Construction and smoothing of triangular Coons patches with geodesic boundary curves

Transcription

1 Construction and smoothing of triangular Coons patches with geodesic boundary curves R. T. Farouki, (b) N. Szafran, (a) L. Biard (a) (a) Laboratoire Jean Kuntzmann, Université Joseph Fourier Grenoble, France (b) Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA Abstract Given three regular space curves r 1 (t), r 2 (t) r 3 (t), t [ 0, 1 ] that define a curvilinear triangle, we consider the problem of constructing a C 2 triangular surface patch R(u 1, u 2, u 3 ) bounded by these three curves, such that they are geodesics of the constructed surface. Results from a prior study [6] concerned with tensor product patches are adapted to identify constraints on the given curves for the existence of such geodesic bounded triangular surface patches. For curves satisfying these conditions, the patch is constructed by means of a cubically blended triangular Coons interpolation scheme. A formulation of thin plate spline energy in terms of barycentric coordinates with respect to a general domain triangle is also derived, and used to optimize the smoothness of the geodesic bounded triangular surface patches. Key words: geodesic curves, surface reconstruction, Coons interpolation 1. Introduction In a prior study [6], sufficient and necessary conditions for the boundaries of a four sided analytic surface patch to be geodesic curves of the surface were identified. A Coons interpolation scheme was then developed [7] to construct polynomial and rational examples of such geodesic bounded surface patches, given boundary curves that satisfy the existence conditions. In the present paper, these results are extended to the case of geodesic bounded triangular surface patches, parameterized in terms of barycentric coordinates. Also, the familiar Cartesian expression for thin plate spline energy is transformed to barycentric coordinates defined on a general triangular domain T. In general, Preprint submitted to Elsevier July 1, 2009

2 the barycentric form contains 21 terms, whose coefficients depend explicitly on the geometry of T. This form is used to optimize the shape of geodesic bounded triangular patches with respect to residual free parameters. The motivation for these studies comes from the problem of constructing computer representations of free form surfaces from positional/orientational measurements obtained with the Morphosense a flexible ribbon like device with embedded microsensors that assumes the shape of a geodesic when laid on a smooth physical surface [15]. By orienting the Morphosense in different directions on the surface, it may be divided into a collection of rectangular and triangular patches. Note that constructing triangular geodesic bounded patches is actually a more fundamental problem, since a closed surface of genus zero cannot be entirely covered by four sided patches. The remainder of this paper will be organized as follows. After reviewing some basic facts concerning differential geometry and barycentric coordinates on triangular domains in Section 2, the problem of constructing triangular surface patches with geodesics as boundary curves is described in Section 3, and the methodology for solving it is presented in Section 4. The properties of the constructed surface patches are then discussed, in the context of the Gauss Bonnet theorem, in Section 5. Section 6 develops the thin plate spline energy in terms of barycentric coordinates on general triangle domains, while Section 7 presents computed examples, with shape optimized in terms of this energy. Finally, Section 8 summarizes and assesses key results of the paper. 2. Preliminaries For linearly independent unit vectors u and v, and a unit vector n such that n u and n v, let (u, v) n denote the oriented angle between u, v in the sense of n. Specifically, the angle A = (u, v) n is defined (see Fig. 1) by n v u sin A = det(u, v, n), cos A = u, v. (1) Fig. 1. Angle measurement. We assume the reader is familiar with the elementary differential geometry of curves and surfaces [3, 16] see also [6] for pertinent background. For a space curve r(t) let (e(t), n(t), b(t)) be the Serret Frenet frame, and 2

3 k(t) and τ(t) the curvature and torsion, defined at non inflectional points by e(t) = r (t) r (t), b(t) = r (t) r (t), n(t) = b(t) e(t), (2) r (t) r (t) k(t) = r (t) r (t) r (t) 3 and τ(t) = det(r (t), r (t), r (t)) r (t) r (t) 2. (3) For a regular curve, satisfying r (t) = 0 for all t, the tangent e(t) is defined at every point. At inflection points (where r (t), r (t) are linearly dependent), the curvature k(t) is zero, and the principal normal n(t), binormal b(t), and torsion τ(t) are undefined. Henceforth, we specifically exclude the possibility of pathological inflections i.e., points where the curvature vanishes, and the left and right limits n and n + of the principal normal are neither parallel nor anti parallel, since no solution to the geodesic interpolation problem can be found in the presence of such points. The barycentric coordinates u = (u 1, u 2, u 3 ) of a point p = (x, y) with respect to a domain triangle T with vertices p i = (x i, y i ), i = 1, 2, 3 are given by u 1 u 2 = 1 y 2 y 3 x 3 x 2 x 2 y 3 x 3 y 2 x y 3 y 1 x 1 x 3 x 3 y 1 x 1 y 3 y, (4) u 3 y 1 y 2 x 2 x 1 x 1 y 2 x 2 y 1 1 where = x 1 x 2 x 3 y 1 y 2 y 3. (5) For distinct points p, q the directional derivative along the vector V = q p is defined [4] by 3 D V = (u i (q) u i (p)). u i i=1 Hence, the derivatives along directions parallel to each side s i of T, defined by u i = 0, is D si = u i 1 u i+1, i = 1, 2, 3 (6) where all indices are reduced modulo 3 to the set {1, 2, 3}. 3

4 p 1 u 1 1 = D s 3 D n 3 D n 2 D s 2 u 0 3 = p 2 D n 1 u 1 = 0 p 3 D s 1 u 2 = 1 u 3 = 1 u 0 2 = Fig. 2. Barycentric coordinates relative to the triangle (p 1, p 2, p 3 ). We then define (see Fig. 2) a derivative D ni each side s i by in the inward direction along D ni = α i 1 (u) D si+1 α i+1 (u) D si 1, i = 1, 2, 3 (7) where the three blending functions defined [8] by satisfy α i (u) = u 2 i (3 2 u i + 6 u i 1 u i+1 ), i = 1, 2, 3 3 α i (u) 1. i=1 Note also that, since α i 1 (u) + α i+1 (u) 1 along the side u i = 0 of T, the derivative D ni coincides with the derivative D si+1 at the point p i 1 and D si 1 at the point p i+1, along the side s i. 3. The three-geodesic interpolation problem Consider, as shown in Fig. 3, three regular space curves r 1 (t), r 2 (t), r 3 (t) with t [ 0, 1 ] such that 4

5 r i+1 (1) = p i = r i+2 (0) (8) for i = 1, 2, 3, where all indices are reduced modulo 3 (so that 0 3, 4 1, 5 2). These curves are assumed sufficiently smooth, and their derivatives at the corner points p i, i = 1, 2, 3 are assumed to be linearly independent, so as to define the tangent plane Π i for the interpolating surface R at p i. r (t) 3 r (t) 1 p 1 r (t) 2 p 2 p 3 Fig. 3. Triangular patch boundaries The geodesic interpolation problem Our intent is to construct an oriented surface patch that interpolates the specified curves as its boundaries, in such a manner that they are geodesic paths of the constructed surface. In terms of the barycentric coordinates u = (u 1, u 2, u 3 ) relative to the triangle with vertices p 1, p 2, p 3 the interpolating surface R(u) must have the following properties. Interpolation property: R(0, 1 t, t) = r 1 (t), R(t, 0, 1 t) = r 2 (t), t [ 0, 1 ]. R( 1 t, t, 0) = r 3 (t), (9) Geodesic property: the principal normal n ( t) at each non inflectional point of the curves r i (t) must be normal to the surface R(u). Let (e i (t), n i (t), b i (t)) denote the Serret Frenet frame and k i (t), τ i (t) the curvature and torsion of each of the curves r i (t), as defined by (2) and (3). Let r(t) be the concatenation of the three boundary curves r i (t), defined by r(t) = r i (t i + 1) for t [i 1, i] and i = 1, 2, 3, and consider the principal normal n(t) of this concatenated curve r(t), defined for t [ 0, 3 ] {0, 1, 2}. This normal n(t) is simply the concatenation of the principal normals n i (t) of the individual boundary curves r i (t). Note that the domain of the parameter t is considered to [ 0, 3 ] modulo 3. Now let N(u) be the unit normal to the interpolating surface R(u), and let N(t) for t [ 0, 3 ] be its restriction to the boundary r(t) of the surface 5

6 patch. For a continuous normal function N(t), we define the crossing angles A i at each corner p i by A i = ( e i+1 (1), e i 1 (0) ) N(p i, i = 1, 2, 3. (10) ) Assuming that the principal normals of the two boundary curves meeting at each corner agree (modulo sign) with the unit surface normal N, we consider the values σ il, σ ir { 1, +1} defined by N(0) = N(p 1 ) = σ 2R n 2 (1) = σ 3L n 3 (0), N(1) = N(p 2 ) = σ 3R n 3 (1) = σ 1L n 1 (0), N(2) = N(p 3 ) = σ 1R n 1 (1) = σ 2L n 2 (0). (11) Fig. 4 shows the Frenet frames (e i, n i, b i ) and the Darboux frames (e i, h i, N), with h i (t) = N(t) e i (t), along the patch boundary curves i = 1, 2, 3. p 2 N e 3 r (t) 3 b n 3 p 1 h 3= 3 r (t) 1 N e 2 h 2 N b h 1 = 1 e 1 n 1 n 2 b 2 r (t) 2 p Existence conditions for an interpolating surface Fig. 4. Vectors along patch boundaries. From Proposition 7 in [6], a surface interpolating the boundary curves as geodesics can be found if and only if they satisfy the following constraints: (C1) osculating constraint: the principal normals of the boundary curves that meet at each corner p i must agree modulo sign. This implies that, at each corner p i, the boundary curves meeting there have osculating planes orthogonal to the surface tangent plane Π i ; (C2) global normal orientation constraint: a continuous unit vector function N(t) must exist, such that N(t) = ± n(t) for all t R ; (C3) corner geodesic crossing constraint: for the crossing angles A i and values σ il, σ ir { 1, +1} defined by (10) and (11), the boundary 6

7 curves r i (t) must satisfy the following curvature and torsion constraints at each corner p i : [ σ i+1,r k i+1 (1) σ i 1,L k i 1 (0) ] cos A i + [ τ i+1 (1) + τ i 1 (0) ] sin A i = 0. (12) 4. Three sided geodesic interpolation Given three space curves free of inflection points, that satisfy conditions (C1) (C3), we wish to construct a triangular patch R(u 1, u 2, u 3 ) interpolating the curves as geodesic boundaries. A cubically blended Coons interpolation scheme [8] will be invoked to construct the surface (see also [6, 9, 11, 15] for related prior work on interpolation of geodesic boundary curves) Triangular Geodesic Coons interpolation p 1 For a triangular patch, the cubic Coons interpolation scheme requires a tangent vector T i (t) to be defined along each of the boundary curves r i (t) see Fig. 5. p2 r (t) 3 T 3 (t) T 1 (t) r (t) 1 r (t) 2 T 2 (t) p 3 Fig. 5. Tangent vectors T i (t) along patch boundary curves. In order to satisfy the geodesic constraint, the tangent vectors T i (t) must lie in the surface tangent plane at each point of r i (t), so these tangent vectors can, in general, be expressed in the form T i (t) = d i (t) [ cos(α i (t)) e i (t) + sin(α i (t)) h i (t) ], i = 1, 2, 3, (13) where the angle functions α i (t) specify the inclination of the vectors T i (t) relative to the boundary curve tangents e i (t). As in [6], we will now show that the vector fields T i (t) must satisfy the following two conditions: Interpolation of the corner derivatives (see Section 4.2): T i (0) = r i 1(1) and T i (1) = r i+1(0), i = 1, 2, 3 (14) 7

8 i..e, the tangent vectors T i must coincide with the derivatives of the two curves r i 1 (t), r i+1 (t) meeting the curve r i (t) at its extremities. Twist crossing constraints at the corners (see Section 4.3): The derivatives of the tangent vectors T i (t) must satisfy the following relations, which define the twist vectors W i at the patch corners. T i 1(0) = T i+1(1) =: W i, i = 1, 2, 3. (15) Consider now the C 1 -triangular Coons patch R(u), as developed in [8]: with R(u) = 3 α i (u) R i (u), (16) i=1 R i (u) = r i 1 (u i+1 ) + u i 1 [ T i 1 (u i+1 ) T i 1 (0) ] + r i+1 (1 u i 1 ) + u i+1 [ T i+1 (1 u i 1 ) T i+1 (1) ] p i u i 1 u i+1 W i. (17) A straightforward computation yields R(u) ui =0 = r i(u i 1 ), i = 1, 2, 3, (18) and D ni R(u) ui =0 = [ α i 1(u) D si+1 α i+1 (u) D si 1 ] R(u) ui =0 [ ] 3 = α i+1 (u) α i 1 (u) α i (u) R i (u) u i u i+1 = T i (u i 1 ) u i 1 α i+1 (u) ui =0 [ r i 1(1) + T i (0) + u i 1 (T i 1(1) + W i+1 ) ] i=1 ui =0 + α i 1 (u) ui =0 [ r i+1(0) T i (1) + u i+1 (T i+1(0) W i 1 ) ]. (19) Assuming that the three vector fields T i (t) satisfy conditions (14) (15), the patch R(u) will interpolate the boundary curves r i (t) and the tangent vectors T i (t) precisely: R(u) ui =0 = r i(u), D ni R(u) ui =0 = T i(u), i = 1, 2, 3. (20) 8

9 which proves with our hypothesis that the patch R(u) interpolates the boundary curves r i (t) as geodesics. Consider now constraints (14) and (15). We will establish conditions on the vector fields T i (t) in order to satisfy these constraints. In particular, we show that the twist constraints (15) can only be satisfied if the conditions (C1) (C3) are satisfied by the curves r i (t) Interpolation of corner derivatives T i (0)= -r' i-1 (1) r i-1 (t) r i+1 (t) T i (1)= r' i+1 (0) h i (0) N(p i+1) A i+1 e i-1 (1) e i (0) h i-1 (1) r i (t) e i+1 (0) N(p i-1) A i-1 e i (1) Fig. 6. Vectors along the curve r i (t). To satisfy the constraints (14) on the vector field T i (t), defined by (13), we introduce the following scalar coefficients d il, d ir, α il, α ir (see Fig. 6) for i = 1, 2, 3: d il := d i (0) = r i 1(1), d ir := d i (1) = r i+1(0), α il := α i (0) = π A i+1, α ir := α i (1) = A i 1. (21) Then, as in [6], we consider the functions α i (t) = α il H 0 (t) + α i1 H 1 (t) + α i2 H 2 (t) + α ir H 3 (t), (22) d i (t) = d il H 0 (t) + d i1 H 1 (t) + d i2 H 2 (t) + d ir H 3 (t), (23) where H i (t) are the cubic Hermite polynomials, and the parameters α i1, α i2, d i1, d i2 will be specified in Section

10 4.3. Twist crossing constraints at corners p i r' i+1 (1) e i-1 (0) r' i-1 (0) e (1) i+1 A i N(p ) i h (1) i+1 r (t) i-1 h (0) i-1 r (t) i+1 Fig. 7. Vectors at corner p i. At corner p i, the vectors e i+1 (1), h i+1 (1), e i 1 (0), h i 1 (0) are coplanar and related (see Fig. 7) as follows: e i+1 (1) = cos A i e i 1 (0) sin A i h i 1 (0), e i 1 (0) = cos A i e i+1 (1) + sin A i h i+1 (1). Hence (noting from Section 3 that A i 0, π) we deduce that h i 1 (0) = cos A i e i 1 (0) e i+1 (1), h i+1 (1) = e i 1(0) cos A i e i+1 (1). sin A i sin A i (24) Then using the definitions (13), (22), (23), relations (11), (21) and the Serret- Frenet relations as in [6], with b i+1 (1) = σ i+1,r h i+1 (1), b i 1 (0) = σ i 1,L h i 1 (0), we express each of the derivatives occurring in (15) with respect to the frame (e i+1 (1), e i 1 (0), N(p i )) as [ T i 1(0) = d i 1(0) + r i+1(1) α i 1(0) cos A ] i e i+1 (1) sin A i r i+1(1) α i 1(0) 1 e i 1 (0) sin A [ i ] + r i+1(1) r i 1(0) cos A i σ i 1,L k i 1 (0) + sin A i τ i 1 (0) N(p i ), 10

11 T i+1(1) = r i 1(0) α i+1(1) 1 e i+1 (1) sin A i [ + d i+1(1) + r i 1(0) α i+1(1) cos A ] i e i 1 (0) sin A [ i ] + r i+1(1) r i 1(0) cos A i σ i+1,r k i+1 (1) + sin A i τ i+1 (1) N(p i ). Thus, the twist crossing constraint (15) at corner p i, together with definitions (21) (23), yields the following system of equations cos A i 1 d i 1,1 + d i 1,L α i 1,1 = d i+1,r α i+1,2, sin A i sin A i d i 1,L α i 1,1 1 sin A i = d i+1,2 d i+1,r α i+1,2 cos A i sin A i, cos A i σ i 1,L k i 1 (0) + sin A i τ i 1 (0) = cos A i σ i+1,r k i+1 (1) sin A i τ i+1 (1). (25) As in [6], identifying components of the twist crossing constraint (15) in the direction of N(p i ) at corner p i yields the geodesic crossing relation (12) at that point, which is satisfied through the assumption that the boundary curves obey conditions (C1) (C3) in Section 3.2. Thus, multiplying equations (25) by sin A i yields the following linear equations in the four unknowns d i 1,1, α i 1,1, d i+1,2, α i+1,2 : { sin Ai d i 1,1 + cos A i d i 1,L α i 1,1 d i+1,r α i+1,2 = 0, (26) d i 1,L α i 1,1 + sin A i d i+1,2 + cos A i d i+1,r α i+1,2 = 0. This linear system is of rank 2, and if S i denotes the set of its solutions, we have dim(s i )=2 (see Section 7 for examples illustrating the influence of the free parameters in S i, i = 1, 2, 3). Thus, we are able to define a twist vector W i at the patch corner p i, and we can now state the following result. Proposition 1. Given three regular space curves, that satisfy the conditions (C1) (C3) specified in Section 3.2, an oriented triangular surface patch R(u) exists, that interpolates these curves as boundaries in such a way that they are geodesics of the surface. Conversely, when any of the conditions (C1) (C3) is not satisfied, such an interpolating surface can not be constructed. 11

12 5. Gauss Bonnet theorem The well known Gauss Bonnet theorem relates the integral of Gaussian curvature over a region of a surface to the integral of the geodesic curvature along the boundary of that region. This theorem is of particular relevance to the geodesic bounded surface patches constructed herein. Theorem 1. Global Gauss-Bonnet Theorem [3, 16]. Let R S be a regular simple region of an oriented surface S and Γ 1,..., Γ k be closed, simple, piecewise regular curves forming its boundary R. Suppose that Γ 1,..., Γ k are all positively oriented, and let A 1,..., A k be the external angles at the junctures of the curves Γ 1,..., Γ k, as defined by (10). Then k i=1 Γ i k i (s) ds + R K dσ + k A i = 2π, (27) where k i (s) is the geodesic curvature of the curve Γ i, s is arc length along it, and K is the Gaussian curvature of the surface S with area element dσ. Since the geodesic curvature vanishes along any geodesic curve, it follows from the above theorem that if we construct a triangular surface patch R(u) as described in Proposition 1, so that its boundary curves are geodesics, the Gaussian curvature K of this surface must satisfy R K dσ = 2π i=1 3 A i, (28) where A 1, A 2, A 3 are the corner angles at the vertices p 1, p 2, p 3. i=1 Fig. 8. Gauss Bonnet triangle. 12

13 For example, given three regular space curves r i (t) satisfying conditions (C1) (C3), such that 3 i=1 A i = 2π, as in Fig. 8, any interpolating surface R(u) constructed as described in Proposition 1 must satisfy K dσ = 0, (29) R which indicates that the interpolating surface must contain both elliptic and hyperbolic points (where K > 0 and K < 0) see Fig. 9. Fig. 9. Two views of a surface interpolating the Gauss Bonnet triangle of Fig. 8 as geodesics. The free parameters d ij and α ij are set equal to zero. Consider the problem of smoothing the interpolating surface. From (28), we see that the integral of the Gaussian curvature will be the same for all geodesic bounded interpolating surfaces R(u). Thus, one natural choice is to minimize the variation of Gaussian curvature over the surface. Another choice is based on minimizing the thin plate spline energy, adapted to the context of triangular patches as described below. 6. Thin plate spline energy for triangular patches For a bivariate function f(x, y) the thin plate spline energy per unit area is defined by E = f 2 xx + 2f 2 xy + f 2 yy. (30) 13

14 This expression is clearly invariant under any translation, since it depends only on the second derivatives of f. One can easily see that it is also invariant under any rotation, specified by [ ] [ ] [ ] x cos θ sin θ x y =. sin θ cos θ y Applying the derivative operators x = x x x y = x y x + y x + y y y = cos θ x sin θ y, y = sin θ x + cos θ y, twice to f, one can verify that f 2 x x + 2f 2 x y + f 2 y y f 2 xx + 2f 2 xy + f 2 yy. Hence, for Cartesian coordinates, the thin plate spline energy per unit area (30) is a universal expression. Consider now the formulation of the thin plate spline energy in terms of the barycentric coordinates (4) with respect to a domain triangle T with the vertices p i = (x i, y i ) for i = 1, 2, 3. To transform derivatives with respect to (x, y) into derivatives with respect to (u 1, u 2, u 3 ) we use the relations x = u 1 + u 2 + u 3 x u 1 x u 2 x u 3 = 1 [ ] (y 2 y 3 ) u1 + (y 3 y 1 ) u2 + (y 1 y 2 ) u3, y = u 1 + u 2 + u 3 y u 1 y u 2 y u 3 = 1 [ ] (x 3 x 2 ) u1 + (x 1 x 3 ) u2 + (x 2 x 1 ) u3. (31) Clearly, these derivative operators depend explicitly on the chosen vertices of the domain triangle T. Consequently, there is no universal barycentric formulation of the expression (30). It was noted above that (30) is invariant under transformations of the Cartesian coordinates that correspond to rigid motions. However, mappings between barycentric coordinates with respect to 14

15 different domain triangles define general affine transformations, not just rigid motions. Hence, each domain triangle T has its own associated expression for the thin plate spline energy, dependent on the vertices of T. We denote the second derivatives of f with respect to (u 1, u 2, u 3 ) by f jk = 2 f u j u k, 1 j, k 3. The expression for E in barycentric coordinates with respect to an arbitrary domain triangle T contains 21 terms the squares of the 6 second derivatives f 11, f 22, f 33, f 12, f 23, f 31 and the ( 6 2) = 15 cross terms arising from their pair wise products. The coefficients of these 21 terms will depend on the chosen vertices of T. Using the partial derivatives (31) and writing x jk = x j x k and y jk = y j x k for 1 j, k 3 we obtain 2 f xx = y 2 23f 11 + y 2 31f 22 + y 2 12f (y 23 y 31 f 12 + y 31 y 12 f 23 + y 12 y 23 f 31 ), 2 f xy = x 32 y 23 f 11 + x 13 y 31 f 22 + x 21 y 12 f 33 + (x 13 y 23 + x 32 y 31 )f 12 + (x 21 y 31 + x 13 y 12 )f 23 + (x 21 y 23 + x 32 y 12 )f 31, 2 f yy = x 2 32f 11 + x 2 13f 22 + x 2 21f (x 32 x 13 f 12 + x 13 x 21 f 23 + x 21 x 32 f 31 ), Defining the sides of the domain triangle T by s 1 = p 3 p 2, s 2 = p 1 p 3, s 3 = p 2 p 1, one can then verify that expression (30) becomes 4 E = s 1 4 f s 2 4 f s 3 4 f [ s 1 2 s (s 1 s 2 ) 2 ] f [ s 2 2 s (s 2 s 3 ) 2 ] f [ s 3 2 s (s 3 s 1 ) 2 ] f (s 1 s 2 ) 2 f 11 f (s 2 s 3 ) 2 f 22 f (s 3 s 1 ) 2 f 33 f f 11 [ s 1 2 (s 1 s 2 ) f 12 + (s 1 s 2 )(s 3 s 1 ) f 23 + s 1 2 (s 3 s 1 ) f 31 ] + 4 f 22 [ s 2 2 (s 1 s 2 ) f 12 + s 2 2 (s 2 s 3 ) f 23 + (s 1 s 2 )(s 3 s 1 ) f 31 ] + 4 f 33 [ (s 2 s 3 )(s 3 s 1 ) f 12 + s 3 2 (s 2 s 3 ) f 23 + s 3 2 (s 3 s 1 ) f 31 ] + 4 [ s 2 2 (s 3 s 1 ) + (s 1 s 2 )(s 2 s 3 ) ] f 12 f [ s 3 2 (s 1 s 2 ) + (s 2 s 3 )(s 3 s 1 ) ] f 23 f [ s 1 2 (s 2 s 3 ) + (s 3 s 1 )(s 1 s 2 ) ] f 31 f 12. (32) 15

16 If l 1 = s 1, l 2 = s 2, l 3 = s 3 are the lengths of the domain triangle sides, and θ 1, θ 2, θ 3 are its interior angles at the vertices p 1, p 2, p 3 so that s 1 s 2 = l 1 l 2 cos θ 3, s 2 s 3 = l 2 l 3 cos θ 1, s 3 s 1 = l 3 l 1 cos θ 2, the expression (32) can also be written as 4 E = l 4 1 f l 4 2 f l 4 3 f l 2 1l 2 2(1 + cos 2 θ 3 ) f l 2 2l 2 3(1 + cos 2 θ 1 ) f l 2 3l 2 1(1 + cos 2 θ 2 ) f l 2 1l 2 2 cos 2 θ 3 f 11 f l 2 2l 2 3 cos 2 θ 1 f 22 f l 2 3l 2 1 cos 2 θ 2 f 33 f l 2 1 f 11 (l 2 l 3 cos θ 2 cos θ 3 f 23 l 1 l 2 cos θ 3 f 12 l 3 l 1 cos θ 2 f 31 ) + 4 l 2 2 f 22 (l 3 l 1 cos θ 3 cos θ 1 f 31 l 1 l 2 cos θ 3 f 12 l 2 l 3 cos θ 1 f 23 ) + 4 l 2 3 f 33 (l 1 l 2 cos θ 1 cos θ 2 f 12 l 2 l 3 cos θ 1 f 23 l 3 l 1 cos θ 2 f 31 ) + 4 l 2 1l 2 l 3 (cos θ 2 cos θ 3 cos θ 1 ) f 31 f l 1 l 2 2l 3 (cos θ 3 cos θ 1 cos θ 2 ) f 12 f l 1 l 2 l 2 3 (cos θ 1 cos θ 2 cos θ 3 ) f 23 f 31. An adaptation of the thin plate spline energy (30) is often invoked in the problem of smoothing a rectangular (tensor product) surface patch R(u, v) for (u, v) [ 0, 1 ] [ 0, 1 ]. Specifically, the expression E = R uu R uv 2 + R vv 2 is employed in lieu of (30). The equivalent expression for a triangular surface R(u 1, u 2, u 3 ) parameterized by barycentric coordinates on a domain triangle T is obtained by replacing f 11, f 12, etc, by R 11, R 12, etc, with the cross terms f 11 f 22, f 12 f 23, etc, replaced by the dot products R 11 R 22, R 12 R 23, etc. Here R jk = 2 R/ u j u k for 1 j, k 3. The three patch corner points p 1 = R(1, 0, 0), p 2 = R(0, 1, 0), p 3 = R(0, 0, 1) are a natural choice for the vertices of the barycentric coordinates domain triangle T. 7. Smoothing methods and computed examples We seek optimal values for the parameters d ij, α ij in the set S = 3 i=1s i which provide smooth interpolating surfaces. For this purpose, we propose to minimize one of the following functionals. 16

17 Criterion A minimization of the thin plate spline energy E expressed in terms of barycentric coordinates relative to the reference domain triangle T, as developed Section 6: min E du. S T Criterion B: minimization of the expression where D 2 s i s j = D si D sj ( ) min D S T 2 s i s j R(u) 2 du, 1 i,j 3 and T is the reference domain triangle. Fig. 10 illustrates the smoothing of geodesic bounded triangular patches using these two measures, for the case of the boundary curves shown in Fig. 8. (1) (2) (3) Criterion A (4) Criterion B (1) (2) 17

18 (3) Criterion A (4) Criterion B Fig. 10. Two differents views of the Gauss Bonnet example of Fig. 8, before and after smoothing: (1) the three boundary curves; (2) the initial reconstructed surface with free parameters d ij equal to zero; (3) the reconstructed surface after smoothing according to criterion A; and (4) the reconstructed surface after smoothing according to criterion B. The optimization is carried out with respect to the parameters d ij the parameters α ij are evaluated from (26). For criterion A and criterion B, the smoothness measure is reduced by a factor of approximately 2. Figs illustrate triangular patches with boundary curves defined by geodesics on some simple quadric surfaces a sphere, cylinder, and cone. Fig. 11. Three geodesics on a sphere and the reconstructed surface. 18

19 Fig. 12. Three geodesics on a cylinder and the reconstructed surface. (a) (b) (c) 19

20 Fig. 13. Geodesic triangle on a cone: (a) the cone and the three geodesic curves; (b) the cone, the three geodesic curves and the interpolating surface; and (c) the three geodesic curves and the interpolating surface alone. 8. Conclusion Given three space curves that define a curvilinear triangle, a method has been presented for constructing a triangular surface patch bounded by these curves, such that they are geodesic curves of the constructed surface. For the construction to be feasible, the given boundary curves must satisfy certain a priori compatibility conditions. In order to optimize the smoothness of the geodesic bounded triangular patches, a formulation of the thin plate spline energy in terms of barycentric coordinates with respect to a general domain triangle was also derived. This formulation is more complicated (containing 21 terms) than the familiar Cartesian expression, and its coefficients depend explicitly on the shape of the domain triangle. The availability of triangular geodesic bounded patches, in conjunction with earlier formulations for rectangular geodesic bounded patches, allows one to reconstruct surfaces from general networks of geodesic paths on them, as physically determined by the Morphosense device. Acknowledgment. Part of this work was accomplished during the visit of the third author to the Department of Mechanical & Aeronautical Engineering, University of California, Davis. References [1] Coons, S., Surfaces for Computer Aided Design, Technical Report, M.I.T., 1964 (available as AD from National Technical Information Service, Springfield, VA 22161). [2] Coons, S., Surface patches and B-spline curves, R. Barnhill and R. Riesenfeld, editors, Computer Aided Geometric Design, Academic Press, [3] Do Carmo, M. P., Differential Geometry of Curves and Surfaces, Prentice Hall, [4] Farin, G., Triangular Bernstein Bézier patches, Computer Aided Geometric Design 3, ,

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