α4R := α4(1) = −A11. (33)

Thus, the angle functions αi(t) used in (24) to define the unit departure/arrival vectors Ti(t) and the correspondingmagnitude functions di(t) must satisfy

αi(0) = αiL, αi(1) = αiR and di(0) = diL, di(1) = diR .

Fig. 9. Binormal vectors (blue) and arrival vectors (red) on the curve r4(t). (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)


Fig. 10. Twist vector constraint at the patch corner p00.

We shall use the functions

αi(t) = αiL H0(t) + αi1 H1(t) + αi2 H2(t) + αiR H3(t), (34)

di(t) = diL H0(t) + di1 H1(t) + di2 H2(t) + diR H3(t), (35)

where the parameters αi1, αi2, di1, di2 will be specified in steps 5 and 6.• Step 5. Geodesic crossing property and twist vectors. From Eq. (27), we observe that the cross-derivative property

Rv(u,0) = d1(u)T1(u) along the curve r1(t) is satisfied if

Ruv(0,0) = d

dvd2(v)T2(v)

∣∣∣∣v=0

and Ruv(1,0) = d

dvd4(v)T4(v)

∣∣∣∣v=0

.

Considering the cross-derivative property along each boundary curve ri(t), we obtain the following constraints and defi-nitions:

d

dud1(u)T1(u)

∣∣∣∣u=0

= d

dvd2(v)T2(v)

∣∣∣∣v=0

=: Ruv(0,0), (36)

d

dvd2(v)T2(v)

∣∣∣∣v=1

= d

dud3(u)T3(u)

∣∣∣∣u=0

=: Ruv(0,1), (37)

d

dud3(u)T3(u)

∣∣∣∣u=1

= d

dvd4(v)T4(v)

∣∣∣∣v=1

=: Ruv(1,1), (38)

d

dvd4(v)T4(v)

∣∣∣∣v=0

= d

dud1(u)T1(u)

∣∣∣∣u=1

=: Ruv(1,0). (39)

Consider for example the “twist constraint” (36) at the corner p00. Since the four unit vectors e1(0) = T2(0), e2(0) =T1(0), h1(0), h2(0) are coplanar (see Fig. 10), we can write

e1(0) = cos A00e2(0) − sin A00h2(0),

e2(0) = cos A00e1(0) + sin A00h1(0). (40)

Hence (noting from Section 4.1 that A00 = 0,π ) we deduce that

h1(0) = − cos A00e1(0) + e2(0)

sin A00, h2(0) = −e1(0) + cos A00e2(0)

sin A00. (41)

Using definitions (24), (34), (35), (22), relations (32), (33), (41), and relations (12) from Lemma 2, we express each ofthe derivatives occurring in (36) with respect to the frame (e1(0),e2(0),N(0,0)), with N(0,0) being the unit normal to thesurface R(u, v) at the corner p00.

d

dud1(u)T1(u)

∣∣∣∣u=0

= d′1(0)T1(0) + d1(0)

d

ducosα1(u)e1(u) + sinα1(u)h1(u)

∣∣∣∣u=0

= d11e2(0) + d1L[−α′

1(0) sinα1(0)e1(0) + α′1(0) cosα1(0)h1(0) + cosα1(0)e′

1(0) + sinα1(0)h′1(0)

]= d11e2(0) + d1Lα11

[− sinα1Le1(0) + cosα1Lh1(0)]

+ d1L∥∥r′

1(0)∥∥[

(cosα1L)σ1Lk1(0) + (sinα1L)τ1(0)]N(0,0)

= −d1Lα111

sin A00e1(0) +

[d11 + d1Lα11

cos A00

sin A00

]e2(0)

+ d1Ld2L[(cos A00)σ1Lk1(0) + (sin A00)τ1(0)

]N(0,0).


In the same way, we obtain

d

dvd2(v)T2(v)

∣∣∣∣v=0

= d′2(0)T2(0) + d2(0)

d

dvcosα2(v)e2(v) + sinα2(v)h2(v)

∣∣∣∣v=0

=[

d21 − d2Lα21cos A00

sin A00

]e1(0) + d2Lα21

1

sin A00e2(0)

+ d2Ld1L[(cos A00)σ2Lk2(0) − (sin A00)τ2(0)

]N(0,0).

Thus, in order to satisfy the constraint (36) at corner p00, and be able to define the twist vector Ruv(0,0), we must findparameters α11, d11, α21, d21 such that the three following relations are satisfied:⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

−d1Lα111

sin A00= d21 − d2Lα21

cos A00

sin A00,

d11 + d1Lα11cos A00

sin A00= d2Lα21

1

sin A00,

(cos A00)σ1Lk1(0) + (sin A00)τ1(0) = (cos A00)σ2Lk2(0) − (sin A00)τ2(0).

(42)

The third equation in (42) is the geodesic crossing relation (13), and is satisfied through the assumption that the boundarycurves obey the relations (23) set up in Section 4.5. Multiplying by sin A00, the system (42) is equivalent to the followinglinear equations in the four unknowns α11, α21, d11, d21:{

α11d1L − α21d2L cos A00 + d21 sin A00 = 0,

α11d1L cos A00 − α21d2L + d11 sin A00 = 0.(43)

This system admits solutions. For example, the parameters d11, d21 can be freely chosen, and the parameters α11, α21 arethen computed from these equations (see examples in Fig. 14 which illustrate the influence of parameters dij ). Hence, weare able to define the twist vector Ruv(0,0) at the patch corner p00. In the same way, the twist vectors Ruv(i, j) at eachcorner pi j for i, j ∈ {0,1} can be defined. We can now state the following result.

Proposition 7. Given four regular space curves, as specified in Section 4.1, satisfying the following conditions:

(C1) the osculating constraints (19);(C2) the global normal orientation constraint set up in Section 4.4;(C3) the crossing constraints (23) at corners pi j;

there exists a regular oriented surface R(u, v) interpolating these four curves in such a way that these curves are geodesics of thesurface. Conversely, if any of the conditions (C1)–(C3) is not satisfied, such an interpolating surface can not be constructed.

• Step 6. Surface smoothing. In this last step we propose to find optimal parameters dij for i = 1,2,3,4 and j = 1,2[parameters αi j being deduced from Eqs. (43)] which provide smooth interpolating surfaces. The criteria involving thecurvature and the torsion of the isoparametric curves, applied in Sprynski et al. (2008), are not efficient here, due thecorner constraints which mainly impose the departure and ending direction of the interpolating curves. So, we propose tominimize the following functionals.

Criterion 1. This criterion minimizes the parametric speed variation along each isoparametric curve.

mindij

[ 1∫0

1∫0

(d

dv

∥∥Rv(u, v)∥∥)2

dv du +1∫

0

1∫0

(d

du

∥∥Ru(u, v)∥∥)2

du dv

].

Criterion 2. Minimization of the Dirichlet energy.

mindij

[ 1∫0

1∫0

(∥∥Ru(u, v)∥∥2 + ∥∥Rv(u, v)

∥∥2)du dv

].

Criterion 3. Minimization of the thin plate spline energy.

mindij

[ 1∫0

1∫0

(∥∥Ruu(u, v)∥∥2 + 2

∥∥Ruv(u, v)∥∥2 + ∥∥Rv v (u, v)

∥∥2)du dv

].

Computed examples are presented and discussed in Section 6.


Fig. 11. Geodesic on a cone.

Fig. 12. Four geodesic curves are evaluated on a circular cone in (1). The Coons patch interpolating these curves as boundary geodesics is shown superposedon the cone in (2). Finally, the Coons patch is shown on its own — with and without the departure/arrival vectors — in (3) and (4).

6. Examples

The following examples illustrate the application of the existence conditions for surfaces interpolating geodesic boundarycurves. A companion paper (Farouki et al., 2008) gives a detailed analysis in the case of Bézier boundary curves, in terms ofconstraints on the control points that ensure satisfaction of these conditions.

6.1. Analytical geodesics from elementary surfaces

We consider here elementary surfaces, which admit analytical expressions for their geodesic curves. These geodesiccurves naturally satisfy the constraints (C1), (C2), (C3). The interpolation of the four boundary curves as geodesics is ac-complished using the procedure described in Section 5, and we compare the constructed surface patch with the originalanalytic surface. For simplicity, the parameters αi1, αi2 and di1, di2 are all set to zero in these examples.


Fig. 13. Left and center: two views of the Coons patch interpolant to four great-circle boundary segments on the sphere. Right: an approximation of thewhole sphere using 6 patches. Each patch interpolates its boundaries in such a way that they are geodesics of that patch. Note that the construction is notsymmetric, since each of the six patches is different from the others.

Fig. 14. Six different surfaces interpolating given geodesic boundary curves, corresponding to different values of the parameters di1, di2 for i = 1,2,3,4.Surface (n) is associated with the parameters di1 = −8(n − 1) and di2 = 8(n − 1) — these are related to αi1, αi2 through Eqs. (43).

• Circular cone. Consider a circular cone generated by rotating a straight line about the z-axis. If this line has inclination awith the z-axis, we can parameterize the cone by R(u, v) = (u cos v, u sin v, Au) for u � 0 and 0 � v < 2π , whereA = cot a. Geodesic curves on this cone may be parameterized in terms of the angle v as

r(v) = u0

cos[(v − v0) sin a]

( cos vsin v

A

)

where (u0, v0) are parameter values such that the geodesic is tangent at r(v0) to the circle of latitude of radius u0on the cone (see Fig. 11). Fig. 12 illustrates the construction of a surface patch bounded by four geodesic curves ona circular cone. Although these boundary curves are geodesics of the constructed surface patch, the patch is not (ingeneral) an exact subset of the cone. In Fig. 12, portions of the patch beneath the cone are shown hatched, while thoseabove the cone are shown smoothly shaded.

• Sphere. Geodesics on the sphere are simply great-circle arcs. Fig. 13 shows a four-sided patch constructed so as toincorporate four great-circle segments as geodesic boundary curves — again, this patch is not (in general) an exactsubset of the sphere. Fig. 13 also illustrates a covering of the entire sphere by six patches.

6.2. Influence of parameters

The examples in Fig. 14 illustrate the influence on the surface shape of varying the parameters di1, di2 for i = 1,2,3,4.These examples show the importance of the smoothing procedures in step 6 of the algorithm, since the shape quality isevidently quite sensitive to these parameters, and ad hoc choices for them can yield surfaces of poor quality. Finally, Fig. 15shows the outcome of the three different smoothing procedures on a surface constructed using the geodesic boundary datain Fig. 4.


Fig. 15. Coons patches interpolating the geodesic boundary curves shown in Fig. 4, as smoothed according to the criteria 1, 2, and 3 in step 6 of thealgorithm (two different views of each smoothed surface are presented).

7. Conclusion

The problem of constructing a rectangular Coons surface patch from given boundary curves was investigated, under thestipulation that these boundary curves correspond to geodesic curves of the constructed surface. The existence of solutionsto this problem was shown to be contingent on the satisfaction of two types of constraints by the given boundary curves.A global (topological) constraint concerns the compatibility of the variation of the boundary curve principal normals withthe normal to an oriented surface. Local constraints, applicable to each of the four patch corners, relate the angles at whichthese curves meet to their curvatures and torsions.

For boundary curve data satisfying the above constraints, the construction of geodesic-bounded tensor-product surfacepatches can be accomplished using the bicubically-blended Coons interpolation scheme. By way of illustration, examplesof the construction of such patches on elementary analytic surfaces were presented. When the boundaries are specified ascompatible free-form polynomial or rational Bézier curves, the resulting surface is not (in general) a polynomial or rationalpatch, because of the need to unitize certain vectors in its construction. In a companion paper (Farouki et al., 2008) we relaxthe requirement of unit arrival/departure vectors, and thereby identify additional constraints on the control points and/orweights of given Bézier boundary curves, so as to obtain polynomial or rational surface patches for which these boundariesare geodesic curves.

Acknowledgements

This work has been accomplished during the visit of the third author to the Department of Mechanical and AeronauticalEngineering, University of California, Davis.

References

Bennis, C., Vézien, J.-M., Iglésias, G., 1991. Piecewise surface flattening for non-distorted texture mapping. Computer Graphics 25 (4), 237–246.Coons, S., 1964. Surfaces for Computer Aided Design. Technical Report, MIT (available as AD 663 504 from National Technical Information Service, Springfield,

VA 22161).Coons, S., 1974. Surface patches and B-spline curves. In: Barnhill, R., Riesenfeld, R. (Eds.), Computer Aided Geometric Design. Academic Press.Do Carmo, M.P., 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.Farin, G., 2002. Curves and Surfaces for CAGD, 5th edition. Academic Press.Farouki, R.T., Szafran, N., Biard, L., 2008. Construction of Bézier surface patches with Bézier curves as geodesic boundaries. Computer-Aided Design, submitted

for publication.Gordon, W., 1983. An operator calculus for surface and volume modeling. IEEE Computer Graphics and Applications 3, 18–22.Paluszny, M., 2008. Cubic polynomial patches through geodesics. Computer-Aided Design 40 (1), 56–61.Sánchez-Reyes, J., Dorado, R., 2008. Constrained design of polynomial surfaces from geodesic curves. Computer-Aided Design 40 (1), 49–55.Sprynski, N., Szafran, N., Lacolle, B., Biard, L., 2008. Surface reconstruction via geodesic interpolation. Computer-Aided Design 40 (4), 480–492.Struik, D.J., 1976. Lectures on Classical Differential Geometry. Dover (reprint).Tucker III, C.L., 1997. Forming of advanced composites. In: Gutowski, T.G. (Ed.), Advanced Composites Manufacturing. Wiley, New York, pp. 297–372.Wang, G., Tang, K., Tai, C.H., 2004. Parametric representation of a surface pencil with a common spatial geodesic. Computer-Aided Design 36 (5), 447–459.

Existence conditions for Coons patches interpolating geodesic … · 2010-11-10 · a local...

Documents

Transcript of Existence conditions for Coons patches interpolating geodesic … · 2010-11-10 · a local...