Computer-Aided Design 40 (2008) 49–55www.elsevier.com/locate/cad

Constrained design of polynomial surfaces from geodesic curves

J. Sanchez-Reyesa,∗, R. Doradob

a Department of Applied Mechanics, University of Castilla-La Mancha, Instituto de Matematica Aplicada la Ciencia e Ingenierıa, ETS Ingenieros Industriales,Campus Universitario, 13071 Ciudad Real, Spain

b Department of Mechanical and Mining Engineering, University of Jaen, Edificio A-3, Campus Las Lagunillas, 23071 Jaen, Spain

Received 2 October 2006; accepted 3 May 2007

Abstract

In a recent work, Wang et al. [Wang G, Tang K, Tai CH. Parametric representation of a surface pencil with common spatial geodesic. Computer-Aided Design 2004;36(5): 447–59] discuss a constrained design problem appearing in the textile and shoe industry for garment design. Given amodel and size, the characteristic curve called girth is usually fixed, and preferably should be a geodesic for manufacturing reasons. The designermust preserve this girth, being allowed to modify other areas according to aesthetic criteria. We present a practical method to construct polynomialsurfaces from a polynomial geodesic or a family of geodesics, by prescribing tangent ribbons. Differently from previous procedures, we identifythe existing degrees of freedom in terms of control points, and our method yields parametric polynomial surfaces that can be incorporated intocommercial CAD programs. The extension to rational geodesics is also outlined.c© 2007 Elsevier Ltd. All rights reserved.

Keywords: Bezier surface; Constrained design; Garment design; Geodesic

1. Introduction: Why constrained design from geodesics

The geodesic [2] between two points P, Q on a surface(Fig. 1) is defined as the curve embedded in the surface thatconnects the points with minimal distance d . In other words,geodesics can be regarded as the metric analogue on surfacesof straight lines on a plane. This property makes geodesics ofinterest in many engineering applications [3]. In particular, ageodesic on a developable surface maps to a straight line afterunfolding the surface onto a plane.

Here we focus our attention on a problem related togeodesics that arises in garment design. Given a model andsize, the characteristic curve called girth [1] is usually fixed(Fig. 2). The designer must preserve this girth, being allowedto modify other areas according to aesthetic criteria. Except forthe case of developable surfaces, when the fabric or leather isflattened to the plane, the development implies some distortion.Formally speaking, a general doubly curved surface does notadmit an isometric mapping to the plane, i.e., a mapping thatpreserves distances. This unavoidable distortion must be hence

∗ Corresponding author. Tel.: +34 926 295463; fax: +34 926 295361.E-mail address: Javier.SanchezReyes@uclm.es (J. Sanchez-Reyes).

0010-4485/$ - see front matter c© 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cad.2007.05.002

minimized using certain optimization criteria. In particular, ifwe choose a geodesic on the surface, we can devise a mappingΦ that avoids distortion along the geodesic, as shown byAzariadis and Aspragathos [4]. The flattening hence preservesthe length of the geodesic, which becomes a straight line(Fig. 1). This property is highly desirable for the girth curve,which means that the girth should be a geodesic on the 3Dsurface [1]. In addition, shoemakers naturally employ geodesicsto cut patterns [5], so the possibility of specifying directly in the3D surface such geodesic seams would be an attractive designoption.

We have thus identified our constrained design problem: howto construct surfaces from a given curve (or several curves),such that the curve becomes a geodesic in the surface. Thisproblem, for the simple case of one curve, was tackled byWang et al. [1], who obtained a representation of the surfacepencil that shares a given geodesic as an isoparametric curve.The solution provided in that seminal work certainly solvesthe mathematical problem, by furnishing the surface pencilin terms of certain sets of shape functions whose parametersthe user must adjust to achieve the desired shape. Since suchparameters lack intuitive geometric meaning, they must becomputed via least square fitting by specifying a set of datapoints to approximate. In addition, no formulation in terms

50 J. Sanchez-Reyes, R. Dorado / Computer-Aided Design 40 (2008) 49–55

Fig. 1. Mapping a geodesic on a surface onto a straight line on a plane.

Fig. 2. Girth curve of a shoe model.

of standard polynomial or rational Bezier form is provided. ACAD tool in a design scenario should rather yield surfaces thatadhere to the NURBS standard [6,7], and employ geometricshape handles, such as control points.

The paper is arranged as follows. In Section 2, after brieflyreviewing some fundamental concepts of differential geometry,we present our approach to overcome the shortcomings ofprevious methods, by constraining the surface through a tangentribbon that specifies cross-boundary derivatives. In Section 3we analyze this basic building block, namely how to designa ruled surface from a given polynomial geodesic. Once suchruled surfaces have been defined, in Section 4 we show howto construct a general surface from a geodesic or familyof geodesics, and discuss examples of garment design. InSection 5 we explain how to carry over these ideas to the case ofrational geodesics. Finally, conclusions are drawn in Section 6.

2. Constraining the surface through a “tangent ribbon”

2.1. Local characterization of geodesics

To solve our constrained design problem, rather thanemploying the metric definition of a geodesic in terms of lengthminimization, we resort to the alternative characterization usinglocal properties. A geodesic p0(u) on a surface is the curvewhose osculating plane, i.e., generated by the tangent t andnormal n to the curve, encompasses the normal vector N to thesurface (Fig. 3). Equivalently, the vectors n and N are alwaysparallel, or the binormal b to the curve is tangent to the surface.

Some textbooks on differential geometry [8] assumegeodesics to be arc-length parameterized, and thus distinguish

Fig. 3. Local definition of a geodesic on a surface.

Fig. 4. Interpreting the geodesic constrain in terms of a tangent ribbon to thesurface.

between geodesics and pregeodesics, related by reparameter-ization. For the sake of simplicity, either will be referred toas geodesics throughout this paper, regardless of their pa-rameterization. In most commercial CAD systems, based onthe NURBS standard, hardly ever are the input curves arc-length parameterized, since the straight line is the only rationalcurve that admits this optimal parameterization, as Farouki andSakkalis [9] noted.

2.2. A two-step approach

We have just concluded that the geodesic condition basicallyconstrains the orientation of the tangent plane to the surfacealong the prescribed geodesic p0(u). To facilitate the designand identify the degrees of freedom available in the surfaceconstruction, the natural approach is to split our constrainedproblem into two steps (Fig. 4):

1. Obtain the pencil of ruled surfaces (tangent ribbons [10])that share p0(u) as boundary. The rulings correspond to thetangent lines in v-direction of the desired surface p(u, v).

2. Identify the construction of p(u, v) as a standard interpo-lation problem from these tangent ribbons determining thecross boundary derivatives.

J. Sanchez-Reyes, R. Dorado / Computer-Aided Design 40 (2008) 49–55 51

Fig. 5. Constructing a ruled surface from a isogeodesic boundary p0(u).

For the case of a polynomial geodesic, the existing degreesof freedom are characterized in terms of control points in theBezier model, and we get polynomial surfaces that can bereadily incorporated into commercial CAD systems.

3. Constructing a ruled surface from a polynomialisogeodesic

3.1. Ruled surface from an isogeodesic

Given an arbitrary parametric curve p0(u), we are tocharacterize those ruled surfaces p(u, v) such that p0(u)

becomes an isogeodesic. This term, coined by Wang et al. [1],denotes a geodesic that is also an isoparametric line (v =

0). Using p0(u) as directrix, such ruled surfaces admit theparameterization:

p(u, v) = p0(u) + vq(u), (1)

where q(u) denotes the direction of the rulings. Not any q(u) isadmissible, because it must guarantee the geodesic condition,namely a binormal vector b(u) along p0(u) that is alwaystangent to the surface. Since the tangent plane (Fig. 5) is thatspanned by q(u) and the tangent t(u) to the curve, q(u) mustbe spanned by t(u) and b(u) [1]:

q(u) = α(u)t(u) + β(u)b(u), β(u) 6= 0. (2)

α(u), β(u) are arbitrary functions that define the pencil ofruled surfaces (1) sharing the isogeodesic p0(u). They controlhow the rulings depart from the isogeodesic on the tangentplane and provide us with two shape handles to control thesurface, namely a tension β(u) and shear-like effect α(u).Note that α(u) = 0 simply yields rulings perpendicular to thegeodesic, whereas β(u) = 0 is not admissible, as it would notguarantee the geodesic condition. For a constant β(u) = 0, thedirections (2) would generate a developable surface with lineof striction [2] p0(u), which instead of a geodesic is just theopposite, an asymptotic curve [8].

Expression (2) holds provided that t(u), b(u) are vectors intangent and binormal directions, not necessarily unitary, just byrescaling the functions α(u), β(u). Therefore, we can employa geodesic with arbitrary parameterization (not necessarily arc-length), and compute the tangent and normal directions directlyfrom the derivatives of p0(u):

t(u) = p′

0(u), b(u) = t(u) ∧ t′(u). (3)

3.2. Polynomial Bezier isogeodesic

The remarkable case of an integral Bezier isogeodesic p0(u)

deserves a detailed study. Restricting ourselves to polynomialfunctions α(u), β(u) (2), then q(u) and hence the ruledsurface p(u, v) are polynomial too. Its Bezier representation isobtained by expressing the vectors t(u), b(u) in Bezier form,and rewriting p(u, v) (1) as the ruled surface with rulingsconnecting two directrixes p0(u), p1(u):

p(u, v) = (1 − v)p0(u) + vp1(u), p1(u) = q(u) + p0(u).

(4)

Clearly, by expressing p0(u), p1(u) in Bezier form of acommon degree, we get the Bezier control points for p(u, v).

Consider a degree-m Bezier curve p0(u), u ∈ [0, 1], withcontrol points pi . Its derivative t(u), of degree m−1, has points:

ti = (pi+1 − pi )m, i = 0, . . . , m − 1.

At first glance, the cross product b(u) (3) seems of degree2m −3. However, a closer inspection [11] reveals that its actualdegree is only 2m − 4. In fact, a 3D polynomial curve ismathematically equivalent to the homogeneous representationof a planar rational curve, where the cross product (3) ofcurve and derivative appears in the conversion of a Beziercurve to its dual form [7]. In consequence, b(u) is efficientlycomputed as the cross product of the two intermediate pointstm−20 (u), tm−2

1 (u) appearing at the last step (m − 2) in the deCasteljau algorithm for the evaluation of t(u):

b(u) = tm−20 (u) ∧ tm−2

1 (u), (5)

which is clearly of degree 2m − 4. As b(u) (5) involves justadditions and products of polynomial functions in Bernsteinform, its list of control points bi admit a straightforwardcomputation via discrete convolutions [12]. Pottmann andWallner [13] also provide explicit expressions. In particular, forthe cubic case (m = 3):

{bi }2i=0 =

{t0 ∧ t1,

12(t0 ∧ t2), t1 ∧ t2

}. (6)

We hence conclude that t(u) and b(u) have degrees m − 1,2m − 4, respectively. If we want a ruled surface (4) of degreeat most (m,1), then the multiplying function α(u) (2) is linear,whereas β(u) is of degree 4 − m. The sought ruled surfacein Bezier form has 2 lines of control points pi0, pi1, (i =

0, . . . , m):

• pi,0 = pi , inherited from the initial isogeodesic p0(u).• pi,1 = pi + qi , inherited from the directrix p1(u) (4).

For the common case of a cubic isogeodesic (m = 3), wheret(u) and b(u) (3) are both quadratic, both α(u) and β(u) can belinear. In Bernstein form:

α(u) = α0(1 − u) + α1u, β(u) = β0(1 − u) + β1u. (7)

For completeness, we list the resulting points of the cubic vectorβ(u)b(u) (2) as a function of points bi (6):{β0b0,

13β1b0 +

23β0b1,

13β0b2 +

23β1b1, β1b2

}.

52 J. Sanchez-Reyes, R. Dorado / Computer-Aided Design 40 (2008) 49–55

Fig. 6. Control points of a ruled Bezier surface from a cubic geodesic p0(u)

(degree m = 3).

The four constants α0, α1, β0, β1 (7) correspond to the fourdegrees of freedom the construction enjoys. We may use themto choose the position of the corner point p01 anywhere onthe tangent plane at the corner p00, i.e., generated by t(0) andb(0), and analogously for p31. Once we have chosen p01, p31,depicted in grey in Fig. 6, the location of the two remainingpoints p11, p21 (drawn as solid points) is already determined.

3.3. Planar isogeodesic

In some cases, surfaces are constructed from planar sections,simpler to draw for a CAD user. A planar isogeodesic p0(u)

simplifies notably the construction, since the binormal isalways constant, and perpendicular to the plane containing theisogeodesic.

In the polynomial case, if we want a ruled surface of degree(m, 1), then the function β(u) (2) must be also of degreem. Expressing it in Bernstein form of coefficients βi (i =

0, . . . , m), we identify them as m+1 degrees of freedom. Usingthe end values β0, βm , along with α0, α1, we choose the cornerpoints p01, pm1, similarly to the preceding case, but we havenow m − 1 inner additional tensions β i (i = 1, . . . , m − 1).

In this planar case the construction extends in astraightforward manner from a single Bezier geodesic to a B-spline curve. Given a B-spline curve p0(u) of degree m, henceCm−1 at internal knots of single multiplicity, its derivative t(u)

has degree m − 1 and Cm−2 continuity at such knots. Thegoal is constructing a ruled surface of degree (m,1), preservingat least this Cm−2 continuity (along the u-direction). Sincesuch a continuity must be shared by the multiplying functionsα(u), β(u) (2):

• β(u) must be a degree-m spline function, sharing theknots of p0(u), optionally with internal knots of at mostmultiplicity two.

• α(u), a linear function, cannot be piecewise as it would enjoyonly C0 continuity. It must be a single linear function all overthe knot sequence, thereby providing, as for the Bezier case,two shape parameters α0, α1. For a closed isogeodesic, theonly alternative is a constant α(u) = α (only one tensionparameter).

Fig. 7 shows an example, a cubic (m = 3) B-splineisogeodesic yielding a C1 surface, where we set a constantfunction β(u) = 1.

4. Surface from a polynomial geodesic or a family of them

4.1. Surface from one isogeodesic

Once we have solved the construction of ruled surfaces froma polynomial isogeodesic, the extension to the real case ofdesigning an arbitrary surface is immediate. We are to define arectangular Bezier surface p(u, v) of degree (m, n), n ≥ 2, withour geodesic p0(u), as boundary v = 0. Along this boundary,the cross-boundary tangent lines in v-direction are those joiningp0(u) and the Bezier curve defined by the line j = 1 of controlpoints pi,1. On the other hand, such tangent lines in v-directionfor the ruled surface (4) are precisely the rulings joining p0(u),p1(u). Therefore, to fulfil the geodesic condition, just chooseas control points pi,0, pi,1 in the Bezier surface those of theruled surface (tangent ribbon). The remaining points pi, j (linesj ≥ 2) can be freely chosen, since they do not influencethe derivative at the boundary. The same applies to a B-splinesurface with a nonparametric v-knot vector starting at v = 0.

Fig. 8(a) illustrates an application in the garment industry (T-shirt), prescribing a geodesic to define the waist. In this surface,of degree (3,2), we have employed a trimming curve around theneck.

4.2. Joining two surfaces along a common isogeodesic

Clearly, reproducing the construction above at both sidesof a given isogeodesic p0(u) yields a G1 piecewise surface,even when the two tangent ribbons do not coincide, since weare enforcing a common surface normal N = n(u) alongthe common isogeodesic. In principle, if a curve separatestwo patches joining with only G1 continuity, instead of G2,then the development of this common boundary would bedifferent with respect to these patches, because its geodesiccurvature is different. However, we do not join the patchesalong arbitrary curves, but very special ones, namely geodesicswith respect to both patches, hence showing a common zerogeodesic curvature.

To achieve parametric C1 continuity, the two tangent ribbonsI, II must join with C1 continuity too. Therefore [10], the threepoints pI

i,1, pi , pIIi,1 must be collinear and in the same ratio for

all i . Once we have chosen one tangent ribbon, this commonratio is the only additional degree of freedom available for theother tangent ribbon. This geometry is shown in the examples ofFig. 8(b), (c) for a Bezier cubic isogeodesic. Fig. 8(b) illustratesa shoe design, whereas in Fig. 8(c) we have prescribed a hipline of a skirt. Note that in the example of Fig. 8(b) the surfacesstitched along the girth have different degrees.

4.3. Surface between two geodesic boundaries

Our method extends easily to the constrained constructionof a surface from two isogeodesics. Consider first the case ofa Bezier surface p(u, v) of degree (m, n), n ≥ 3, between two

J. Sanchez-Reyes, R. Dorado / Computer-Aided Design 40 (2008) 49–55 53

Fig. 7. B-spline ruled surface from C2 planar isogeodesics.

Fig. 8. Examples of constrained shoe and garment design construction from an isogeodesic.

Fig. 9. Bicubic Bezier surfaces that share common geodesic boundaries p0(u), pn(u).

degree-m isogeodesic boundaries p0(u) (v = 0) and pn(u) (v =

1). Simply compute ruled surfaces for both boundaries, whichfurnishes the lines j = {0, 1, n−1, n} of control points pi, j . Theremaining inner lines (for n > 3) can be freely chosen. A simi-lar method applies to a B-spline surface of degree (m, n), n ≥ 2,with a nonparametric v-knot vector between v = 0 and v = 1.

Fig. 9 illustrates the bicubic Bezier case. Once we havechosen both isogeodesics, we may still place the 4 greyboundary points anywhere on their corresponding tangentplanes. The tangent ribbons are then defined, and determine the4 inner control points. This restriction is the price to pay to solveour constrained design problem. A higher degree in v-direction

54 J. Sanchez-Reyes, R. Dorado / Computer-Aided Design 40 (2008) 49–55

Fig. 10. C2 bicubic B-spline surfaces built from 3 planar cubic isogeodesics.

would furnish additional lines of control points that could befreely located.

4.4. Surface from a family of isogeodesics

Suppose now that we are given a one-parameter family ofisogeodesics, and we want to construct a skinning surface thatinterpolates all those curves. By repeating the construction ofthe preceding section between each pair of curves, we get aG1 surface. If you want a higher continuity, for instance C2,first impose the same tangent ribbon at both sides of each innerisogeodesic. Second, apply along the v-direction any standardinterpolation method for curves that accepts as input points andtangents, for instance the cubic interpolating described by Piegland Tiller [6], where points and tangents are provided by thetangent ribbons. Fig. 10 plots an example for the case of cubicplanar isogeodesics.

5. Extension to rational geodesics

In this section, we outline the case when the prescribedisogeodesic is a rational Bezier curve p0(u), of degree m withcontrol points pi . Once again, the key problem is how tocompute the tangent and binormal vectors to p0(u).

Regarding the tangent t(u), the scaled hodograph [14] of adegree-m rational curve has degree 2m − 2. However, such adegree is unnecessarily high, as concluded after consideringthe problem in projective space. Adhering to the notationintroduced in Farin’s book [7], underlined points will denotehomogeneous coordinates. It is well known that the derivativep′

0(u) of p

0(u) lies on the tangent line. Therefore, an arbitrary

point t(u) on such a line (Fig. 11) is written as:

t(u) = p0(u) + α(u)p′

0(u), (8)

where α(u) indicates an arbitrary function. Observe that theaffine projection p′

0(u) in general does not always lie on the

same side of the tangent line. If w(u) denotes the weightfunction (last homogeneous coordinate) in p

0(u), then a

sign flipping in its derivative w(u) makes t(u) pass throughp

0(u). To avoid this annoying phenomenon, w(u) should be

monotone, which can be achieved by reparameterization ofp

0(u).Let us compute now the binormal to the isogeodesic. Given

a 3D Bezier curve p(u), its dual is the osculating plane b(u) =

〈p, p′, p′′〉 defined by the curve and its derivatives, where the

Fig. 11. Constructing a ruled surface from a rational isogeodesic boundaryp0(u).

operator 〈 〉 [7] denotes the analogue of cross product in 4Dspace. Such a plane is more efficiently computed from the 3intermediate points appearing at the step m − 2 (before last) inthe de Casteljau algorithm for the evaluation of p(u):

b(u) = 〈pm−20

(u), pm−21

(u), pm−22

(u)〉, (9)

and has thus degree 3m − 6.We are now ready to define the directrix p

1(u) of a

ruled surface with p0(u) as isogeodesic. Writing b(u) =

{b(u), wb(u)} (9) in terms of its weight function wb(u), thisdirectrix is given by:

p1(u) = t(u) + β(u) {b(u), 0} , (10)

where β(u) denotes an arbitrary shape function.For a cubic geodesic p

0(u), to keep the degree of p

1(u) (10)

at most cubic, the function α(u) (8) must be at most linear,whereas β(u) = β must be constant.

6. Conclusions

In the approximate flattening of surfaces, we can devise amapping that avoids distortion along a geodesic on the surface,transforming it into a straight line and preserving its length.Such a property is highly convenient in the shoe and garmentindustry, where the designer must preserve certain curves.This restriction leads to a constrained design problem: how toconstruct a surface from one or several geodesics.

The geodesic condition constrains the orientation of thetangent plane to the surface along the prescribed geodesic.The key idea is first to generate the pencil of ruled surfaces

J. Sanchez-Reyes, R. Dorado / Computer-Aided Design 40 (2008) 49–55 55

(tangent ribbons) sharing as boundary the prescribed geodesicand then construct the sought surface from this tangent ribbon.Differently to previous works, as the resulting surface mustbe transferable to CAD systems, we have restricted ourselvesto the case of polynomial curves, which yield polynomialsurfaces. We have adhered to the Bezier model and identifiedthe degrees of freedom in terms of control points.

The basic building block is thus the ruled surface that has asboundary the prescribed geodesic. In Bezier form, whereas oneline of control points is inherited from the geodesic, in the otherline some control points are partially or totally constrained fromthe geodesic condition. This tangent ribbon furnishes two linesof control points in the sought Bezier or B-spline surface. Theremaining control points can be freely chosen. The extension tothe construction of a surface interpolating several isogeodesicsis a straightforward problem. Finally, the case of a rationalgeodesic is easily tackled in homogeneous space.

Acknowledgements

This work is supported by the Spanish Ministerio deEducacion y Ciencia, under research grants DPI2003-02598and DPI2006-09091. We thank Dr. Marco Paluszny for severalhelpful discussions on geodesics and pregeodesics during hisstay at the University of Castilla-La Mancha.

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