A Reeneable Space of Spline Surfaces of Arbitrary Topological Genus Projekt 03-ho7stu-2 Modelle F Ur Freiformmm Achen Beliebiger Topologischer Struktur a Reeneable Space of Spline Surfaces a Reeneable Space of Spline Surfaces of Arbitrary Topological Genus

It is shown that parametrical smoothness conditions are su cient for modeling smooth spline surfaces of arbitrary topology if degenerated surface segments are accepted. In general, degeneracy, i.e. vanishing partial derivatives at extraordinary points, is leading to surfaces with geometrical singularities. However, if the partial derivatives of higher order satisfy certain conditions, the existence of a regular smooth reparametrization can be guaranteed. So, degeneracy is no fundamental obstacle to generating surfaces which are smooth in the sense of di erential geometry. Besides its striking simplicity the approach presented here admits the construction of spline spaces which have a natural re nement property. Thus, various algorithms based on subdivision of tensor product B-spline surfaces become available for surfaces of general type. Keywords subdivision, arbitrary topology, extraordinary vertex, D-patch, degenerated surface segment, quasi control point, piecewise polynomial surface, n-sided hole 3 A Re neable Space of Spline Surfaces of Arbitrary Topological Genus Abstract It is shown that parametrical smoothness conditions are su cient for modeling smooth spline surfaces of arbitrary topology if degenerated surface segments are accepted. In general, degeneracy, i.e. vanishing partial derivatives at extraordinary points, is leading to surfaces with geometrical singularities. However, if the partial derivatives of higher order satisfy certain conditions, the existence of a regular smooth reparametrization can be guaranteed. So, degeneracy is no fundamental obstacle to generating surfaces which are smooth in the sense of di erential geometry. Besides its striking simplicity the approach presented here admits the construction of spline spaces which have a natural re nement property. Thus, various algorithms based on subdivision of tensor product B-spline surfaces become available for surfaces of general type. Introduction Commonly, parametrical smoothness conditions are considered to be too restrictive for modeling spline surfaces of arbitrary topological genus. So, the concept of geometrical smoothness has been developed [1,2,4-8,10,15] which is exible enough to admit smooth joints of arbitrarily connected surface segments. Besides requiring polynomials of high degree, the major drawback of this approach is the absence of a natural re nement property as provided by subdivision algorithms for tensor product spline surfaces. In this report we present a di erent way of constructing smooth spline surfaces of arbitrary topological type. The idea is to use exclusively parametrical smoothness conditions and to overcome the related well known di culties by introducing surface patches which are degenerated in the sense that their partial derivatives vanish at extraordinary vertices. This method was proposed in [11,12] and [13] and veri ed strictly in [14] for polynomial patches. Here it is shown that the method can be generalized to degenerated patches based on arbitrary analytical basis functions thus extending the range of applicability to most spline models currently in use. Besides its striking simplicity the major bene t of this approach is that it admits the construction of spline spaces which have a natural re nement property. This should facilitate the of many powerful tools and algorithms based on subdivision of tensor product B-splines to spline surfaces of arbitrary topology. To enumerate only a few of them, think of surface rendering and determining cross sections of spline surfaces in computer aided design or the recently developed concepts of wavelets and hierarchical basis in the eld of approximation theory. The paper is organized as follows. In the rst section su cient regularity conditions for degenerated analytical surface patches are derived. Roughly speaking, a surface segment is degenerated if the partial derivatives of order (1; 0); (0; 1) and (1; 1) are zero at one of 4 its corners and in general, this will cause a cusp-like geometrical singularity. However, regularity, i.e. the existence of a regular smooth re-parametrization, can be guaranteed if the partial derivatives of order (2; 0); (2; 1); (1; 2) and (0; 2) are coplanar and properly arranged at this point. Surface segments of this type are called D-patches. Further, the behavior of the main curvatures near the singular point is studied. In the second section the conditions for D-patches are applied to polynomial patches in Bernstein-B ezier form. It turns out that degeneracy is equivalent to four coalescing B ezier points while regularity requires certain neighboring B ezier points to be coplanar. In the third section a space S of spline surfaces incorporating D-patches is introduced. It is based on bicubic polynomial patches joining parametrically smooth of rst order. The elements of S can be described in a geometrical intuitive way by control points, whose topological structure is a natural generalization of the tensor product arrangement. Unlike proper control points some of the control points assigned to patches sharing extraordinary vertices cannot be chosen arbitrarily but have to ful ll certain constraints related to the conditions imposed on Dpatches. Therefore, they will be referred to as quasi control points. In the fourth section the re nement property of the spline space S is established in terms of a linear map acting on control points. This map is uniform in the sense that it uses the same mask of weights equally on the regular and the extraordinary parts of the surface. In the fth section a linear subspace S S is speci ed by replacing the coplanarity condition for quasi control points by xed linear dependencies. Thus, by providing linearity, one of the major prerequisites for various applications is ful lled. In the sixth section methods for projecting arbitrarily chosen control points to the space of quasi control points are given. On one hand, this is of particular importance for design purposes where control points are assumed to be manipulable without restrictions. But on the other hand, this allows the construction of a family of real valued B-spline functions spanning a space of smooth spline surfaces and providing most of the favorable properties of ordinary B-splines. Finally, in the seventh section an application to the problem of lling the n-sided hole is brie y discussed. By using B-splines a nicely shaped solution minimizing a certain fairness functional can be found easily using standard tools from linear algebra. 1. Regularity of Degenerated Patches Consider n 2 INnf1; 2g analytical surface segments over the unit square ! := [0; 1]2 xj : ! 3 u := (u; v) 7! 1 X p;q=0Ajpq upvq 2 IR3 ; j = 1; : : : ; n (1:1) joining parametrically smooth according to xj(0; t) = xj+1(t; 0) ; xju(0; t) = xj+1 v (t; 0) ; t 2 [0; 1] : (1:2) Here and subsequently, the index j runs from 1 to n and has to be understood modulo n. By uniqueness of Taylor series, the smoothness conditions imply Aj0r = Aj+1 r0 ; Aj1r = Aj+1 r1 ; r 2 IN0 : (1:3) 5 For r 2, these equations are decoupled and no problems arise. However, for r 2 f0; 1g we obtain a cyclic system of equations, Aj00 = Aj+1 00 ; Aj10 = Aj+1 01 Aj01 = Aj+1 10 ; Aj11 = Aj+1 11 : (1:4) The rst equation simply de nes the common center M := A100 = : : : = An00 whereas the other three equations are leading to Aj01 = Aj+2 01 = Aj+4 01 Aj10 = Aj+2 10 = Aj+4 10 Aj11 = Aj+1 11 = Aj+2 11 : (1:5) For all n 2 IN, this system has the trivial solution Aj10 = Aj01 = Aj11 = 0 ; j = 1; : : : ; n (1:6) corresponding to patches xj , which are degenerated in the sense that they are singularly parametrized at M and consequently not necessarily smooth. For n even, there exist additional non-trivial solutions, which are 2-periodic for n=2 odd and 4-periodic for n=2 even. However, taking into account that the geometry of the con guration requires nperiodic solutions, it turns out that only the case n = 4 admits reasonable non-trivial solutions. Therefore, it is commonly called the regular case. This observation was made frequently before [e.g. 6,8], and usually one tries to solve this problem by introducing socalled geometrical smoothness conditions. They are weaker than (1.2) but still guarantee smooth joints in the sense of di erential geometry. The idea of accepting the trivial solution (1.6) can be found in [13] and [11,12], however, without being veri ed rigorously. To do so we temporarily con ne ourselves to the examination of a single singularly parametrized surface segment and start with the following de nitions: De nition 1.1 An analytical surface segment x of type x : ! 3 u := (u; v) 7! 1 X p;q=0Apq upvq 2 IR3 (1:7) is degenerated if A10 = A01 = A11 = 0 : (1:8) A degenerated surface segment x is called D-patch if there exist constants ; 2 IR and ; 2 IR+ such that A21 A12 = A20 A02 : (1:9) A D-patch is said to be generic if A20 and A02 are linearly independent. 6 Theorem 1.2 A generic D-Patch x is regular at A00, that is there exists a regular smooth parametrization representing x, locally. The tangent plane passing through A00 is spanned by A20 and A02. Proof Since A20 and A02 are assumed to be linearly independent, one can chose a coordinate system such that A00 = o is the origin and (A20;A02) = (e1; e2) are the rst two unit vectors. Using the fact that the coe cients ; ; ; do not depend on the particular choice of coordinates we obtain A20 = (1; 0; 0) ;A21 = ( ; ; 0) ;A02 = (0; 1; 0) ;A12 = ( ; ; 0) and x(u) =:0@x(u) y(u) z(u)1A =0@u2 + uv2 +O(u2v + u3 + v3 + (u+ v)4) v2 + u2v +O(uv2 + u3 + v3 + (u+ v)4) O(u3 + v3 + (u+ v)4) 1A : (1:10) The xy-plane is expected to be the tangent plane at the origin, so we try to represent x as the graph of some function h(x; y) near the origin, i.e. (x; y) 7! (x; y; h(x; y)) 2 x : (1:11) This is possible if the projection of x to the xy-plane is locally injective or, equivalently, if the function x : U" 3 u 7! (x(u); y(u)) is invertible for U" := fu 2 ! : kuk < "g and " small enough. Denote the Jacobian of x by J and the symmetrized Jacobian by ~ J := (J + JT )=2 ; (1:12) then a short computation yields trace ~ J(u) = 2(u+ v) +O((u + v)2) det ~ J(u) = 2( u3 + v3 + 2uv) +O(u2v + uv2 + (u+ v)4) : (1:13) Both expressions are positive for u 2 U"n(0; 0) and " su ciently small. For the trace, this is obvious, and for the determinant it can be shown as follows: The inequalities u3 + v3 + 2uv 2uv u3 + v3 + 2uv (u+ v)3 (1:14) with := minf ; ; 1=3g hold for all ; > 0 and u 2 !. Thus, det ~ J(u) 2uv +O(u2v + uv2) + (u+ v)3 +O((u + v)4) 2uv + (u+ v)3 1 +O(u+ v) : (1:15) Trace and determinant being positive, the matrix ~ J(u) is positive de nite, i.e. r J(u) rT = r ~ J(u) rT > 0 (1:16) 7 for every row-vector r 2 IR2n(0; 0) and u 2 U"n(0; 0). Now, in order to show injectivity of x, consider two points u1; u2 2 U" with x(u1) = x(u2). De ne r := u2 u1 and s(t) := r x(u1 + tr) ; t 2 [0; 1] ; (1:17) then s(0) = s(1). Consequently, by the mean value theorem, there is a 2 (0; 1) with s0( ) = r J(u1 + r) rT = 0 : (1:18) Either u1 = u2 = (0; 0) or the argument of the Jacobian is an element of the convex set U"n(0; 0) and r = (0; 0) by (1.16). Hence, x is injective for " su ciently small and the patch x can be parametrized near the origin according to (1.11) with h(x; y) := z(x 1(x; y)) ; (x; y) 2 x(U") =: V" : (1:19) x 1 is continuous on V" and, moreover, it is also continuously di erentiable on V"n(0; 0). This follows from the inverse function theorem and the inequality det J(u) det ~ J(u) > 0 ; u 2 U"n(0; 0) ; (1:20) where the rst estimate is valid for arbitrary 2 2-matrices satisfying (1.12). For the gradient of h we obtain using the chain rule and the inequalities (1.14) lim (x;y)!(0;0)krh(x; y)k = lim u!(0;0) rz(u)J(u) 1 = = lim u!(0;0)O(u2v + uv2 + (u+ v)4) det J(u) lim u!(0;0) O(u2v + uv2 + (u+ v)4) 2uv + (u+ v)3 = 0 : (1:21) As an immediate consequence of the mean value theorem, this implies that h is continuously di erentiable on the entire domain V" and rh(0; 0) = (0; 0). So, (1.11) is a regular smooth parametrization of x near the origin and the tangent plane at the origin is spanned by A20 and A02 as stated. Although not always stated explicitly, D-patches are assumed to be generic throughout this paper. The exceptional case of non-generic D-patches is of minor importance and not considered here. The conditions imposed on D-patches are necessary for regularity in the following sense: Assume that (A20;A02); (A20;A21) and (A02;A12) are linearly independent, respectively, and de ne the normal vector n(u) of x at a regularly parametrized point x(u) by n(u) := xu(u) xv(u) kxu(u) xv(u)k : (1:22) 8 Then we obtain for di erent paths of parameters approaching the origin lim t#0 n(t; t) = A20 A02 kA20 A02k lim t#0 n(t; 0) = A20 A21 kA20 A21k lim t#0 n(0; t) = A12 A02 kA12 A02k : (1:23) Regularity implies equality of all three expressions, and so, obviously, A20;A02;A21 and A12 must be coplanar. Using the representation (1.9) we obtain lim t#0 n(t; 0) = sign ( ) A20 A02 kA20 A02k ; lim t#0 n(0; t) = sign ( ) A20 A02 kA20 A02k (1:24) and consequently, and must be positive. Considering the main curvatures 1; 2 of D-patches it turns out that, in general, they diverge near the singular point. A suitable measure for the rate of divergence, which is independent of the particular parametrization, is the set of exponents p 2 IR+ for which j 1;2jp is integrable over x near the singular point. De nition 1.3 The semi-norm k kp;"; p 1; " > 0, for functions f over x is given by kfkp;" := Zx" jf jp dS 1=p ; (1:25) where x" denotes the restriction of x to the domain U". Theorem 1.4 k ikp;"; i 2 f1; 2g is nite for 1 p < 4 and " su ciently small. Proof A detailed proof of this theorem is rather technical without revealing deeper insight into the problem. So we con ne ourselves to an outline of the crucial estimates. Consider the patch x in canonical form (1.10) and introduce polar coordinates according to (u; v) = r(cos(t); sin(t)); (r; t) 2 [0; ") [0; =2]. Then we obtain for the rst fundamental form g := xu xu xu xv xu xv xv xv = O(r2) (1:26) and, using the inequalities (1.14) again, det g = kxu xvk2 = O(r4) (det g) 1 = O(1=r4) : (1:27) 9 The normal vector n converges to the third basis vector e3 according to n e3 = O(r). This implies for the second fundamental form h := n xuu n xuv n xuv n xvv = O(r) : (1:28) Now, we nd for the mean curvature H and the Gaussian curvature K 2H = 1 + 2 = trace (h g 1) = O(1=r) K = 1 2 = det h=det g = O(1=r2) (1:29) and consequently, the main curvatures are of order i = O(1=r); i 2 f1; 2g. Finally, we obtain k ikpp;" = Z =2 0 Z " 0 j ijp kxu xvk r drdt = Z " 0 O(r3 p) dr (1:30) and niteness of the integral for 1 p < 4. As can be shown by examples, the given range of exponents is sharp in the sense that the integral is not necessarily nite for p = 4. 2. Degenerated B ezier Patches The results derived in the preceding section suggest that D-patches are suitable for generating smooth surfaces using various types of basis functions, including polynomial, rational, trigonometric, or exponential splines. However, the further development of the theory will be restricted to a case of particular interest, namely modeling surfaces by polynomial patches. As a common practice in computer aided design we replace the monomial representation (1.7) of a polynomial patch by the more convenient Bernstein{B ezier form (2.2). To this end denote the Bernstein polynomials of degree d 2 by bpd(u) := dp up(1 u)d p ; p = 0; : : : ; d bd(u) := [b0d(u); : : : ; bdd(u)] : (2:1) The B ezier patch xB corresponding to a (d+1) (d+1)-matrixB of B ezier points Bpq 2 IR3 is de ned by xB(u; v) = bd(u)B bd(v)T ; (u; v) 2 [0; 1]2 : (2:2) The following Lemma provides necessary and su cient criteria for B ezier patches matching De nition 1.1. Here and subsequently it is assumed that the singular point is located at (u; v) = (0; 0). Due to the inherent symmetries of the Bernstein{B ezier representation this is no loss of generality. 10 Lemma 2.1 A B ezier patch xB is degenerated if and only if the B ezier points B00 = B10 = B01 = B11 coalesce. xB is a D-patch if and only if, in addition, there exist constants ; 2 IR and ; 2 IR+ such that B21 B00 B12 B00 = B20 B00 B02 B00 : (2:3) Proof Expanding (2.2) yields xB(u; v) = B00 + d (B10 B00)u+ (B01 B00) v + + d2 (B11 +B00 B10 B01)uv + h:o:t: (2:4) and comparison with (1.8) shows that xB is degenerated if and only if B00 = B10 = B01 = B11. Using this identity, we obtain xB(u; v) = B00 + d(d 1) (B20 B00)u2 + (B02 B00) v2 =2 + + d2(d 1) B21 B20)u2v + (B12 B02)uv2 =2 + h:o:t: (2:5) and xB is a D-patch if and only if there exist constants ; 2 IR and ; 2 IR+ with d2(d 1) B21 B20 B12 B02 = d(d 1) B20 B00 B02 B00 : (2:6) The latter equation is equivalent to B21 B00 B12 B00 = B21 B20 B12 B02 + B20 B02 B02 B00 = =d+ 1 =d =d =d+ 1 B20 B02 B02 B00 (2:7) and renaming the matrix entries by ; ; ; , respectively, completes the proof. Corollary 2.2 The tangent plane of a D-patch in Bernstein-B ezier form (2.2) at the singular point is spanned by B00; B20 and B02. Degenerated bilinear patches were excluded a priori since, evidently, they are shrunk to single points rather than being proper surface segments. But there is a further unwelcome phenomenon raising the bi-degree d which is necessary for generating reasonable D-patches. If d = 2 the boundary curves xB(t; 0) and xB(0; t) emanating from the singular point degenerate to straight lines thus restricting the shape of patches inadmissibly. In general, choosing d 3 is su cient to avoid this e ect, except for one case. It is conceivable that a patch has not only one but several singular points. If d = 3 and if besides xB(0; 0) also one of the adjacent corners xB(1; 0) or xB(0; 1) is singular then the boundary curve connecting the singular points is a straight line, again. 11 3. A Spline Space Incorporating D-Patches The spline space S := fx : 7! IR3g is a class of functions over some domain = f!i IR2; i 2 I INg formed by a set of compact subdomains !i provided with a connectivity relation C. The topological structure of the domain can be visualized conveniently by a two-dimensional mesh as indicated in Figure 3.1. The elements of S are called spline surfaces over . The restrictions xi of x 2 S to !i are called patches or segments and the connectivity relation describes how to link them. Points on the graph of a spline function which are uniquely assigned to one segment are called interior points, points which are common to exactly two segments form edges, and points shared by n > 2 segments are called vertices of order n. Here, the segments are assumed to be quadrilateral B ezier patches and so the domain is uniform in the sense that it consists of copies of only one set, namely the unit square ! := [0; 1]2, hence = ! I. According to Section 1, vertices of order 4 are called regular. Otherwise, they are called extraordinary. If all vertices are regular S is called regular, but as a consequence of Eulers theorem for polyhedra regular spline spaces only admit the modeling of surfaces which are homeomorphic to a torus or a plane or parts of it. This well known observation justi es the necessity of irregular spline spaces including extraordinary vertices. An irregular spline space is called semi-regular if the extraordinary vertices are separated, i.e. if there is no edge connecting any two of them. Figure 3.1 shows a part of a semi-regular domain with two extraordinary vertices of order three and ve, which are separated by a regular vertex of order four. In order to obtain smooth surfaces a spline space must be provided with a set of smoothness conditions. One part of these smoothness conditions requires the segments to be parametrized by smooth functions. The other and less trivial part establishes rules how to join adjacent patches. The spline space S to be considered here is de ned over some semiregular domain and the patches are bicubic polynomials joining parametrical smooth according to (1.2). In view of the observations made in the preceding section, this choice is the simplest one avoiding unwelcome geometrical restrictions and was merely made for the sake of clarity. Generalizations to irregular domains and patches of arbitrary degree are straightforward. By the way, any irregular domain can be made semi-regular by splitting each subdomain into four quadratic pieces. So, semi-regularity is no essential restriction. An important property of S is that it can be represented by a set of control points. They are arranged according to the structure of patches and give an intuitive geometrical description of the corresponding spline surface. Thus, one of the major bene ts of modeling with B-splines is preserved. Dealing with bicubic patches joining with smoothness of rst order it seems reasonable to choose control points compatibly to bicubic tensor product B-splines with equally spaced double knots. Firstly, this means that four control points are assigned to the interior of each patch. Secondly, passing from control points to B ezier form is characterized by the fact that the control points themselves are B ezier points from which the remaining ones assigned to the edges are simply computed by averaging of direct neighbors. This arrangement of control points turns out to be equally suitable for repre12 Figure 3.1: Domain | structure of patches and control points senting S despite of its higher topological complexity. To state this fact more precisely we start with considering a regular patch xi, i.e. xi incorporates no extraordinary vertices. The corresponding 4 4-matrix Ci of control points is assembled by the control points assigned to xi and its eight neighbors as indicated by the following gure: ................................................................................................ ................................................................................................ .............................................................................................. .............................................................................................. xi =) Ci =0B@ 1CA Figure 3.2: Matrix Ci | regular case In order to represent xi in Bernstein-B ezier form (2.2) the 4 4-matrix Bi of B ezier points has to be generated from Ci. As mentioned above, this is done by keeping the four inner entries and averaging the remaining ones, Bi := ACiAT ; A :=0B@ 1=2 1=2 0 0 0 1 0 0 0 0 1 0 0 0 1=2 1=21CA : (3:1) This is exactly the procedure how to convert B-spline control points corresponding to degree d = 3 and equally spaced double knots into Bernstein{B ezier form. Near an extraordinary vertex of order n the situation is only slightly di erent. Again, four control points fCj11;Cj12;Cj21;Cj22g are assigned to each of the n patches xj ; j = 1; : : : ; n, and as in the regular case they are assumed to coincide with the inner B ezier points of the corresponding patch. According to the results of the rst section, the patches xj must be bicubic D-patches implying that the control points cannot be chosen completely arbitrarily. Unlike ordinary control points they have to satisfy certain constraints stemming 13 from Lemma 2.1. In order to emphasize this di erence the points Cj11;Cj21;Cj12 will also be referred to as quasi control points in contrast to proper control points which are free of restrictions. First of all, Lemma 2.1 enforces that the n innermost quasi control points coalesce, i.e. C111 = : : : = Cn11 =:M : (3:2) Exploiting this fact the smooth joint of two adjacent patches xj ;xj+1 according to (1.2) can be achieved using quite the same averaging process as in the regular case. De ning ............................................................................................................................. ................ ............................................................................................................................. ................. ........................................................................................................................................... .......................................................................................................................................... Cj21 Cj22 Cj12 Cj 1 12  Cj+1 21 M =) Cj = 0BB@ M M Cj+1 21 M M Cj12 Cj 1 12 Cj21 Cj22 1CCA Figure 3.3: Matrix Cj | extraordinary case the 4 4-matrices Cj as indicated by Figure 3.3 the matrices Bj of B ezier points are again given by Bj := ACjAT = 0BB@ M M (Cj12 +Cj+1 21 )=2 M M Cj12 (Cj21 +Cj 1 12 )=2 Cj21 Cj22 1CCA : (3:3) Note that the de nition of Cj di ers from the regular case only by the somewhat arti cial assignment of the rst entry. Comparison of (3.3) with Lemma 2.1 yields Corollary 3.1 The quasi control points Cj11;Cj12;Cj21 must satisfy the following constraints: i) The points C111 = = Cn11 =:M coalesce. ii) There exist constants j ; j 2 IR and j; j 2 IR+ such that Cj21 M Cj12 M = 1 2 j j j j Cj21 +Cj 1 12 2M Cj+1 21 +Cj12 2M : (3:4) By enforcing n control points to coalesce the rst condition actually a ects the topological structure of the control net. The essential part of the second condition is that the points 14 Cj12;Cj21 and M must be coplanar. The additional requirement concerning positivity of j ; j is indispensable, however, it will be ful lled by every reasonable choice of quasi control points. This is due to the fact that generally, in order to model surfaces, control points (and with them B ezier points) provide a "natural" geometric structure rather than being randomly distributed in space. Evidently, this structure requires the B ezier points to be located in a common plane so that not only the coe cients ; but also ; are positive. 4. The Re nement Property The major bene t of the approach presented in the previous sections is that the spline space S is re neable in the following sense: A re ned domain ~ is obtained by splitting every subdomain (!; i) of into four smaller squares and then rescaling the new subdomains to original size. Thus, the original domain = ! I is transformed to ~ = ! (I f1; 2; 3; 4g). It is provided with the new connectivity relation ~ C which is simply obtained from C by two steps. Firstly, each pair of neighbors is converted into two pairs according to the split of the corresponding subdomains. Secondly, the relations between any four new subdomains stemming from the split of the same original subdomain have to be added. The spline space S and its re nement S~ are similar in the sense that number and order of the extraordinary vertices coincide, that is, all new inserted vertices are regular. The re nement process is illustrated in Figure 4.1. xi ~ xi;1 ~ xi;2 ~ xi;4 ~ xi;3 Figure 4.1: Re ning a domain 7! ~ . 15 The generation of S~ from S is exclusively based on topological facts and actually rather trivial. However, and this is the crucial point, there exists an induced analytical transformation acting on spline functions which is characterized by the invariance of graphs. Theorem 4.1 There exists a canonical embedding R : S ,! S~ such that for all spline surfaces x 2 S and ~ x := R(x) 2 S~ the graphs x( ) and ~ x(~ ) coincide when regarded as point sets in IR3. Proof The proof is constructive. Consider a segment xi : (u; v) 7! b3(u)Bi b3(v)T (4:1) of the spline surface x 2 S . Then the four corresponding segments ~ xi;1 : (u; v) 7! b3(u) ~ Bi;1 b3(v)T := xi(u=2; v=2) ~ xi;2 : (u; v) 7! b3(u) ~ Bi;2 b3(v)T := xi((u+ 1)=2; v=2) ~ xi;3 : (u; v) 7! b3(u) ~ Bi;3 b3(v)T := xi(u=2; (v + 1)=2) ~ xi;4 : (u; v) 7! b3(u) ~ Bi;4 b3(v)T := xi((u+ 1)=2; (v + 1)=2) (4:2) of ~ x := R(x) are generated by the well known procedure of subdividing B ezier patches. By symmetry, it is su cient to specify only one formula for computing ~ Bi;k, e.g. ~ Bi;1 := SBiST ; S := 0B@ 1 0 0 0 1=2 1=2 0 0 1=4 1=2 1=4 0 1=8 3=8 3=8 1=81CA : (4:3) By construction, [k~ xi;k(!) = xi(!) and thus applying subdivision to all patches implies ~ x(~ ) = x( ) as required. Further, it is evident that the new patches ~ xi;k join parametrically smooth. So it remains to show that subdivision applied to D-patches yields D-patches, again. To this end consider n D-patches xj sharing the extraordinary pointM. Specifying only the relevant entries of the subdivided matrix of B ezier points we nd ~ Bj;1 = 1 8 0BB@ 8M 8M 6M+ 2Bj02 8M 8M 6M +Bj02 +Bj12 6M+ 2Bj20 6M+Bj20 +Bj21 1CCA : (4:4) The four entries in the upper left corner coincide and a short computation shows that Bj21 M Bj12 M = j j j j Bj20 M Bj02 M (4:5) implies ~ Bj21 M ~ Bj12 M = 1 2 j + 1 j j j + 1 ~ Bj20 M ~ Bj02 M : (4:6) 16 Consequently, the coe cients in question are positive and ~ xj;1 is a D-patch. As discussed in the preceding section switching between the representation by B ezier points and control points is particularly simple. In one direction, it is done by averaging and in the other direction by deleting the B ezier points assigned to the edges. Thus, the re nement process described in terms of B ezier points has a counterpart in the space of control points. Starting with the original control points the corresponding B ezier points have to be computed rst by (3.1). Then subdivision is carried out in the space of B ezier points according to (4.3), and nally the new control points are obtained by inverting (3.1). Since (3.1) is equally valid for regular and extraordinary patches we obtain a single subdivision formula combining these three steps, ~ Ci;1 = TCiTT ; T := (A 1SA) = 1 8 0B@ 6 2 0 0 2 6 0 0 1 5 2 0 0 2 5 11CA : (4:7) Due to symmetry the corresponding expressions for k 2 f2; 3; 4g are not required. Actually, (4.7) contains a lot of redundancy since applying it to all patches will cause multiple evaluation of many new control points. Evidently, it is su cient to restrict the result of (4.7) to the inner 2 2-submatrix ~ Ci23 of ~ Ci which contains only those new control points assigned to the patch ~ xi;1. Moreover, it turns out that ~ Ci23 depends only on the upper left 3 3-submatrix ~ Ci13 of Ci and we obtain the reduced formula ~ Ci;1 23 = TrCi13TT r ; Tr := 1 8 2 6 0 1 5 2 : (4:8) ............................................................................................................................. ................................ ............................................................................................................................. ................................ ........................................................................................................................................................... ........................................................................................................................................................... ............................................................................................................................. ................................ ............................................................................................................................. ................................ .......................................................................................................................................................... .......................................................................................................................................................... .......................................................................................................................... ............ ................................................................................................................................... Ci13 7 ! ~ Ci;1 23 Figure 4.2: Application of the reduced subdivision formula (4.8) 17 Finally, some remarks: When the subdivision algorithm is iterated it will generate a sequence Cm;m 2 IN of control nets converging to the originally de ned spline surface x. The rate of convergence is the same as for repeated subdivision of bicubic B ezier patches, namely O(4 m) [3,9]. The subdivision algorithm presented here is the rst one working on meshes of arbitrary topology which converges to an explicitly known limit. It is a noteworthy fact that the subdivision formula (4.8) is equally valid for regular and degenerated patches. This uniformity should be advantageous for implementations. 5. A Linear Subspace The spline space S as de ned in the third section reveals two drawbacks stemming from the conditions imposed on quasi control points. The rst is that these conditions are not linear implying that S is a nonlinear space. This could be a serious problem for further conceivable developments such as approximation of surfaces, constructing wavelets or applications in the theory of nite elements. The second concerns the fact that the sheer presence of conditions is a fundamental obstacle to applications in computer aided design systems. Methods for overcoming these di culties will be presented in this and the next section. The problem of non-linearity can be solved by identifying a linear subspace S S . To this end the conditions listed in Corollary 3.1 must be modi ed. The rst condition is linear and can be kept. The second condition becomes linear when the constants j ; j ; j ; j are not merely assumed to exist but speci ed explicitly and xed. Thus, these constants play the role of shape parameters, for instance just like relative knot spacings in B-spline spaces. When choosing the constants j ; j; j ; j one must take into consideration that they have to satisfy a certain consistency condition related to the periodic structure of (3.4). Lemma 5.1 Assume that a set of quasi control points satis es the constraints speci ed in Corollary 3.1. Then n Y j=1 j = E ; j := 1 j ~ j 1 ~ j ~ j j j ~ j ; (5:1) with ~ j := 1 j ; ~ j := 1 j , E denoting the identity, and the product sign addressing multiplication from the left. Proof Introducing the quantities pj := Cj21 +Cj 1 12 2M Cj21 Cj 1 12 (5:2) 18 a short computation shows that (3.4) implies pj+1 = jpj . Thus, by periodicity, p1 = pn+1 =Qnj=1 j p1. Since p1 = 1 0 ~ 1 1 C121 +C012 2M C221 +C112 2M (5:3) the two vectors forming p1 are linearly independent if the patch x1 is generic and consequently the product of the matrices j has to be the identity. De nition 5.2 A set of real constants := f j ; j ; j ; j ; j = 1; : : : ; ng is called feasible if j ; j > 0 and if Qnj=1 j = E. If S is a spline space incorporating M extraordinary vertices of order n ; = 1; : : : ;M the subspace S is characterized by M feasible sets of constants := f 1; : : : ; Mg via the following constraints on quasi control points: i) The points C111 = = Cn11 =:M coalesce. ii) The points Cj21;Cj12 form a solution of the periodic system Cj21 M Cj12 M = 1 2 j j j j Cj21 +Cj 1 12 2M Cj12 +Cj+1 21 2M : (5:4) Theorem 5.3 S S is a linear space of dimension dimS = 3 4#I 3 M X =1(n 1)! (5:5) where #I denotes the total number of patches. Proof Evidently, S is linear. The expression in brackets is the number of scalar degrees of freedom which must be multiplied by three since the coe cients lie in IR3. 4#I is the total number of control points and thus it has to be shown that 3(n 1) is the number of linearly independent conditions assigned to an extraordinary vertex of order n . Clearly, the rst constraint yields n 1 independent conditions. As shown in the proof of Lemma 5.1 the second constraint is equivalent to pj+1 = jpj providing 2n equations for the 2n unknowns pj. However, exactly two of them can be chosen arbitrarily. If, for instance, p1 is given then the other variables are uniquely determined by the iteration pj+1 = jpj; j = 1; : : : ; n 1. The remaining equation p1 = npn is ful lled automatically since npn = n n 1 Y j=1 j p1 = n Y j=1 j p1 = p1 : (5:6) Thus, the rank of the second constraint is 2n 2 and the total e ective number of conditions is 3n 3 as required. 19 From a topological point of view the structure of patches forming a spline surface reveals certain local symmetries near an extraordinary vertex of order n. Roughly speaking there is an n-fold rotational symmetry and an additional invariance under re ection. A case of particular interest is that the spline space itself re ects these symmetries. In other words, there is no reason why to treat one of the n D-patches sharing an extraordinary point di erently from the others or why to give preference to a particular direction of rotation around the center. So the set of quasi control points should be invariant under the shift S : Cj11;Cj21;Cj12 7! Cj+1 11 ;Cj+1 21 ;Cj+1 12 (5:7) and the re ection R : Cj11;Cj21;Cj12 7! C j 11 ;C j 12 ;C j 21 : (5:8) This feature can be achieved readily by a special choice of constants. De nition 5.4 A set of constants = f j ; j ; j ; j ; j = 1; : : : ; ng is called symmetric if 1 = 1 = = n = n =: and 1 = 1 = = n = n =: . The spline space S is called symmetric if = f 1; : : : ; Mg consists of feasible symmetric sets. Theorem 5.5 A symmetric set of constants is feasible if > 0 and = 1 cos'n ; 'n := 2 =n : (5:9) Proof Choosing = 1 cos'n implies that the eigenvalues of := 1 = n are exp( i'n). Hence, Qnj=1 is the identity. From a purely analytical point of view the conditions speci ed in Theorem 5.5 are not necessary. In general, choosing = 1 cos(2 k=n); k 2 f1; : : : ; n 1gnfn=2g yields eigenvalues exp( 2 ki=n) and hence a feasible set. The cases k = 0 and, for n even, k = n=2 are leading to double eigenvalues and can be ruled out by inspection. However, one can show using discrete Fourier analysis that k = 2 f1; n 1g is leading to inadequately arranged quasi control points corresponding to surfaces with local self-intersections near the extraordinary point. In the sense of di erential geometry such surfaces are not smooth and thus (5.9) turns out to be both necessary and su cient. Finally, let us discuss the re nement of S induced by R : S 7! S~ . It turns out that the re ned space S ~ ~ := R(S ) is linear, again. However, the process is not uniform in the way that the sets of constants and ~ are di erent. Theorem 5.6 Consider the space S with := f 1; : : : ; Mg and := f j ; j ; j ; j ; j = 1; : : : ; ng; = 1; : : : ;M . Re nement by R as de ned in the proof of Theorem 4.1 yields the linear space S ~ ~ := R(S ) S~ where ~ j ~ j ~ j ~ j := 1 2 j + 1 j j j + 1 : 20 The symmetric case results in ~ := =2 and ~ := ( + 1)=2 = 1 ~ cos'n. Proof The proof follows immediately from (4.5) and (4.6). 6. Control Points and B-Splines The features of the quasi control point construction are ambivalent. On one hand their topological structure is a natural generalization of the familiar tensor product setup. On the other hand the constraints imposed on them make them somewhat awkward to deal with. In particular, they will not be accepted by designers willing to model intuitively rather than to solve equations. One way out of this dilemma would be to identify the free parameters explicitly and to use them exclusively as control points. For the linear space S this is quite simple. For instance, C111;C121;C112 can be chosen arbitrarily and then all other quasi control points will be determined uniquely as outlined in the preceding section. Thus, 3n quasi control points could be replaced by three proper control points assigned to one of the n D-patches sharing an extraordinary vertex. However, the fundamental drawback of this and all similar procedures is that the topology of the resulting control mesh does not match the patch structure uniformly. By assigning four control points to one D-patch and only one control point to each of the remaining ones the natural equivalence of patches would be violated inadmissibly. So, we propose a di erent approach, which is both satisfactory from a theoretical point of view and useful for applications. The idea is to start with arbitrarily located proper control points cj11; cj21; cj12 which have exactly the same topological structure as the quasi control points but do not necessarily comply with the constraints. The next step is to project these points to the space of quasi control points and then one can proceed as described above. For instance, a convenient projection can be de ned by selecting those quasi control points which minimize the distance to the given points with respect to some norm. This projection can be computed numerically or even analytically, if the Euclidian norm is used. However, we shall not elaborate on this problem in full generality but con ne ourselves to a more detailed discussion of the linear symmetric case, which is of particular importance for applications. The given proper control points cj11; cj21; cj12 and the quasi control points Cj11;Cj21;Cj12 which have to be determined are collected in vectors, e.g. c11 := [c011; : : : ; cn 1 11 ]T and analogously for all others. What we are looking for is an a ne invariant projection P :0@ c11 c21 c121A 7!0@C11 C21 C121A = 0@P1 P2 P3 P4 P5 P6 P7 P8 P91A 0@ c11 c21 c121A (6:1) mapping proper control points to the space of quasi control points. For symmetry reasons it is required that P commutes with both the shift S and the re ection R. SP = PS implies that all n n-matrices Pm;m = 1; : : : ; 9 are cyclic, that is there are vectors pm := [p0m; : : : ; pn 1 m ]T such that P jk m = pj k m ; j; k = 0; : : : ; n 1. Further, RP = PR yields pj1 = p j 1 ; pj2 = p j 3 ; pj4 = p j 7 ; pj5 = p j 9 ; pj6 = p j 8 : (6:2) 21 The discrete Fourier transformation p 7! p̂ and its inverse are given by p̂k := n 1 Xj=0 ! jk n pj ; pj = 1 n n 1 Xk=0 !jk n p̂k ; !n := exp(i'n) : (6:3) Applying it to the space control points splits (6.1) into n decoupled 3 3-systems P̂ k : 0@ ĉk11 ĉk21 ĉk121A 7! 0@ Ĉk11 Ĉk21 Ĉk121A =0@ p̂k1 p̂k2 p̂k3 p̂k4 p̂k5 p̂k6 p̂k7 p̂k8 p̂k91A 0@ ĉk11 ĉk21 ĉk121A (6:4) and the symmetry conditions (6.2) imply p̂k1 = p̂k1 ; p̂k2 = p̂k3 ; p̂k4 = p̂k7 ; p̂k5 = p̂k9 ; p̂k6 = p̂k8 : (6:5) The constraints on quasi control points as speci ed in De nition 5.2 are transformed to 0@ 1 !k n 0 0 1 ( + !k n)=2 1 ( ! 1 n + )=2 1 ( !k n + )=2 ( + ! 1 n )=2 11A0@ ĉk11 ĉk21 ĉk121A = 0 : (6:6) Denoting the matrix by Q̂k the projection P has to satisfy Q̂kP̂ k = 0. With = 1 cos'n we nd det Q̂k = (1 !k n)(cos(k'n) cos('n)) (6:7) and the kernel of Q̂k is non-trivial if and only if k 2 f0; 1; n 1g, thus P̂ 2 = = P̂n 2 = 0 : (6:8) For k = 0 we obtain q̂0 := ker Q̂0 = [1; 1; 1]T and P̂ 0 := q̂0 [a00; a01; a02]. The symmetry conditions (6.5), the a ne invariance of P and the fact that P̂ 0 has real entries imply a01 = a02 = (1 a00)=2 ; a00 2 IR : (6:9) For k = 1 we obtain q̂1 := ker Q̂1 = [0; exp(i ); exp( i )]T ; := arg (1+i sin'n)! 1=2 n and P̂ 1 := q̂1 [a10; a11; a12]. The symmetry conditions (6.5) imply that a10 2 IR and a11 = a12 =: r exp(i ) ; (r; ) 2 IR+0 [0; 2 ) : (6:10) For k = n 1 we obtain q̂n 1 := ker Q̂n 1 = q̂1 and setting P̂n 1 = P̂ 1 ensures that P is a real matrix. So the projection P is characterized by four real-valued parameters a00; a10; r; 22 and the vectors p1; : : : ; pm given by pj1 = a00=n pj2 = (1 a00)=2n pj3 = (1 a00)=2n pj4 = (a00 + 2a10 cos(j'n))=n pj5 = ((1 a00) + 4r cos( + + j'n))=2n pj6 = ((1 a00) + 4r cos( + j'n))=2n pj7 = (a00 + 2a10 cos(j'n))=n pj8 = ((1 a00) + 4r cos( j'n)=2n pj9 = ((1 a00) + 4r cos( + j'n))=2n : (6:11) How to choose the parameters? A natural criterion is to minimize the distance n 1 Xj=0 kĈj11 ĉj11k2 + kĈj21 ĉj21k2 + kĈj12 ĉj12k2 ! min : (6:12) Some elementary calculus yields a00 = 1=3; a10 = 0; r = 1=2; = and 0@ pj1 pj2 pj3 pj4 pj5 pj6 pj7 pj8 pj91A = 1 3n 0@ 1 1 1 1 1 + 3 cos(j'n) 1 + 3 cos(2 + j'n) 1 1 + 3 cos(2 j'n) 1 + 3 cos(j'n) 1A : (6:13) Some of the coe cients are negative and consequently the quasi control points and with them the spline surface will not necessarily lie in the convex hull of the proper control points. The projection complies with the convex hull property if the parameters satisfy 0 a00 1; ja10j a00=2; r (1 a00)=4. If this is desired a reasonable choice is a00 = a10 = 0; r = 1=4; = yielding 0@pj1 pj2 pj3 pj4 pj5 pj6 pj7 pj8 pj91A = 1 2n 0@ 0 1 1 0 1 + cos(j'n) 1 + cos(2 + j'n) 0 1 + cos(2 j'n) 1 + cos(j'n) 1A : (6:14) As mentioned above, providing unrestricted control points is of particular importance for design purposes. As usual, the designer speci es proper control points and the result is a smooth spline surface. All intermediate steps, and in particular the projection to the space of quasi control points, can be hidden in a black box. Actually, the same process turns out to be equally useful from a more theoretical point of view. Recall the generation of a point x(p); p 2 = ! I on the spline surface x corresponding to a set of control points c = fci; i = 1; : : : ; 4#Ig. Firstly, quasi control points C are computed by applying the projection P to the proper control points c. Then the quasi control points are completed 23 to a set of B ezier points by averaging as described in the third section, and nally x(p) is obtained by evaluating the appropriate B ezier patch. Combining all three steps yields a single linear and a ne invariant map acting an control points according to B : c 7! x 2 S ; x(p) = 4#I Xi=1 Bi(p)ci : (6:15) The functions Bi : 7! IR are B-splines in the following sense: They are real valued, piecewise polynomial, compactly supported, form a partition of unity and coincide with ordinary B-splines on the regular parts of the domain. Non-negativity can be achieved using particular projections P , for instance (6.14). Further, they provide built-in smoothness by generating smooth surfaces when combined linearly with arbitrary spatial control points. They span the complete space S but do not form a basis since they are linearly dependent as a consequence of the rank de ciency of the projection matrix P . 7. Variational Surface Design | An Application In this section we want to demonstrate brie y how the theory applies to a problem encountered occasionally in computer aided design|the lling of an n-sided hole. Some of the details appearing here are problems in there own right but a more profound investigation is beyond the scope of this paper. The n-sided hole problem addresses the following: The surface x to be determined is de ned over some bounded domain = ! I incorporating a single extraordinary vertex of order n. Further, x has to match the prescribed boundary data (p;n) consisting of a closed spatial curve p and normal vectors n attached pointwise to it. That is x(@ ) = p ; nx(@ ) = n ; (7:1) where nx denotes the normal vector eld of x. In order to select a nicely shaped surface from the in nite set of surfaces complying with (7.1) some fairness functional F is speci ed and the solution sought after is expected to minimize it. Here we consider the thin plate energy de ned by F (x) := 4#I Xi=1 Z!(kxiuuk2 + 2kxiuvk2 + kxivvk2) dudv : (7:2) The solution of the variational problem F (x)! min ; x(@ ) = p ; nx(@ ) = n (7:3) can be approximated by an element x := 4#I Xp=1 Bpcp (7:4) 24 of the spline space S . In order to obtain a quadratic optimization problem with linear side conditions the boundary data (p;n) are approximated rst. Denote the index set of all control points in uencing boundary curve and normal by @I then all these control points can be determined using familiar univariate approximation techniques yielding c0p; p 2 @I. The energy becomes a positive semi-de nite quadratic form of the control points F (x) = 4#I X p;q=1(Bp; Bq) cp cq ; (7:5) where the inner product ( ; ) corresponding to the thin plate energy (7.2) is de ned by (Bp; Bq) := 4#I Xi=1 Z!(Bi p;uuBi q;uu + 2Bi p;uvBi q;uv +Bi p;vvBi q;vv) dudv : (7:6) Now, x is obtained as solution of the optimization problem F (x) = 4#I X p;q=1(Bp; Bq) cp cq ! min ; cp = c0p for p 2 @I (7:7) which can be determined quite easily using standard tools from linear algebra and sparse matrix techniques. Due to the linear dependence of the B-spline functions Bp the solution of (7.7) is not unique. However, all solutions de ne the same spline surface and so, any of them can be selected. The following gures show an example which was generated using boundary data kindly provided by the Mercedes Benz AG. In Figure 7.1 the topological structure of the domain and the control points is visualized. According to the arc length of the ve pieces of the boundary curve the sides are partitioned into di erent numbers of segments. The control points cp; p 2 @I in uencing the boundary are marked by whereas the remaining ones which have to be computed as solution of the optimization problem (7.7) are marked by . Figure 7.2 shows the given boundary data (p;n) and the control points cp; p 2 @I approximating it. In Figure 7.3 we see the complete set of control points solving (7.7) and the corresponding patch boundaries, where the B-splines Bp were computed using the projection (6.13). Finally, Figure 7.4 shows the resulting shaded spline surface. Conclusion Degenerated surface patches are suitable for modeling smooth spline surfaces. The major bene t of this approach is the capability of modeling a re neable spline space of arbitrary topological genus provided with a uniform set of parametrical smoothness conditions. The existence of a non-trivial linear subspace spanned by a family of real-valued compactly supported B-spline functions favors various applications in both computer aided geometric design and approximation theory. Disadvantages of the methods are that it requires polynomials of bi-degree three for modeling C1-surfaces and that the main curvatures diverge near singularly parametrized surface points.25 Figure 7.1: Domain and structure of control points Figure 7.2: Boundary data and approximation by control points 26 Figure 7.3: Optimal control points and corresponding patch boundaries27 References1. Barsky, B.A., DeRose, T.D., Geometric continuity, shape parameters, and geometric con-structions for Catmull-Rom splines, ACM Trans. on Graphics 7 (1988), 1 412. Bohm, W., Visual continuity, Computer Aided Design 20 (1988), 307 3113. Cohen, E., Schumaker, L.L., Rates of convergence of control polygons, CAGD 2 (1985),229 2354. 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