Construction of rational surface patches bounded by lines of curvature
Luc Biard a, Rida T. Farouki b,∗, Nicolas Szafran 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 17 November 2009Received in revised form 8 March 2010Accepted 11 March 2010Available online 17 March 2010
Keywords:Darboux frameAngular velocityRotation-minimizing frameLines of curvaturePythagorean-hodograph curveRational surface patch
The fact that the Darboux frame is rotation-minimizing along lines of curvature of asmooth surface is invoked to construct rational surface patches whose boundary curvesare lines of curvature. For given patch corner points and associated frames defining thesurface normals and principal directions, the patch boundaries are constructed as quinticRRMF curves, i.e., spatial Pythagorean-hodograph (PH) curves that possess rational rotation-minimizing frames. The interior of the patch is then defined as a Coons interpolant,matching the boundary curves and their associated rotation-minimizing frames as surfaceDarboux frames. The surface patches are compatible with the standard rational Bézier/B-spline representations, and G1 continuity between adjacent patches is easily achieved.Such patches are advantageous in surface design with more precise control over the surfacecurvature properties.
As an extension of earlier studies (Farouki et al., 2009c, 2009d) concerned with constructing rational surface patchesthat have given geodesic boundary curves, we consider here the construction of patches with lines of curvature as boundarycurves. The procedure exploits the fact that the Darboux frame is rotation-minimizing along surface lines of curvature, andthe recent identification (Farouki, 2009; Farouki et al., 2009a) of curves that possess rational rotation-minimizing frames(RRMF curves).
By interpolating a set of RRMF curves as surface patch boundaries, with their associated rotation-minimizing frames de-scribing the variation of the surface Darboux frame along them, one can construct rational surface patches whose boundarycurves are automatically lines of curvature in a geometrically intuitive manner. This approach greatly extends the class ofrational surface patches, bounded by lines of curvature, that can be constructed.
The first systematic investigation of surface patches bounded by lines of curvature was described in the work of R.R. Mar-tin (Martin, 1982, 1983), who called them principal patches, since the tangents to the boundary curves correspond to principaldirections on the surface — i.e., the orthogonal directions of the extremal principal curvatures at each point of a surface. Mar-tin showed that the existence of such patches was contingent upon certain position matching and frame matching conditionsalong the patch boundary curves.
The construction of patches satisfying these conditions, from prescribed variations of the normal and geodesic curvaturesalong the boundaries, is a rather difficult process that does not ordinarily yield rational surface patches, although certain“simple” surfaces — such as generalized cylinders, surfaces of revolution, and the Dupin cyclides — permit easy constructionof principal patches (Martin, 1982, 1983). The Dupin cyclides (a special class of rational biquadratic surfaces that includesthe plane, cylinder, cone, sphere, and torus as special cases) are characterized by the fact that all their lines of curvature are
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simply circular arcs. However, the vertices of a four-sided Dupin cyclide principal patch are necessarily coplanar, a rathersevere constraint in piecing them together for free-form surface design applications.
A different approach to constructing free-form surfaces bounded by lines of curvature is proposed herein, based on thefact that a space curve equipped with an adapted orthonormal frame — of which the curve tangent is one of the frame vectors,and the other two span the curve normal plane — will be a line of curvature for any surface in which it is embedded, ifone ensures that the frame is rotation-minimizing with respect to the curve tangent (i.e., its angular velocity vector has nocomponent in the direction of the tangent) and it is coincident with the surface Darboux frame. The ability to constructrational surfaces bounded by lines of curvature then follows from standard Coons interpolation schemes (Coons, 1964, 1974;Farin, 2002; Gordon, 1983).
Note that all RRMF curves must be Pythagorean-hodograph (PH) curves (Farouki, 2008), since only PH curves possessrational unit tangent vectors. They may be characterized as proper subsets of the spatial PH curves by certain algebraic con-straints on the coefficients of the quaternion and Hopf map representations (Farouki, 2009; Farouki et al., 2009a). Actually,any spatial PH curve admits an essentially exact rotation-minimizing frame (RMF) computation, by integration of a ratio-nal function (Farouki, 2002). However, these RMFs do not ordinarily exhibit a rational dependence on the curve parameter,since rational functions do not in general have rational integrals. Thus, in order to obtain rational surface patches boundedby lines of curvature, we employ only RRMF curves as patch boundaries. A key step in the construction is the computationof RRMF curves interpolating given initial/final positions together with corresponding adapted frames.
The ability to construct rational patches (compatible with the standard Bézier/B-spline representations of contemporaryCAD systems) bounded by lines of curvature is valuable in applications where smooth surfaces must be designed with moreprecise a priori control over surface curvature properties. For such patches, the normal curvature is extremal (with respectto direction) along the patch boundaries and orthogonal to them. Composite surfaces may thus be designed by laying outa grid of “feature lines” that are prescribed as lines of curvature, and using the procedures described below to constructthe individual patches delineated by these feature lines. The patch construction scheme can easily ensure G1 continuitybetween adjacent patches, and may also be extendable to accommodate G2 continuity.
The remainder of this paper is structured as follows. Section 2 reviews the relevant background on adapted frames,rotation-minimizing frames, PH curves, and the characterizations of RRMF curves and their construction as geometric Her-mite interpolants. Concepts from surface differential geometry are then discussed in Section 3, including the Darboux frame,the principal curvatures/directions and lines of curvature, and the characterization of lines of curvature as loci along whichthe Darboux frame is rotation-minimizing. Section 4 then develops a construction for rational surface patches that matchgiven RRMF boundary curves and their associated frames through a Coons interpolation scheme, and Section 5 presentscomputed examples of these rational patches bounded by lines of curvature. Finally, Section 6 summarizes key results ofthe paper, and identifies topics for further investigation.
2. Rational rotation-minimizing frame curves
Before proceeding to the problem of rational surface patches bounded by lines of curvature, we review below essentialbackground information concerning rotation-minimizing adapted frames, PH curves, and RRMF curves.
2.1. Adapted frames on space curves
An adapted frame (t,u,v) on a space curve r(ξ) is a set of three orthogonal unit vectors, defined at each curve point,such that t = r′/|r′| is the curve tangent and u, v span the curve normal plane with u × v = t. The variation of such a framecan be specified through its angular velocity ω(ξ) by
dt
dξ= σω × t,
du
dξ= σω × u,
dv
dξ= σω × v, (1)
where σ(ξ) = |r′(ξ)| is the parametric speed of the curve r(ξ). The magnitude and direction of the angular velocity define theinstantaneous angular speed ω = |ω| and rotation axis a = ω/|ω| of the adapted frame (t,u,v). There are infinitely manyadapted frames on a given space curve r(ξ) — if a particular adapted frame (t,u,v) is chosen as a reference, another adaptedframe can be defined (Bishop, 1975) as (t, cos φu + sin φv,− sin φu + cosφv) for any scalar function φ(ξ), corresponding toa rotation of the reference frame normal-plane vectors u, v through angle φ(ξ) at each point of r(ξ).
Since (t,u,v) define a basis for R3 we can write ω = ω1t + ω2u + ω3v, and the characteristic property of a rotation-
minimizing adapted frame on r(ξ) is that the angular velocity component ω1 along t vanishes, i.e., ω · t ≡ 0. This impliesthat u and v have no instantaneous rotation about t — they vary only because t varies on r(ξ), and they must remainorthogonal to it. Alternatively, the frame (t,u,v) is rotation-minimizing with respect to t if the derivatives of u and v arealways parallel to t.
Perhaps the most-familiar adapted frame is the Frenet frame, defined by
t = r′′ , p = r′ × r′′
′ ′′ × t, b = r′ × r′′′ ′′ , (2)
|r | |r × r | |r × r |
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where (Kreyszig, 1959) the principal normal p points toward the center of curvature, and b = t × p is the binormal. Its angularvelocity is given by the Darboux vector
ω = κb + τ t, (3)
where the curvature κ and torsion τ are defined by
κ = p · t′
σ= |r′ × r′′|
|r′|3 and τ = −p · b′
σ= (r′ × r′′) · r′′′
|r′ × r′′|2 .
2.2. Rotation-minimizing adapted frames
The Frenet frame (2) is not an RMF, since its angular velocity (3) contains the term τ t which is non-zero for non-planarcurves. The basis vectors (u,v) of an RMF can be obtained by a rotation of (p,b) through a suitable angle function φ(ξ).Klok (1986) showed that these vectors satisfy certain differential equations, and Guggenheimer (1989) subsequently notedthat the solutions of these equations correspond to a rotation of (p,b) through the angle1
φ(ξ) = φ0 −ξ∫
0
τ (u)σ (u)du, (4)
the free integration constant φ0 indicating the existence of a one-parameter family of RMFs on a given space curve, differingfrom each other by just a constant angular displacement �φ in the normal plane at each curve point. The basic differencebetween the Frenet frame and an RMF is that the latter omits the term τ t from (3), so its angular velocity is just ω = κb.
Since the integral in (4) does not ordinarily admit closed-form reduction for polynomial and rational curves, manyschemes for the approximation of RMFs have been proposed (Farouki and Han, 2003; Jüttler, 1998; Jüttler and Mäurer,1999a, 1999b; Wang and Joe, 1997; Wang et al., 2008). The spatial Pythagorean-hodograph (PH) curves are an exception(Farouki, 2002) — for these curves the integrand reduces to a rational function, which may be integrated by a partial fractiondecomposition (but this usually incurs transcendental terms).
2.3. Pythagorean-hodograph curves
The parametric speed of a space curve r(ξ) = (x(ξ), y(ξ), z(ξ)) is the function
σ(ξ) = ∣∣r′(ξ)∣∣ =
√x′2(ξ) + y′2(ξ) + z′2(ξ) (5)
specifying the derivative ds/dξ of arc length s with respect to the parameter ξ . Hence, arc length derivatives may beexpressed as
d
ds= 1
σ(ξ)
d
dξ.
A Pythagorean-hodograph (PH) curve is a polynomial curve characterized by the fact that its parametric speed is a polynomialin ξ . Hence, the hodograph (derivative) r′(ξ) = (x′(ξ), y′(ξ), z′(ξ)) components must satisfy
x′2(ξ) + y′2(ξ) + z′2(ξ) ≡ σ 2(ξ) (6)
for some polynomial σ(ξ). A sufficient-and-necessary condition for satisfying the Pythagorean identity (6) is that x′(ξ),y′(ξ), z′(ξ), σ(ξ) are expressible (Dietz et al., 1993) in terms of four polynomials u(ξ), v(ξ), p(ξ), q(ξ) in the form
x′(ξ) = u2(ξ) + v2(ξ) − p2(ξ) − q2(ξ),
y′(ξ) = 2[u(ξ)q(ξ) + v(ξ)p(ξ)
],
z′(ξ) = 2[v(ξ)q(ξ) − u(ξ)p(ξ)
],
σ (ξ) = u2(ξ) + v2(ξ) + p2(ξ) + q2(ξ).
This structure is conveniently embodied in two algebraic models for spatial PH curves, introduced by Choi et al. (2002). Inthe quaternion representation, a Pythagorean hodograph is generated from a quaternion2 polynomial A(ξ) = u(ξ) + v(ξ)i +p(ξ)j + q(ξ)k by the expression
1 An incorrect sign before the integral is given by Guggenheimer (1989).2 Calligraphic characters denote quaternions, the scalar and vector components (Roe, 1993) of a quaternion A being indicated by scal(A) and vect(A).
When pure scalars or vectors are juxtaposed with quaternions, the quaternion product is imputed.
362 L. Biard et al. / Computer Aided Geometric Design 27 (2010) 359–371
r′(ξ) = A(ξ)iA∗(ξ), (7)
A∗(ξ) = u(ξ) − v(ξ)i − p(ξ)j − q(ξ)k being the conjugate of A(ξ). Note that the expression on the right in (7) is a quater-nion with zero scalar part — i.e., a vector in R
3. The Hopf map representation, on the other hand, generates a Pythagoreanhodograph from two complex polynomials3 α(ξ) = u(ξ) + iv(ξ), β(ξ) = q(ξ) + ip(ξ) through the expression
r′(ξ) = (∣∣α(ξ)∣∣2 − ∣∣β(ξ)
∣∣2,2 Re
(α(ξ)β(ξ)
),2 Im
(α(ξ)β(ξ)
)). (8)
The equivalence of (7) and (8) may be seen by taking A(ξ) = α(ξ) + kβ(ξ), the imaginary unit i being identified with thequaternion basis element i.
2.4. Rational rotation-minimizing frames
The possibility of constructing polynomial curves that have rational rotation-minimizing frames (RRMF curves) has recentlybeen demonstrated — these are necessarily PH curves, since only PH curves have rational unit tangents. Hence, RRMF curvescan be characterized by identifying constraints on the coefficients of PH curves that are sufficient and necessary for theexistence of a rational RMF. In this context, the Frenet frame (2) is not a good reference for identifying rational RMFs,because it is inherently non-rational and can exhibit singular behavior (p and b suddenly reverse) at inflections.
To remedy these problems, Choi and Han (2002) identified a rational adapted frame on spatial PH curves, the Euler–Rodrigues frame (ERF), defined by
t = A(ξ)iA∗(ξ)
|A(ξ)|2 , f = A(ξ)jA∗(ξ)
|A(ξ)|2 , g = A(ξ)kA∗(ξ)
|A(ξ)|2 .
This frame is not intrinsic (it depends upon the chosen coordinate system) — but it is always rational and non-singular.Han (2008) subsequently identified an algebraic criterion characterizing RRMF curves. Namely, the hodograph (7) defines anRRMF curve if and only if two relatively prime polynomials a(ξ), b(ξ) exist, such that the components u(ξ), v(ξ), p(ξ), q(ξ)
of A(ξ) satisfy
uv ′ − u′v − pq′ + p′qu2 + v2 + p2 + q2
= ab′ − a′ba2 + b2
. (9)
If this is satisfied, the RMF (t,u,v) can be obtained from the ERF (t, f,g) through the rational rotation specified by
u = a2 − b2
a2 + b2f − 2ab
a2 + b2g, v = 2ab
a2 + b2f + a2 − b2
a2 + b2g. (10)
It was shown by Han (2008) that all RRMF cubics are degenerate (i.e., planar curves, or curves with non-primitivehodographs). The existence of non-degenerate RRMF quintics was first constructively demonstrated in Farouki et al. (2009a),in which they were identified by one real and one complex constraint on the six complex coefficients that specify the Hopfmap form of spatial PH quintics. A much simpler (and more symmetric) characterization of these RRMF quintics was subse-quently derived by Farouki (2009), including a formulation in terms of the more commonly-used quaternion representationof spatial PH curves.
The following propositions state the key results concerning identification of quintic RRMF curves in the quaternion andHopf map representations of spatial PH curves — see Farouki (2009) for complete details and proofs.
Proposition 1. A spatial PH quintic specified in the quaternion form (7) by the quadratic quaternion polynomial
A(ξ) = A0(1 − ξ)2 + A12(1 − ξ)ξ + A2ξ2 (11)
has a rational RMF if and only if the coefficients of this polynomial satisfy
vect(
A2iA∗0
) = A1iA∗1. (12)
Proposition 2. A spatial PH quintic specified in the Hopf map form (8) by quadratic complex polynomials
α(ξ) = α0(1 − ξ)2 + α12(1 − ξ)ξ + α2ξ2,
β(ξ) = β0(1 − ξ)2 + β12(1 − ξ)ξ + β2ξ2, (13)
has a rational RMF if and only if the coefficients of these polynomials satisfy
3 Bold font symbols are used to denote both complex numbers and vectors in R3 — the meaning should be clear from the context.
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2.5. Geometric Hermite interpolation
In order to employ quintic RRMF curves as surface patch boundaries that are lines of curvature, it is necessary to havealgorithms for geometric design with these curves. In the standard C1 Hermite interpolation algorithm (Farouki, 2002;Farouki et al., 2008) for spatial PH quintics, there are two residual scalar freedoms. The conditions (12) or (14) that identifythe RRMF quintics among all spatial PH quintics amount to three scalar constraints. Hence, by relaxing from C1 to G1 data,it should be possible to construct RRMF quintic interpolants to specified end points and (unit) tangents, with one residualscalar freedom.
A preliminary study of G1 Hermite interpolation by RRMF quintics was presented in Farouki et al. (2010), and a morecomplete analysis was subsequently described in Farouki et al. (2009b). To embed an RRMF quintic as a surface line ofcurvature, its RMF must coincide with the surface Darboux frame (see Section 3 below). Since computing RMFs is an initialvalue problem, the surface normal at one end of an RRMF quintic intended as a line-of-curvature patch boundary can befreely chosen. The residual scalar freedom must then be used to match the desired surface normal at the other end, i.e., thecurve must interpolate given end points and its RMF must interpolate given end frames.
In Farouki et al. (2009b) it is shown that, after satisfying the RRMF condition, the problem of interpolating the end framesalways has a unique real solution, and the problem of interpolating the end points can be reduced to finding the positivereal roots of a degree six polynomial. Although the highly non-linear nature of the problem makes it difficult to establishthe existence of such roots in the case of arbitrary end points/frames, their existence has been verified for data sampledasymptotically from a smooth analytic curve. In cases where no RRMF quintic interpolant exists, it should be possible toensure existence of interpolants by introducing additional degrees of freedom. For example, the hodograph of an RRMFquintic can be modulated by a scalar polynomial, without compromising its rational RMF property — alternatively, one mayappeal to RRMF curves of higher degree.
3. Lines of curvature on a surface
Consider a curve r(ξ) = s(u(ξ), v(ξ)) on the surface s(u, v) specified by the parameter functions u(ξ), v(ξ). Denotingthe surface partial derivatives by su and sv , the Darboux frame (t,h,n) along this curve is defined by
t = u′su + v ′sv
|u′su + v ′sv | , h = su × sv
|su × sv | × t, n = su × sv
|su × sv | . (15)
Here t is the tangent to the curve r(ξ), n is the surface normal along r(ξ), and h = n × t is called the tangent normal, sinceit lies in the surface tangent plane and is orthogonal to the curve tangent t at each point. We now show that the Darbouxframe is rotation-minimizing along a surface curve if and only if that curve is a line of curvature, a fact that allows RRMFcurves to be used in constructing rational patches bounded by lines of curvature.
The variation of the Darboux frame along the curve r(ξ) = s(u(ξ), v(ξ)) is described (Kreyszig, 1959; Struik, 1988) bythe equations⎡
⎣ t′h′n′
⎤⎦ = σ
⎡⎣ 0 κg κn
−κg 0 −τg−κn τg 0
⎤⎦
⎡⎣ t
hn
⎤⎦ , (16)
where
σ = ∣∣u′su + v ′sv∣∣, κn = n · t′
σ, κg = h · t′
σ, τg = h · n′
σ
are the parametric speed, normal curvature, and the geodesic curvature and geodesic torsion of r(ξ). Geodesic curvature mea-sures the deviation between the principal normal p of the curve and the surface normal n along it — a geodesic curve ischaracterized by the fact that p ≡ n, and thus κg ≡ 0. The geodesic torsion in any chosen direction at a given surface point isthe torsion of the geodesic in the chosen direction at that point. For a geodesic curve, the geodesic torsion is thus identicalto the “ordinary” torsion, τg ≡ τ .
Now the normal curvature along r(ξ) = s(u(ξ), v(ξ)) is defined (Kreyszig, 1959) by
κn = − Lu′2 + 2Mu′v ′ + N v ′2
Eu′2 + 2F u′v ′ + G v ′2, (17)
E = su ·su , F = su ·sv , G = sv ·sv and L = n ·suu , M = n ·suv , N = n ·sv v being coefficients of the first and second fundamentalforms. At each point of r(ξ) = s(u(ξ), v(ξ)) expression (17) gives the curvature of the section of the surface by the planespanned by t and n at that point. The normal curvature κ is a function of the ratio u′ : v ′ that defines the orientation t ofthe section plane about the surface normal n. The extrema of (17) with respect to this ratio, the principal curvatures at thepoint under consideration, are given by
κmin = H −√
H2 − K and κmax = H +√
H2 − K , (18)
364 L. Biard et al. / Computer Aided Geometric Design 27 (2010) 359–371
where the mean curvature H = 12 (κmin + κmax) and Gaussian curvature K = κminκmax are defined at each surface point by
H = 2F M − E N − GL
2(EG − F 2)and K = LN − M2
EG − F 2. (19)
The directions in which the principal curvatures are attained — i.e., the principal directions of the surface at each point —are given by the ratios
u′ : v ′ = −(κp F + M) : κp E + L = −(κpG + N) : κp F + M, (20)
with κp = κmin or κmax. These principal directions are always orthogonal. Exceptionally, they become indeterminate at pointswhere κmin = κmax, since the normal curvature is then independent of direction — the surface is of spherical shape in theneighborhood of such umbilic points.
The lines of curvature on a surface are two families of curves, each tangent to one of the principal directions at everypoint, forming an “orthogonal net” (except at umbilic points) that covers the surface. It can be shown (Kreyszig, 1959) thata surface curve is a line of curvature if and only if the geodesic torsion along the curve satisfies τg ≡ 0. Now from (16), theangular velocity of the Darboux frame along r(ξ) = s(u(ξ), v(ξ)) can be expressed as
ω = −τgt − κnh + κgn, (21)
and hence it is rotation-minimizing if and only if τg ≡ 0. Since vanishing of the geodesic torsion along a surface curve issufficient and necessary for the curve to be a line of curvature, and for the Darboux frame to be rotation-minimizing alongit, we deduce the following result.
Proposition 3. A curve on a smooth surface is a line of curvature of that surface if and only if the Darboux frame is rotation-minimizingwith respect to the curve tangent.
Of course, lines of curvature on free-form polynomial or rational surfaces are generally not polynomial/rational curves.Thus, the ability to construct rational patches with lines of curvature as boundaries is quite extraordinary.
4. Lines of curvature as patch boundaries
According to Proposition 3, when an RRMF curve r(ξ) with an associated rational rotation-minimizing adapted frame(t(ξ),u(ξ),v(ξ)) is incorporated as a boundary of a rational surface patch, such that the rotation-minimizing frame de-fines the surface Darboux frame on r(ξ), the boundary curve will be a line of curvature of the constructed surface. Bymeans of geometric Hermite interpolation with RRMF curves, suitable boundary curves r(ξ) and their RMFs (t(ξ),u(ξ),v(ξ))
can be constructed (Farouki et al., 2009b), that match given end points r(0) and r(1) and frames (t(0),u(0),v(0)) and(t(1),u(1),v(1)).
4.1. Construction of 4-sided patches
The initial data consists of the four patch corner points pi j and corresponding orthonormal frames (ti j,hi j,ni j) identifyingthe surface principal directions and normals for i, j = 0,1. The first step is to construct RRMF curves rk(ξ), k = 1, . . . ,4defining a curvilinear rectangle (see Fig. 1), such that
r1(0) = p00, r1(1) = p10,
r2(0) = p00, r2(1) = p01,
r3(0) = p01, r3(1) = p11,
r4(0) = p10, r4(1) = p11. (22)
Also, the rational rotation-minimizing adapted frames associated with these curves, denoted by (tk(ξ),hk(ξ),nk(ξ)) wheretk(ξ) = r′
k(ξ)/|rk(ξ)|, are to be identified with the surface Darboux frames along each boundary curve rk(ξ), as in Fig. 1.Noting that the lines of curvature are orthogonal at each point, the initial and final frames (tk(0),hk(0),nk(0)) and(tk(1),hk(1),nk(1)) for each curve rk(ξ) must satisfy
(t1(0),h1(0),n1(0)
) = (t00,h00,n00),(t1(1),h1(1),n1(1)
) = (t10,h10,n10),(t2(0),h2(0),n2(0)
) = (h00,−t00,n00),(t2(1),h2(1),n2(1)
) = (h01,−t01,n01),
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Fig. 1. Orientation of Darboux frames (RMFs) along patch boundaries.
(t3(0),h3(0),n3(0)
) = (t01,h01,n01),(t3(1),h3(1),n3(1)
) = (t11,h11,n11),(t4(0),h4(0),n4(0)
) = (h10,−t10,n10),(t4(1),h4(1),n4(1)
) = (h11,−t11,n11). (23)
The four boundary curves rk(ξ), k = 1, . . . ,4 are constructed as RRMF quintic interpolants to the end points and endframes, defined in terms of the initial data by (22) and (23). If we construct a surface patch bounded by these curves, suchthat their rotation-minimizing frames coincide with the Darboux frame, the geodesic torsion will vanish along them, andthey will be lines of curvature on the surface.
The construction of the interpolating surface s(u, v) will ensure that the four boundary curves r1(ξ), r2(ξ), r3(ξ), r4(ξ)
correspond, respectively, to the surface isoparametric curves s(u,0), s(0, v), s(u,1), s(1, v). Furthermore, for each fixedvalue u∗ and v∗ of u and v , the two families of isoparametric curves s(u, v∗) and s(u∗, v) will meet the “boundary lines ofcurvature” orthogonally — i.e., in the complementary principal direction.
4.2. Coons interpolation scheme
In addition to the four boundary curves rk(ξ), k = 1, . . . ,4 the cubic Coons interpolation scheme requires a field of“departure/arrival” vectors Tk(ξ) to be defined in the surface tangent plane along each curve. These vectors have the generalform
where xk(ξ) and yk(ξ) are polynomial functions, defined below. As in Farouki et al. (2009c, 2009d) the vector fields Tk(ξ)
must satisfy the following conditions:
• Interpolation of the corner derivatives (Section 4.3.1):
T1(0) = r′2(0) and T1(1) = r′
4(0), (25)
T2(0) = r′1(0) and T2(1) = r′
3(0), (26)
T3(0) = r′2(1) and T3(1) = r′
4(1), (27)
T4(0) = r′1(1) and T4(1) = r′
3(1), (28)
i.e., the vectors (24) must coincide with the end derivatives of the two curves meeting at each of the four surface patchcorners.
• Twist vector constraints at the patch corners (Section 4.3.2) — the vector fields Tk(ξ) must satisfy the following com-patibility constraints, from which the twist vectors suv(i, j) for i, j = 0,1 are deduced:
d
duT1(u)
∣∣∣∣u=0
= d
dvT2(v)
∣∣∣∣v=0
=: suv(0,0), (29)
d
dvT2(v)
∣∣∣∣v=1
= d
duT3(u)
∣∣∣∣u=0
=: suv(0,1), (30)
d
duT3(u)
∣∣∣∣u=1
= d
dvT4(v)
∣∣∣∣v=1
=: suv(1,1), (31)
d
dvT4(v)
∣∣∣∣ = d
duT1(u)
∣∣∣∣ =: suv(1,0). (32)
v=0 u=1
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4.3. Bicubic Coons interpolant
Consider the three surfaces s13(u, v), s24(u, v), s0(u, v) defined by
s13(u, v) = [C0(v)C1(v)C2(v)C3(v)
]⎡⎢⎢⎣
r1(u)
T1(u)
T3(u)
r3(u)
⎤⎥⎥⎦ ,
s24(u, v) = [C0(u)C1(u)C2(u)C3(u)
]⎡⎢⎢⎣
r2(v)
T2(v)
T4(v)
r4(v)
⎤⎥⎥⎦ ,
s0(u, v) = [C0(u)C1(u)C2(u)C3(u)
]⎡⎢⎢⎢⎣
p00 r′2(0) r′
2(1) p01
r′1(0) suv(0,0) suv(0,1) r′
3(0)
r′1(1) suv(1,0) suv(1,1) r′
3(1)
p10 r′4(0) r′
4(1) p11
⎤⎥⎥⎥⎦
⎡⎢⎢⎣
C0(v)
C1(v)
C2(v)
C3(v)
⎤⎥⎥⎦ ,
where C0(ξ), . . . , C3(ξ) are the cubic Hermite basis functions (see Appendix A). The desired interpolating surface is thendefined by
s(u, v) = s13(u, v) + s24(u, v) − s0(u, v). (33)
4.3.1. Interpolation of corner derivativesTo satisfy the conditions (23) and (25)–(28), we must choose xk(0) = xk(1) = 0 for k = 1, . . . ,4 in the general form (24)
of the departure/arrival vectors Tk(ξ). In fact, in the present context it is desirable to assume that xk(ξ) ≡ 0 for k = 1, . . . ,4since this guarantees that all the intermediate isoparametric (constant u and constant v) curves meet the “boundary lines ofcurvature” orthogonally — i.e., in the complementary principal direction — and are thus (locally) also good approximationsto lines of curvature.
Consistent with these choices, the departure/arrival vectors (24) may be expressed using cubics for y1(ξ), . . . , y4(ξ) as
4.3.2. Twist crossing constraints at cornersConsider, for example, the “twist constraint” (29) at the patch corner p00. From the Darboux relations (16), and using
the fact that the geodesic torsion vanishes identically along lines of curvature, we have
T′1(0) = d
duT1(u)
∣∣∣∣u=0
= a11h1(0) + σ2(0)h′1(0) = a11h1(0) + σ2(0)
[−σ1(0)κg,1(0)]t1(0),
where κg,i is the geodesic curvature of ri(ξ). Similarly, we obtain
T′2(0) = d
dvT2(v)
∣∣∣∣v=0
= −a21h2(0) − σ1(0)[−σ2(0)κg,2(0)
]t2(0).
Hence, from relations (23), we see that the parameters a11, a21 must be specified by
a11 = σ1(0)σ2(0)κg,2(0), a21 = −σ1(0)σ2(0)κg,1(0)
in order for the twist constraint (29) to be satisfied at the corner p00, and the twist vector suv(0,0) to be uniquely defined.By considering analogous constraints at the other patch corners, so that the twist constraints (29)–(32) are satisfied, and
the twist vectors suv(i, j) are uniquely defined at each corner pi j , all the coefficients akl are determined if we use (34) asthe departure/arrival vectors. Thus, with the cubic Coons scheme, no residual freedoms remain for smoothing the surface ifit is not initially of good shape. As an alternative, a “pseudo-biquintic” Coons scheme is described below, that provides a setof free parameters for surface smoothing.
L. Biard et al. / Computer Aided Geometric Design 27 (2010) 359–371 367
4.4. Pseudo-biquintic Coons interpolant
As in the cubic case, we choose xk(ξ) ≡ 0, k = 1, . . . ,4 in (24) to ensure that the isoparametric curves cross the patchboundaries orthogonally. Consider the four quintic polynomials A0(ξ), A1(ξ), A4(ξ), A5(ξ) specified on [0,1] by
where αi j are free parameters. The first and last two columns of the matrix on the right indicate that these polynomialscan replace the usual cubic Hermite basis in the bicubically blended Coons interpolation scheme. We also define the fourpolynomials B0(ξ), B1(ξ), B4(ξ), B5(ξ) in an analogous manner, but with free parameters βi j in place of αi j . The interpola-tion scheme based on these polynomials is called “pseudo-biquintic” because it does not use the true quintic Hermite basis(which permits C2 rather than just C1 interpolation).
Consider the three surfaces s13(u, v), s24(u, v), s0(u, v) defined by
s13(u, v) = [B0(v)B1(v)B4(v)B5(v)
]⎡⎢⎢⎣
r1(u)
T1(u)
T3(u)
r3(u)
⎤⎥⎥⎦ ,
s24(u, v) = [A0(u)A1(u)A4(u)A5(u)
]⎡⎢⎢⎣
r2(v)
T2(v)
T4(v)
r4(v)
⎤⎥⎥⎦ ,
s0(u, v) = [A0(u)A1(u)A4(u)A5(u)
]⎡⎢⎢⎣
p00 r′2(0) r′
2(1) p01r′
1(0) suv(0,0) suv(0,1) r′3(0)
r′1(1) suv(1,0) suv(1,1) r′
3(1)
p10 r′4(0) r′
4(1) p11
⎤⎥⎥⎦
⎡⎢⎢⎣
B0(v)
B1(v)
B4(v)
B5(v)
⎤⎥⎥⎦ ,
where αi j , βi j for i = 0,1,4,5 and j = 1,2 are free parameters. The desired interpolating surface is then defined by
s(u, v) = s13(u, v) + s24(u, v) − s0(u, v). (35)
4.4.1. Departure and arrival vector fieldsChoosing xk(ξ) ≡ 0 for k = 1, . . . ,4 again, we use quintics for y1(ξ), . . . , y4(ξ) in (24). We then have
Q 0(ξ), . . . , Q 5(ξ) being the quintic Hermite basis functions (see Appendix A). In this case, eight of the coefficients aij remainas free parameters for surface smoothing after specification of the corner twist vectors below.
4.4.2. Twist crossing constraints at cornersConsider the twist constraint (29) at the patch corner p00. From the Darboux relations (16), and using the fact that the
geodesic torsion vanishes identically along lines of curvature, we have
T′1(0) = d
duT1(u)
∣∣∣∣u=0
= a11h1(0) + σ2(0)h′1(0) = a11h1(0) + σ2(0)
[−σ1(0)κg,1(0)]t1(0),
where κg,i is the geodesic curvature of ri(ξ). Similarly, we obtain
T′2(0) = d
dvT2(v)
∣∣∣∣v=0
= −a21h2(0) − σ1(0)[−σ2(0)κg,2(0)
]t2(0).
Hence, from relations (23), we see that the parameters a11, a21 must be specified by
a11 = σ1(0)σ2(0)κg,2(0), a21 = −σ1(0)σ2(0)κg,1(0)
368 L. Biard et al. / Computer Aided Geometric Design 27 (2010) 359–371
Fig. 2. Left: surface patch corner points and associated frames, defining the principal directions and surface normal. Right: interpolation of the patch cornerdata by four quintic RRMF curves, that define the patch boundaries.
in order for the twist constraint (29) to be satisfied at the corner p00, and the twist vector suv(0,0) to be uniquely defined.By considering the analogous constraints at the other corners, the twist constraints (29)–(32) are satisfied, and the twistvectors suv(i, j) are uniquely defined at each corner pi j .
Note that, in the above surface construction process, the eight coefficients ak2, ak3 for k = 1, . . . ,4 and the 16 quantitiesαi j , βi j for i = 0,1,4,5 and j = 1,2 remain as free parameters for the surface smoothing process.
Remark 1. By inspection of the above interpolation procedures, one can easily verify that s(u, v) defined by (33) or (35)is a rational surface. Since the rotation-minimizing frames of RRMF quintics are rational functions of degree 8 in the curveparameter (Farouki et al., 2009b), we observe that these expressions define rational Coons interpolants of bi-degree 11and 13, respectively.
Another way to obtain surface smoothing parameters is to assume non-zero polynomials xk(ξ) in the departure/arrivalvectors (24). As noted above, however, we consider this approach less desirable, since the orthogonality of the intermediateisoparametric curves to the patch boundaries makes them good (local) approximations to the surface lines of curvature.
5. Computed examples
Example 1. As the patch corner points pi j and associated frames (ti j,hi j,ni j) for i, j = 0,1 we take
This initial data is shown in Fig. 2, together with the four quintic RRMF boundary curves that interpolate it. Fig. 3 showsthe unit surface normals and tangent normals (i.e., the normal components of the rotation-minimizing frames) associatedwith these curves — these are exactly specified as rational vector fields in the curve parameters, in the degree 8 Bernsteinbasis.
In this case, a smooth surface was obtained using the simple cubic Coons interpolation scheme described in Section 4.3— there is no need to introduce additional free parameters (see Section 4.4) for surface smoothing purposes. Fig. 4 showsthe resulting rational surface patch.
L. Biard et al. / Computer Aided Geometric Design 27 (2010) 359–371 369
Fig. 3. Variation of the surface normal n (left) and the tangent normal h (right) along the surface patch boundaries. Together with the curve tangent t,these vectors define the Darboux frame (t,h,n) along each boundary, and by construction the vectors h, n are rotation-minimizing with respect to t.
Fig. 4. The rational surface patch bounded by lines of curvature (quintic RRMF curves) generated from the boundary data shown in Figs. 2 and 3.
Fig. 5. Left: rational surface patch bounded by lines of curvature before smoothing. Right: the same surface patch after minimization of thin-plate splineenergy with respect to the free parameters described in Section 4.4.
Example 2. Figs. 5 and 6 show two more examples, using the same corner points as in Example 1 but different frames tospecify the surface normal and principal directions at those corner points. In these cases, the initial surfaces obtained withthe cubic Coons scheme of Section 4.3 were of unsatisfactory shape, so the pseudo-biquintic Coons scheme of Section 4.4was used instead. The figures show the surfaces before and after minimization of the thin-plate spline energy
E =1∫
0
1∫
0
∣∣suu(u, v)∣∣2 + 2
∣∣suv(u, v)∣∣2 + ∣∣sv v(u, v)
∣∣2du dv
with respect to the free parameters ak2, ak3 for k = 1, . . . ,4 and αi j , βi j for i = 0,1,4,5 and j = 1,2 introduced in Sec-tion 4.4 (a numerical optimization scheme was used for this purpose).
370 L. Biard et al. / Computer Aided Geometric Design 27 (2010) 359–371
Fig. 6. Left: rational surface patch bounded by lines of curvature before smoothing. Right: the same surface patch after minimization of thin-plate splineenergy with respect to the free parameters described in Section 4.4.
6. Closure
A method for creating rational surface patches bounded by lines of curvature has been developed, based on a Coonsinterpolation scheme. The initial data comprises a set of patch corner points with associated orthonormal frames, that definethe surface normal and principal curvature directions. The patch boundary curves are then defined as quintic PH curves withrational rotation-minimizing frames (quintic RRMF curves). Since the surface Darboux frame is rotation-minimizing (withrespect to the tangent) along lines of curvature, Coons interpolation of these RRMF boundary curves and associated framesensures that they are lines of curvature on the constructed surface.
The line-of-curvature boundary property of these patches is not invariant under the usual de Casteljau subdivisionalgorithm, since the boundaries of the sub-patches are not necessarily RRMF curves. However, it is noteworthy that, inconstructing individual patches, the transverse surface isoparametric curves are orthogonal to the line-of-curvature bound-aries, and are thus also excellent (local) approximations to lines of curvature. Although the resulting rational surface patchesare of rather high degree, this is not problematic if the construction is systematically performed using the numerically-stableBernstein form. It is expected that the proposed scheme will be advantageous when surfaces are to be designed with precisecontrol over their curvature properties (e.g., by the interpolation of prescribed “feature” lines).
The intent of this paper was to demonstrate feasibility and methodology for the construction of free-form rational surfacepatches bounded by lines of curvature. The ideas presented herein can be further refined to facilitate the development ofa robust practical surface design system. Issues that deserve further study include extending the set of known solutions tothe problem of geometric Hermite interpolation using RRMF curves, and the incorporation and utilization of free smoothingparameters in the Coons scheme, to produce surfaces of optimum shape quality.
Appendix A. Hermite basis functions
The cubic and quintic Hermite bases on the interval ξ ∈ [0,1] are employed in the Coons interpolation scheme. The cubicbasis functions, defined by
C0(ξ) = 1 − 3ξ2 + 2ξ3,
C1(ξ) = ξ − 2ξ2 + ξ3,
C2(ξ) = −ξ2 + ξ3,
C3(ξ) = 3ξ2 − 2ξ3,
satisfy the boundary conditions⎡⎢⎢⎢⎣
C0(0) C ′0(0) C ′
0(1) C0(1)
C1(0) C ′1(0) C ′
1(1) C1(1)
C2(0) C ′2(0) C ′
2(1) C2(1)
C3(0) C ′3(0) C ′
3(1) C3(1)
⎤⎥⎥⎥⎦ =
⎡⎢⎢⎣
1 0 0 00 1 0 00 0 1 00 0 0 1
⎤⎥⎥⎦ .
Similarly, the quintic basis functions are defined by
Q 0(ξ) = −6ξ5 + 15ξ4 − 10ξ3 + 1,
Q 1(ξ) = −3ξ5 + 8ξ4 − 6ξ3 + ξ,
Q 2(ξ) = 1 (−ξ5 + 3ξ4 − 3ξ3 + ξ2),
2
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Bishop, R.L., 1975. There is more than one way to frame a curve. Amer. Math. Monthly 82, 246–251.Choi, H.I., Han, C.Y., 2002. Euler–Rodrigues frames on spatial Pythagorean-hodograph curves. Comput. Aided Geom. Design 19, 603–620.Choi, H.I., Lee, D.S., Moon, H.P., 2002. Clifford algebra, spin representation, and rational parameterization of curves and surfaces. Adv. Comp. Math. 17, 5–48.Coons, S., 1964. Surfaces for computer aided design. Technical report, M.I.T. (available as AD 663 504 from National Technical Information Service, Spring-
field, VA 22161).Coons, S., 1974. Surface patches and B-spline curves. In: Barnhill, R., Riesenfeld, R. (Eds.), Computer Aided Geometric Design. Academic Press, New York.Dietz, R., Hoschek, J., Jüttler, B., 1993. An algebraic approach to curves and surfaces on the sphere and on other quadrics. Comput. Aided Geom. Design 10,
211–229.Farin, G., 2002. Curves and Surfaces for CAGD, 5th ed. Academic Press, San Diego, CA.Farouki, R.T., 2002. Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves. Graph. Models 64, 382–395.Farouki, R.T., 2008. Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable. Springer, Berlin.Farouki, R.T., 2009. Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves. Adv. Comp.
Math., in press.Farouki, R.T., Han, C.Y., 2003. Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. Comput. Aided Geom.
Design 20, 435–454.Farouki, R.T., al-Kandari, M., Sakkalis, T., 2002. Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves. Adv. Comp. Math. 17,
369–383.Farouki, R.T., Giannelli, C., Manni, C., Sestini, A., 2008. Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures. Comput.
Aided Geom. Design 25, 274–297.Farouki, R.T., Giannelli, C., Manni, C., Sestini, A., 2009a. Quintic space curves with rational rotation-minimizing frames. Comput. Aided Geom. Design 26,
580–592.Farouki, R.T., Giannelli, C., Manni, C., Sestini, A. 2009b. Design of rational rotation-minimizing rigid body motions by Hermite interpolation, preprint.Farouki, R.T., Szafran, N., Biard, L., 2009c. Existence conditions for Coons patches interpolating geodesic boundary curves. Comput. Aided Geom. Design 26,
599–614.Farouki, R.T., Szafran, N., Biard, L., 2009d. Construction of Bézier surface patches with Bézier curves as geodesic boundaries. Comput. Aided Design 41,
772–781.Farouki, R.T., Giannelli, C., Sestini, A., 2010. Geometric design using space curves with rational rotation-minimizing frames. In: Daehlen, M., et al. (Eds.),
Lecture Notes in Computer Science, vol. 5862. Springer, Berlin, pp. 194–208.Gordon, W., 1983. An operator calculus for surface and volume modeling. IEEE Comput. Graph. Appl. 3, 18–22.Guggenheimer, H., 1989. Computing frames along a trajectory. Comput. Aided Geom. Design 6, 77–78.Han, C.Y., 2008. Nonexistence of rational rotation-minimizing frames on cubic curves. Comput. Aided Geom. Design 25, 298–304.Jüttler, B., 1998. Generating rational frames of space curves via Hermite interpolation with Pythagorean hodograph cubic splines. In: Geometric Modeling
and Processing ’98. Bookplus Press, Seoul, pp. 83–106.Jüttler, B., Mäurer, C., 1999a. Cubic Pythagorean hodograph spline curves and applications to sweep surface modelling. Comput. Aided Design 31, 73–83.Jüttler, B., Mäurer, C., 1999b. Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. J. Geom. Graphics 3, 141–159.Klok, F., 1986. Two moving coordinate frames for sweeping along a 3D trajectory. Comput. Aided Geom. Design 3, 217–229.Kreyszig, E., 1959. Differential Geometry. University of Toronto Press, Toronto.Martin, R.R., 1982. Principal patches for computational geometry. PhD thesis, Cambridge University.Martin, R.R., 1983. Principal patches — A new class of surface patch based on differential geometry. In: ten Hagen, P. (Ed.), Eurographics ’83. North-Holland,
Amsterdam, pp. 47–55.Roe, J., 1993. Elementary Geometry. Oxford University Press, Oxford.Struik, D.J., 1988. Lectures on Classical Differential Geometry. Dover Publications, New York (reprint).Wang, W., Joe, B., 1997. Robust computation of the rotation minimizing frame for sweep surface modelling. Comput. Aided Design 29, 379–391.Wang, W., Jüttler, B., Zheng, D., Liu, Y., 2008. Computation of rotation minimizing frames. ACM Trans. Graph. 27 (1), Article 2, 18 pp.
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