Chandrajit L. Bajaj Jindon Chen Guoliang Xu. Purdue University. West Lafayette, Indiana. January 22, 1996.

Transcription

1 Modeling with C Quintic A-patches Chandrajit L. Bajaj Jindon Chen Guoliang Xu Department of Computer Sciences Purdue University West Lafayette, Indiana fbajaj,jdc,xuguog@cs.purdue.edu January, 996 Abstract We present an ecient algorithm to construct C smooth meshes of quintic A-patches that interpolate or approximate the vertices of a given triangulated polyhedron P. An A-patch is a smooth and functional zero-contour patch of a trivariate polynomial in Bernstein-Bezier (BB) form dened within a tetrahedron. A simplicial hull is constructed based on P. A quintic A-patch is then constructed within each tetrahedron of so that the piecewise C algebraic surface interpolates or approximates the vertices of P with given C data. We guarantee that the algebraic surface is smooth and functional, namely, fully connected and free of singularity, unwanted branches. Introduction In this paper, we present an ecient algorithm to construct a C smooth mesh with quintic A- patches to interpolate or approximate the vertices of a given polyhedron P with given C data. The A-patch is a smooth and functional zero-contour patch of a trivariate polynomial in Bernstein- Bezier (BB) form dened within a tetrahedron[bcx95a], where \A" stands for algebraic. Solutions have been given to the problem of constructing a C mesh of algebraic patches which interpolate the vertices of a simplicial polyhedron P, by [Dah89] using quadric patches, [Guo9b, DTS93, Guo93, BCX95a] using cubic patches and [BI9b] using quintic for convex P(all faces are triangular) and degree seven patches for arbitrary P. While papers [Dah89, Guo9b, BI9b, DTS93, Guo93] provide heuristics based on monotonicity and least square approximation to circumvent the multiple sheeted and singularity problems of implicit patches, [BCX95a] introduces new suciency conditions for the BB form of trivariate polynomials within a tetrahedron, such that the zero contour of the polynomial is guaranteed functional and non-singular surface within the tetrahedron (the A-patch). The conditions are no more complicated than linear inequalities. [BX9] can be regarded as the two This work was supported in part by NSF grants CCR 9-67, DMS 9-0, AFOSR grants F , F , NASA grant NAG--73 and a gift from AT&T.

2 dimensional version of [BCX95a], where C k continuous piecewise real algebraic curves(a-splines) are used to achieve local interpolation and approximation. In [BCX9, BCX95b], new schemes are given to construct an \inner" hull of a polyhedron P, such that the A-patch surface approximates P instead of interpolating its vertices. In this paper, we present ascheme for building a C patch complex with quintic surface patches. Like the C scheme [BCX95a], a pair of tetrahedra are built on each side of each face of a triangulated polyhedron P. We call them the face tetrahedra. A pair of tetrahedra are built to ll in the gap between each pair of face tetrahedra that share an edge and on the same side of P. A quintic BB polynomial is dened within each tetrahedron so that they are C continuous across their share boundaries. The zero contour of the piecewise C polynomial C interpolates the vertices of P. We guarantee that the zero contour surface is non-singular and topologically equivalent to P. If we use the schemes described in [BCX9, BCX95b] to construct the simplicial hull, then the nal C spline surface approximates P instead of interpolates it. Related papers which approximate scattered data using algebraic patches are [Baj9, BBX9, BI9a, BIW93, MW9, Pra87, Sed90] and a classication of data tting using parametric surface patches is given in [Pet90]. The rest of this paper is as follows. Section gives some preliminary facts about Bernstein- Bezier (BB) representations, A-patches, the geometry of simple polyhedra and a denition of a simplicial hull. Section 3 presents details of the C continuity schemes for quintic A-patches. Section discuss a set of sucient conditions which guarantee that the zero contour surface is functional. Section 5 provides some implementation details. And nally, Section 6 concludes the paper. Notation and Preliminary Details. Bernstein-Bezier Representation and A-Patches Let fp ;:::p j gir 3. Then the convex hull of these points is dened by [p p :::p j ]=fp IR 3 : p = P j i= ip i ; i 0; P j i= i =g. Let p ; p ; p 3 ; p IR 3 be ane independent. Then the tetrahedron(or three dimensional simplex) with vertices p ;p ;p 3, and p,isv =[p p p 3 p ]. For any p = X i= i p i V, =( ; ; 3 ; ) T is the barycentric coordinate of p. Let p =(x; y; z) T, p i =(x i ;y i ;z i ) T. Then the barycentric coordinates relate to the Cartesian coordinates via the following relation x x y 6 x x 3 x 7 z 5 = y 6 y y 3 y 7 z z z 3 z (:) 3 5 Any polynomial f(p) of degree m can be expressed in Bernstein-Bezier(BB) form over V as f(p) = P jj=m b B m(); Z + where B m() = m!!!3!! 3 3 are the trivariate Bernstein polynomials for jj = P i= i with =( ; ; 3 ; ) T. Also =( ; ; 3 ; ) T is the barycentric coordinate of p, b = b 3 (as a subscript, we simply write as 3 ) are called control points, and Z+ stands for the set of all four dimensional vectors with nonnegative

3 0300 B A nonnegative control point free control point zero control point nonpositive control point 00 D p nonnegative control point free control point 00 zero control point 00 p p C nonpositive control point 0300 p (a) (b) Figure.: (a) A three sided cubic patch tangent atp ;p ;p 3 (b) A degenerate four sided cubic patch interpolates p and p 3 α α β integer components. Let Figure.: Three-sided and four-sided patches F () = X jj=m b B m (); jj =; (:) P be a given polynomial of degree m on the tetrahedron S = f( ; ; 3 ; ) T IR : i= i = ; i 0g. The surface patch within the tetrahedron is dened by S f S : F ( ; ; 3 ; )=0. Denition. Three-sided patch. Let the surface patch S F be smooth on the boundary of the tetrahedron S. If any open line segment (e j ; ) with S j = f( ; ; 3 ; ) T : P j =0; i > 0; i6=j i =g intersects S F at most once(counting multiplicities), then we call S F a three-sided j-patch (see Figure.). Denition. Four-sided patch. Let the surface patch S F be smooth on the boundary of the tetrahedron S. Let (i; j; k; `) be a permutation of (,, 3, ). If any open line segment ( ; ) with (e i e j ) and (e k e`) intersects S F at most once(counting multiplicities), then we call S F a four-sided ij-k`-patch (see Figure.). It is easy to see that if S F is a four-sided ij-k`-patch, it is then also a ji-`k-patch, a `k-ji-patch, and so on. The Appendix contains proofs of the lemmas and theorems below. 3

4 Lemma. The three-sided j-patch and the four-sided ij-k`-patch are smooth (non-singular). Proof. See [BCX95a]. 3 Theorem. Let F () = P jj=n b B n () satisfy the smooth vertex and smooth edge conditions and j( j ) be a given integer. If there exists an integer k(0 k<n) such that b 3 0; j =0; ;:::;k, ; (:3) b 3 0; j = k +;:::;n (:) P and jj=n b > 0 P if k>0, jj=n b < 0 for at least one m(k <m n), then S F j =0 j =m j-patch. is a three-sided Theorem. Let F () = P jj=n b B n () satisfy the smooth vertex and smooth edge conditions and (i; j; k; `) beapermutation of (,, 3, ). If there exists an integer k(0 k<n) such that b 3 0; i + j =0; ;:::;k, ; (:5) b 3 0; i + j = k +;:::;n (:6) P and jj=n b > 0 P if k >0, jj=n b < 0 for at least one m(k <m n), then S F i + j =0 i + j =m four-sided ij-k`-patch. Proof. See [BCX95a]. 3 is Lemma. Let f(p) = P jj=n b B n () be dened on the tetrahedron [p p p 3 p ], then b (n,)ei +e j = b nei + n (p j, p i ) T rf(p i ); j =; ; 3; ; j 6= i (:7) b (n,)ei +e j +e k =,b nei + b (n,)ei +e j + b (n,)ei +e k + n(n,) (p j, p i ) T r f(p i )(p k, p i ); j 6= i; k 6= i (:8) (.7) can be found in [Guo9a](p.3). (.8) is derived from directional derivative formulas (see [Far90] p.30). Lemma.3 ([Far90] p.38) Let f(p) = P jj=n a B n() and g(p) =P jj=n b B n () be two polynomials dened on two tetrahedra [p p p 3 p ] and [p 0 p p 3 p ],respectively. Then (i) f and g are C 0 continuous at the common face [p p 3 p ] if and only if a = b ; for any =0 3 ; jj = n (:9) (ii) f and g are C continuous at the common face [p p 3 p ] if and only if (.9) holds and b 3 = a 3 + a a a (:0)

5 (iii) f and g are C continuous at the common face [p p 3 p ] if and only if (.9)-(.0) holds and b 3 = a 3 + a a a a a a a a a (:) where =( ; ; 3 ; ) T are dened by the following relation p 0 = p + p + 3 p 3 + p ; jj = In Lemma.3, if = 3 = 0, that is p 0 ;p and p are collinear, then (.0) and (.) become a = a 3 + b 3 (:) a 3, a = b 3, b (:3) respectively, where =, ; =, that is p = p + p 0.. Simplicial Hull Denition.3 Edge convexity Let [p i p j ] be anedge of a polyhedron P with endpoint vertex normals n i and n j.if(p j,p i ) T n i (p i, p j ) T n j 0, then the edge is convex. Otherwise, it is nonconvex. If the edge satises the convex condition. and at least one of (p j, p i ) T n i and (p i, p j ) T n j is positive, then we say the edge [p i p j ] is positively convex. Ifboth of them are zero then we say it is zero convex. Ifatleast one of them is negative, the edge is negatively convex. Denition. Face convexity Let [p i p j p k ] be a triangular face ofapolyhedron P. If its three edges are non-negatively (positively or zero) convex and at least one of them is positive convex, then we say the face [p i p j p k ] is positively convex. If all the three edges are zero convex then the face is zero convex. If its three edges are non-positively (negatively or zero) convex and at least one of them is negatively convex, the face is negatively convex. Otherwise, [p i p j p k ] is non-convex. Note, that here we are overloading the term convex to characterize the relations between the vertex normals and edges of faces. We distinguish between convex and non-convex faces in the simplicial hull below where we build one tetrahedra for convex faces and double tetrahedra for non-convex faces. Denition.5 Face tetrahedra A face-tetrahedron [p i p j p k q l ] is a tetrahedron that is built based on a triangular face [p i p j p k ] P. A face-tetrahedron [p i p j p k q l ] is fpositively j zero j negative j non-)g convex if the face [p i p j p k ] is fpositively j zero j negative j non-)g convex. Denition.6 Tangent containment Aconvex face-tetrahedron [p p p 3 p ] is tangent-containing if the tangent planes at the three interpolatory vertices p, p and p 3 intersect with [p p p 3 p ];Apair of non-convex face-tetrahedra [p p p 3 p q ] is tangent-containing if the tangent planes at the three interpolatory vertices p, p and p 3 intersect with either [p p p 3 p ] or [p p p 3 q ]; 5

6 Denition.7 Simplicial hull A simplicial hull of triangulation T, denoted as=(s f ; S e ; R tf ; R fe ) is dened as () S f = f[p i p j p k q l ]g is a collection of face tetrahedra. () S e = f[p i p j q k s l ]g is a collection of edge tetrahedra. (3) R tf = T S f is a relation between T and S f, which can be described as (i) (single sided) there isonetangent-plane-containing face-tetrahedron [p i p j p k q l ] S f is built on a convex face [p i p j p k ] T and (ii) (double sided) there areatangent-plane-containing pair of nonconvex face-tetrahedra [p i p j p k q l ]; [p i p j p k q l ] S f are built on a nonconvex face [p p p 3 ] T, one on each side; () R fe = S f S f S e S e is a relation between a pair of neighboring face-tetrahedra and a pair of edge tetrahedra, which can be described as (i) (non-intersection) two face-tetrahedra [p i p j p k q l ]; [p i p j p m q n ] S f that share acommon edge [p i p j ] T does not intersect each other and (ii) a pair of edge-tetrahedra [p i p j q l s r ]; [p i p j q n s r ] S e where s r = q l +(, )q n ); 0 <<, are built between the pair of edge-sharing face-tetrahedra that are on the same side of triangulation T. 3 C Mesh of Quintic Patches The input of our algorithm is a triangulated polyhedron P, with optional C data at eachvertex. C data means that each vertex p is given a gradient vector f(p) and a Hessian matrix f(p). The gradient vector and/or Hessian matrix can be specied by a plane or a conic surface F p (x; y; z)=0 goes through p, For each vertex that has incomplete C data, a preprocessing step computes reasonable ones by tting a conic surface to the vertex and its neighboring vertices. We then construct a simplicial hull P as in [BCX95a], except that we impose a stronger tangent containment condition: For a tetrahedral simplex [p p p 3 p ] built on polyhedral face [p p p 3 ], jn p [p p ]j >; (3:) where N p is the normal at p and is a given small positive constant. We will show later that we need this stronger condition to guarantee that piecewise surface is connected within. This stronger condition can usually be enforced by raising p. Once we have established a simplicial hull P for the given polyhedron P and a set of point normals N, we construct a C trivariate piecewise polynomial function F within P such that F has the given C data at each vertex. We then further set the free polynomial coecients so that they are C continuous to each other. We adapted the C and functional cubic scheme from [BCX95a] to quintic piecewise polynomial. Please refer to it for a full description of the C scheme. Figure 3.3 illustrates between the weights of neighboring tetrahedral simplex in. V = [p p p 3 p ]; V = [p 0 p p 3 p 0 ]; W = [p 00 p p 3 p ]; W = [p 00 p p 3 p 0 ]; V 0 = [p p p 3 q ]; V 0 = [p 0 p p 3 q 0 ]; W 0 = [q 00p p 3 p ]; W 0 = [q 00p p 3 p 0 ]; and the polynomials f i over V i, g i over W i, fi 0 over V i 0 and g0 i over W i 0 be expressed in Bernstein- Bezier forms with coecients a i ;bi and ci, respectively. Nowwe shall determine these coecients 6

7 0005 p 6 b () () p" b p free weight dependent weight () a a () p p p p c () c () q q q" d () d () Figure 3.3: Adjacent Tetrahedra, Quintic Functions and Control Points for two Non-Convex Adjacent Faces 7

8 step by step. Denote p 00 p 00 p 00 q 00 = p + p + 3p 3 + p ; = = p0 + p + 3 p 3 + p0 ; = = p + p 0 ; + = = q + q 0 ; + = (3:5) Algorithm 3. C quintic scheme(see Figure 3.3). The number 0 weights are given by the function values at the vertices. For examples, a () 5e i = f(p i );i=; ; 3.. The number weights are determined by formula (.7). 3. The number weights are determined by formula (.8).. The number 3 weights, that is a (i) 0 ;a(i) 0 and a(i) 0, are free. 5. The number weights are determined by C condition (.0). 6. The number 5 and 6 weights have to be determined simultaneously. In determining these weights, we need to consider all the C and C conditions related to the tetrahedra surrounding the vertex p. Suppose there are k triangles(hence k edges) around p. Along edge [p p 3 ], there are a \clique" of 8 sets of weights with respect of the 8 tetrahedra around [p p 3 ]. W (i), 0 W (i), 30 W (i), 00 W (i), 0 W (i), 030 W (i), 030 W (i), 03 W (i), W (i), 0 W (i), 0 i = ;, W = a; b; c; d. Actually each of the 8 sets is just a dierent BB-form with respect to a dierent basis. Further studying their relations, we gure that there is a linear relation between a () and a(), regardless of the number6weights or weights c(i). However, a number 5 weight, for example a, is shared by two \cliques" around [p p 3 ] and [p p ]. Therefore, around vertex p, there is a k k linear system. One can show that this system is of rank k, and hence always solvable. a (i) are set by solving the system. The other number 5weights and number 6 weights are set by C condition. Please note that the symmetric linear system with respect to c (i) on the other side is trivially satised by the solution of. See Appendix A for further details. a (i) 7. The number 7 weights are similarly determined as that of number 6 weights. 8. The number 8 weight a (i) are free. 9. The number9weights are determined by C and C conditions. Both the number of equations and the number of unknowns are 6k. See Appendix B for details. 0. For the number 0 weights, we have six equations parallel to the equations (B.53){(B.56) with all the index changed by the rule the index of the number0weright = the index of the number 9, e + e 3 (3:6) and seven independent weights. By choosing one of them, say b (i) 30, to be a free parameter, the entire system can be solved. 8

9 . The number weights are determined in the same way as the that of number 9 weights.. The number and 3 weights are free. The numberweights are determined by C and C conditions. That is b (i) 03 are dened by (.0), and b(i) 0 are dened by (.). For b(i), 30 we haveby (.) and (.3) b () + 30 b () = 30 b() 00 Hence, b () 30 + b () 30 = b () 0, b () 0 b () 30 = b() 00, b() 0 + b() 0 b () 30 = b() 00 + b() 0, b() 0 3. The number 5 weights are similar to that of number, the index is changed by the rule (3.6).. The number 6 weights are free, the number 7's are determined by C and C conditions. 5. The remaining weights with index 3 are determined by C and C conditions (.0) and (.) for and freely chosen for > In summary, the construction steps { are according to the C and C conditions across the common tetrahedra faces that are over or below the original triangulation. Step 5 is according to the C and C conditions across the common tetrahedra faces that are on the original triangulation. Therefore, the composite polynomial function is global C continuous. In the algorithm, there are quite some free weights need to be set. whatever values they are assigned to have impacts on the shape of the surface. Instead of an optimization analysis, we proposed two kinds of cheap heuristics. The rst method is the following. We rst construct a C A-patch surface with cubic patches. After we make sure that the surface looks good, we degreeraised the whole simplicial hull to quintic patches. Now whatever weights a quintic patch has will be used as default values if needed. The second method is to always set the free weight so that the polynomial approximate a polynomial of lower degree. We observe that the rst method works better. Functional Zero Contouring Quintics This section gives a set of sucient conditions to ensure that each quintic is an A-patch. Namely, besides C continuous between adjacent tetrahedra, the zero contouring surface is singly-connected, non-singular and topologically equivalent to T. In the C cubic scheme, one is able to arrange the weights of the BB polynomial so that there is a layer of weights that separates the nonpositive weights and nonnegative. Such an arrangement turns out to be a simple sucient condition for an A-patch. However, in the C quintic scheme, after meeting the C conditions, the signs of the weights could alternate through the the layers. In particular, the weights at the st, nd and 3rd layers can be of any sign. We show in the following 9

10 Free weights () Weights of subpolynomial f3 Fixed weights () Weights of subpolynomial f3 -t t -t t Figure.: Making sure a cubic polynomial is positive by looking into its two subpolynomials text that, one can arrange the th, 5th layers to be nonnegative and large enough in absolute value, so that the zero contouring surface is an A-patch. We rst discuss some sucient conditions to guarantee that a BB polynomial is positive in its domain. Then we discuss some sucient conditions to make sure a univariate BB polynomial has exactly one root between (0; ). Then we show that the sucient functional zero contouring conditions for a quintic is an integration of the single-rooted and non-rooted conditions at every rays, for a three-sided patch, shooting from the top vertex p of the tetrahedron to the bottom face, or for a four sided patch, from edge [p ;p 00] of the tetrahedron to edge [p ;p 3 ].(for a four See Figure Enforcing positivity of a BB polynomial.. Univariate A necessary condition for a univariate polynomial F () to be positive in[0; ] is that F (0) = b m0 > 0 and F () = b 0m > 0. Actually, ifwe are given F (0; ) = b m0 >", for some small constant ">0, and some other weights excluding F (; 0) = b 0m,we can always set b 0m and other free weights to be large enough so that F () > 0; [0:]. In stead of giving a full proof, we check out several cases that are going to be used in weight setting processes. For a cubic BB polynomial F 3() =b 30 B 3 30()+b B 3 ()+b B 3 ()+b 03 B 3 03() (:7) 0

11 Free weights Fixed weights () Weights of subpolynomial f () Weights of subpolynomial f t -t t -t Figure.5: Making sure a quartic polynomial is positive by looking at its two subpolynomials where b 30 > 0, b are given, set b and b 03 so that F 3() > 0in[0; ]. From the subdivision properties of BB polynomial, if F 3() > 0, then there must exist a subdivision of the BB polynomial where the weights of each piece are all positive. Based on this observation, we subdivide the univariate cubic polynomial F 3() into two pieces, F 3 () = b () 30 B3 30 ()+b() B3 ()+b() B3 ()+b() 03 B3 03 () (.8) F 3 () = b () 30 B3 30()+b () B3 ()+b () B3 ()+b () 03 B3 03() (.9) upon =(, t; t) (See gure.). We now just need to choose t carefully so that b () =(, t)b 30 + tb > 0 (:0) The other weights are positive function of b and b 03 and hence can be set to be positive by setting b and b 03 large enough. Quantitative details are given in appendix C. For a quartic BB polynomial, F () = b 0 B 0()+b 3 B 3()+b B ()+b 3 B 3()+b 0 B 0() (.) where b 0 >", b 3 and b are given, set b 3 and b 0 so that F () > 0in[0; ]. We subdivide the univariate quartic polynomial F () into two pieces, F () () = b () 0 B 0 + b () 3 B 3 + b () B + b () 3 B 3 + b () 0 B 0 (.) F () () = b () 0 B 0 + b () 3 B 3 + b () B + b () 3 B 3 + b () 0 B 0 (.3)

12 Free weights Fixed weights () Weights of subpolynomial f Weights of subpolynomial f () t -t t -t Figure.6: Making sure a quintic polynomial is positive by looking at its two subpolynomials upon =(, t; t) (see gure.5). By choosing t carefully, one makes sure that b () 3 = (, t)b 0 + tb 3 > 0 (.) b () = (, t) b 0 + (, t)tb 3 + t b > 0: (.5) The rest of the control points is a positive function of b 3, b 0 and hence can be set to be positive. Quantitative details are given in appendix C For a quintic BB polynomial, F 5() = b 50 B 5 50 ()+b B 5 ()+b 3B 5 3 ()+b 3B 5 3 ()+b B 5 ()+b 05B 05 () (.6) where b 50 >", b, b 3 and b 3 are given, set b and b 05 so that F 5() > 0in[0; ]. We subdivide the univariate quintic polynomial F 5() into two pieces, F 5 () () = b () 50 B b () B5 + b () 3 B5 3 + b () 3 B5 3b () B5 + b () 05 B5 05 (.7) F 5 () () = b () 50 B b () B5 + b () 3 B5 3 + b () 3 B5 3b () B5 + b () 05 B5 05 (.8)

13 upon =(, t; t (see gure.6). By choosing t carefully, one makes sure that b () = (, t)b 50 + tb > 0 (.9) b () 3 = (, t) b 50 + (, t)tb + t b 3 > 0 (.30) b () 3 = (, t) 3 b (, t) tb + 3(, t)t b 3 + t 3 b 3 > 0 (.3) The rest of the control points is a positive function of b, b 05 and hence can be set to be positive. Quantitative details are given in appendix C.. Bivariate Given a bivariate degree m BB polynomial F () in[p p p 3 ], if F () ^ 3 = 0 is xed and larger than a small constant ", we can set b 00m and any other free weights so that F () > 0 in the domain. A sucient algorithm to achieve this is given as follows. We rst construct a \worst case" univariate degree m polynomial Fm, () =b, m0 Bm m0 ()+::: + b, (m,j)j Bm (m,j)j ()+b, 0m Bm 0m (): (:3) where, for the jth layer F () ^ 3 = j, 0<j<m, if a minimum value can be determined from the xed weights assign it to b,. Otherwise mark (m,j)j b, as a free weights and layer j as free (m,j)j layer. We then set the free weights to make sure the univariate polynomial Fm, > 0. Then we make sure that the subpolynomial in each free layer is larger than the corresponding free weight in Fm,. By doing that, we make sure that the univariate subpolynomial in any line [pp 3 ], p [p ;p ], is positive. If we have more time or our computer is more powerful, we may obtain more relax result by rst subdividing [p p ]into n pieces, [p (i) ;p(i) ], p() = p, p (n) = p. Then applying the above algorithm to each triangular simplex [p (i) ;p(i) p 3], and then in each layer, setting the free weights so that it is no less than what is required for each subdivided subpolynomial. Similarly, ifb 00m is xed and larger than a small constant ",we can set the weights of F ()^ 3 = 0 and any other free weights so that F () > 0 in the domain...3 Trivariate There are two kinds of congurations with trivariate BB polynomials. The rst case is analog to the bivariate case. For a trivariate degree m BB polynomial F () in[p p p 3 p ], if F () ^ =0 is xed and larger than a small constant ", we can always set b 000m and any other free weights so that F () > 0 for any over the domain simplex. If b 000m >">0 is given, we can always set the weights of subpolynomial in [p p p 3 ], namely F () ^ ( = 0) and any other free weights to be large enough so that F () > 0. The second case, if F () ^ = = 0 is xed and larger than a small constant ", we can always set the weights of F () ^ = 3 = 0 and any other free weight to be large enough so that F () > 0. In particular, the worst case polynomial Fm, () =b, m0 Bm m0()+::: + b, (m,j)j Bm (m,j)j ()++:::::: + b, 0m () (:33) where b, (m,j)j is the minimum of F ()^ + = j, if the minimum can be determined. Otherwise b, (m,j)j is marked free. 3

14 Summing up, in a simplex tetrahedron, if the subpolynomial in a face is given positive, we can adjust the subpolynomial at its opposite vertex so that the entire polynomial over the simplex is positive and vice versa. If the subpolynomial in an edge is given positive, we can adjust that subpolynomial at its opposite edge so that the entire polynomial over the simplex is positive. The rst case is for the three sided patch while the second case is for the four sided case. Similarly, relax result can be obtained by subdividing [p p p 3 p ]. For the rst case (three sided patch), [p p p 3 p ] is subdivided into [p (i) p(i) p(i) 3 p ]'s, where [p (i) p(i) p(i) ] is a sub-triangle of 3 [p p p 3 ]. For the second case (four sided patch), [p p p 3 p ] is subdivided into [p s i ps i+ ps i p s i+ 's, where [ps i ps i+ is a subsegment of[p p ], and [p s i p s is a subsegment i+ of[p p 3 ].. Sucient conditions for single-rooted univariate quintic BB polynomials We now tackle the following problem. Give a univariate quintic BB polynomial F 5() =b 50 B 5 50 ()+b B 5 ()+b 3B 5 3 ()+b 3B 5 3 ()+b B 5 ()+b 05B 5 05 () (:3) where b 50, b, b 3, and b 3 are given, set b and b 05 so that the polynomial has exactly one root within the interval of (0; ) if () b 50 < 0, or () b 50 = 0 and b >, where >0 is a constant. We rst assume that b 50 < 0. Such a quintic BB polynomial can be classied into the 8 cases in terms of the signs of b, b 3 and b 3, shown in gure.7. If we denote each case by a triple consists of the signs of b, b 3 and b 3 respectively, then in cases (,;,;,), (,;,; +), (,; +; +) and (+; +; +), there is only one sign change in the weights, which is a sucient condition to guarantee that there is exactly one root in [0; ]. We call these four cases category (0). The other four cases are divided into the following two categories in terms of the dierence between b 50 and b : () b 50, b <,", where " is a small positive number, which can be set as b 50 =00; and () otherwise. Case (+; +;,), (+;,; +) and (+;,;,) fall into case (), while case (,; +;,) is further divided into () and () (See gure.8). For a polynomial of category (), we set b and b 05 to make sure that the rst derivative of F 5(), F 5 () = 5((b, b 50 )B 0()+(b 3, b )B 3()+(b 3, b 3 )B ()+ (b, b 3 )B 3()+(b 05, b )B 0()) (.35) is positive in[0; ]. It follow that F 5() is monotonic and hence single-rooted. The problem is hence reduced to the quartic case we have discussed in last subsection. For a polynomial of case (), we set b and b 05 to make sure that the second derivative off (), F 5 () = 0((b 3, b + b 50 )B 3 30()+(b 3, b 3 + b )B 3 ()+ (b, b 3 + b 3 )B 3 ()+(b 05, b + b 3 )B 3 03()) (.36) is positive. The positivity off () ensure that F () is monotonic, which ensure that F () is single-rooted, providing that F 5(0) = b 50 < 0 and F 5() = b 05 > 0. As b 50 < 0, b < 0, b 3 > 0 and b 50, b " we have b 3, b + b 50 > 0. Hence the problem is reduced to the cubic case we discuss in last subsection. In the case that b 50 and b >>0, the polynomial has a root at =(; 0). As b >>0, we can set b and b 05 large enough so that there is no other roots.

15 (a) (,;,;,) (b)(,;,; +) (c)(,; +; +) (d)(+; +; +) (e)(,; +;,) (f)(+; +;,) (g)(+;,;,) (h)(+;,; +) Figure.7: Quintic BB polynomials are subdivided into 8 cases according to the signs of weights b, b 3 abd b 3. 5

16 ε (a) (b) Figure.8: case (,; +;,) is further subdivided. (a) b 50, b <,". (b) Otherwise.3 Sucient conditions for quintic A-patches From denition, a three sided quintic patch is an A-patch if a line segment [pp ] connecting the top vertex p and any point p on the bottom [p p p 3 ]intersects the surface at most once. In other words, the univariate subpolynomial in [pp ] has at most one root in [0; ]. In the case of a four sided patch, a ray polynomial is the univariate subpolynomial in [p p 3 ], where p [p p ], p 3 [p p 3 ]. We rst assume that, at the bottom face [p p p 3 ] (where = 0), F () > 0. Then [pp ] intersects the surface exactly once. We call such a subpolynomial a ray polynomial of the trivariate polynomial in [p p p 3 p ]. Note that how many times [pp ]intersects with the surface are determined by the ray polynomial in [pp ]. Recalling that, in section 3, the weights at the th and 5th are free in the C continuity weight setting algorithm, each ofsuch univariate quintic polynomials can be made single-rooted as we discussed above. So if one can check all [p p]'s for all p in [p p p 3 ], to make sure that the ray polynomial has exactly one root in [0; ], we make sure that the surface is an A-patch. But obviously, it is non-practical to keep track of the innite number of ray polynomials. Hence, we instead subdivide the bottom face [p p p 3 ]into small triangles [p (i) p(i) p(i) 3 ], such that the subpolynomial in [p (i) p(i) p(i) 3 p ] is simpler in the sense that the ray polynomials fall in the same category that we discussed above. We are then able to make sure that all ray polynomials in [p p] are single-rooted by making sure that the \worst case" is single-rooted. Similarly, in the case of a four sided patch [p p p 3 p ], we subdivide edge [p p ] and edge [p p 3 ] to subdivide the tetrahedron into smaller four sided patches, within each of which we enforce that the surface is an A-patch by treating a \worst" case ray polynomial. However, it is not practical to subdivide the polynomial into into subpolynomials that exactly fall into individual categories. Instead, we subdivide the polynomial until the individual subpolynomial can be treated the same way. To see that is possible, in last subsection, we classify the univariate quintic subpolynomials into 3 categories. Category (0) can be trivially made singlesheeted by setting b and b 05 to be positive. So, if a ray polynomial of a trivariate polynomial is either in category (0) or (), we can treat them all as if they are all in category (). Similarly, when aray polynomial of a trivariate polynomial is either in category (0) or (). The algorithm that guarantees a category () subpolynomial single-rooted rely on the fact that b 50 + b 3, b > 0, which can apply to some member of categories (0) and () as well. Hence if we subdivide the polynomial deep enough, we can get subpolynomial whose ray polynomials are in category (), or (0), or () but with b 50 + b 3, b > 0. Based on the above observation, we layout the subdivision scheme in the following way. 6

17 Let F 0 () =F () ^ ( = 0) be the bivariate polynomial in the bottom face, or the 0th layer, F () =F () ^( = ) be the quartic bivariate polynomial in the st layer, F () =F () ^( = ) be the cubic bivariate polynomial in the nd layer. We dene the following two polynomials in [p p p 3 p ] F () = F 0 (), F () (.37) F () = F 0 (), F () + F () (.38) We denote the weights of polynomial F () asb(f ). b(f ) <cmeans the weights of F are all less than c. Then we keep subdividing [p p p 3 p ]until for each subpolynomial [p (i) p(i) p(i) 3 p ] satises one of the following cases. (0) There is only one sign change; () b(f ) <,", which implies that for every univariate subpolynomial in [p p], b 50, b <,"; () b(f ) < 0 ^ (b(f ) _ b(f ) < 0 _ b(f 3 ) > 0) which implies that for every univariate subpolynomial in [p p], b < 0 and b 50, b + b 3 > 0, or can be classied into category (0). where In cases () and (), we dene the worst univariate subpolynomial F worst (x) = b, 50 B b, B5 + b, 3 B5 3 ++b, 3 B5 3 + b, B5 + b, 50 B5 05 (.39) b, 50 = Minfb(F 0 )g b, = Minfb(F )g b, 3 = Minfb(F )g Enforcing the worst univariate subpolynomial to be single-rooted, we guarantee that all the univariate subpolynomial over [p p] is single-rooted, which implies that the surface is an A-patch. In the case that F 0 () is not always positive, we subdivide face [p p p 3 ]until in each subpolynomial F (i) 0 () in[p(i) p(i) p(i) 3 ] is one of the following. Positive. Set the free weights of F (i) () so that the trivariate polynomial has no root.. Negative. Set the free weights of F (i) () so that the trivariate polynomial has exactly one root. 3. The subpolynomial in [p (i) p(i) p(i) (i) 3 ] (denoted as F ()) is an A-spline [BX9]. A sucient condition is that, there exists an integer j( j 3) an integer k(0 <k<5) such that b 0; j =0; ; :::;k, (.0) b 0; j = k +; :::; 5 (.) In this case, say j =,we set the free weights of F (i) () so that the bivariate subpolynomial in [p (i) p(i) 3 p ] is positive, and furthermore, large enough so that any ray subpolynomial in pp, where p [p (i) p(i) 3 p ], is single-rooted. 7

18 At a corner of face [p p p 3 ], say p or =(; 0; 0; 0), the weight b 5000 =0. From the stronger tangent containment condition 3. that we have enforced, b 00 >for some small constant >0. Thus there exists a small neighborhood ; j,j <, where F ( ) > 0. Hence it can be treated as an instance of category (). In short, the functional zero contour enforcing scheme can be embed into the C scheme as follows. Steps and 8 replace steps and 8 in scheme 3.. Step ' is inserted between steps and. Algorithm. Functional zero contouring. If face [p p p 3 ] is convex, set a (i), 0 a(i), 0 a(i) 0 to be positive. 8. Set number8weight, a (i) to be positive. '. Set number weight to be large enough so that (a) we are sure the polynomial in [p p p 3 p ] is a three sided A-patch, and (b) when we compute number 3, and 5 weights later in step, they are large enough so that we are sure the polynomial in [p 00 p p 3 p ] is a four sided A-patch.. Summary In summary, we give an outline of the algorithm. depending on the implementation. The last step can be incorporated into one, Algorithm.3 Construction of C A-patch surface INPUT: a triangulated polyhedron P. Compute the normal at each vertex of P. Construct C cubic simplicial hull and C A-patches Degree raising the cubic simplicial hull to quintic. Used as default values. Estimate C data (Hessian matrix) at each vertex. interpolate them with the polynomials. Construct piecewise C quintic polynomials. Set functional zero contouring conditions. OUTPUT: C A-patch surface. 8

19 (a) (b) (c) (d) Figure 5.9: A simple C example. (a) Drawn in dierent shades to show the piecewise structure. (b) C : Showing Gaussian curvature. (c) C : Showing Mean curvature. (d) C : Showing Gaussian curvature. 9

20 (a) (b) (c) (d) Figure 5.0: A C smooth icosahedron. (a) Drawn in dierent shades to show the piecewise structure. (b) Gaussian curvature. (c) Mean curvature. (d) Mean curvature. The surface is modied locally around the patch facing up. 0

21 (a) (b) (c) (d) (e) (f) Figure 5.: Interactive control of C A-patches. Starting from a sphere, the quintic piecewise surface is deforming toward a cube. (a) Sphere. (b) The surface is dragged toward a vertex of the cube. Mean curvature map. (c) Toward an edge of the cube. Gaussian curvature. (d) Toward an face of the cube. Mean curvature. (e) Toward the cube. Gaussian curvature. (f) Toward the cube. Shaded in pattern to show the piecewise structure.

22 5 Implementation We have presented algorithms for interpolating a three dimensional polyhedron with C quintic A-patches. These algorithms have been implemented in the SplineX and Shilp toolkits of Shastra, a distributed and collaborative geometric design environment [AB93]. Shilp is an X- based, interactive solid modeling system and is used to create a simplicial (face triangulated) polyhedral model of the desired shape. This model could also be the triangulation of an arbitrary surface in three dimensions. This triangulation is C smoothed by a client/server call to a SplineX computation using inter process communication. SplineX is a an X- based, interactive surface modeling toolkit for arbitrary algebraic surfaces (implicit or parametric) in BB form. It allows for the creation of simplex chains (as for example the simplicial hull of the triangulation) and the interactive change of control points and weights of the A-patches for shape control. SplineX also has the ability to distribute its rendering tasks (for the display of the individual A-patches) on a network of workstations, to achieve maximal display parallelism. Figure 5.9 shows a simple example, quintic A-patches based on triangles sharing a vertex. Figure 5.0 shows quintic A-patches built on top of a icosahedron. Note that the sizes of edge patches in pairs are not even, as an convention simplicial hull based on an icosahedron, with equal sized edge patch pair, would cause case degeneracy(see Appendix B). Figure 5. shows interactive control of C quintic A-patches. Starting from a sphere, we drag the surface towards the vertices of a cube by changing the weights a 03, a 03 and a 03 of the quintic face simplexes. Note that the simplicial hull in Figure 5. suers case degeneracy(see Appendix B). Hence we keep weight a as is, but instead change a 03, a 03 and a Conclusion We give piecewise C quintic A-patch scheme to C interpolate or approximate each vertex of a given polyhedron. However, as a rst step toward C algebraic splines in BB form, the scheme still has many problems. Compared to the C cubic A-patch scheme [BCX95a], this scheme is more complicated in the following aspects. First, we have to solve larger linear system to achieve C continuity. Secondly, it is far more dicult to guarantee that the surface is fully connected, free of singularity and unwanted branches, as in general, the \one sign change" principle is not practical. Thirdly, there are quite many degenerate cases, where the linear systems are singular and we have modied the simplicial hull to remove the degeneracies. However, it seems to us there is not much room that we can play with quintic surface for C continuity, under the same simplicial hull structure. Further improvements can be made by studying more about the degenerated cases, using higher degree surfaces, or dierent simplicial hull structures. For example, if we use degree 7 patches, which interpolates some given or precalculated C 3 data at each vertex, and C elsewhere, we are able to break up those big systems around the vertices into smaller ones around the edge. Namely, the systems involve no more than 8 simplexes, same level as in the C cubic case. Besides all these problem, the scheme inherits some common problem of surface tting, as shown in Figure 5.0, although C smooth, the surface looks bumpy. An optimization scheme is hence needed to construct fair surface with respect to this particular problem.

23 References [AB93] [Baj9] [BBX9] [BCX9] V. Anupam and C. Bajaj. Collaborative Multimedia Scientic Design in SHASTRA. In Proc. of the First ACM International Conference on Multimedia, ACM MULTIMEDIA 93, pages 7{56. ACM Press, 993. C. Bajaj. Surface tting with implicit algebraic surface patches. In H. Hagen, editor, Topics in Surface Modeling, pages 3 { 5. SIAM Publications, 99. C. Bajaj, F. Bernardini, and G. Xu. Reconstruction of Surfaces and Surfaces-on-Surfaces from Unorganized Weighted Points. Computer Science Technical Report, CS-93-00, Purdue University, 99. Appeared in SIG '95. C. Bajaj, J. Chen, and G. Xu. Smooth Low Degree Approximations of Polyhedra. Computer Science Technical Report, CSD-TR-9-00, Purdue University, 99. [BCX95a] C. Bajaj, J. Chen, and G. Xu. Modeling with Cubic A-Patches. ACM Transactions on Graphics, (), April 995. [BCX95b] C. Bajaj, J. Chen, and G. Xu. Smooth Low Degree Approximations of Polyhedra (Part ). Manuscript, September 995. [BI9a] C. Bajaj and I. Ihm. Algebraic surface design with Hermite interpolation. ACM Transactions on Graphics, 9():6{9, January 99. [BI9b] C. Bajaj and I. Ihm. C Smoothing of Polyhedra with Implicit Algebraic Splines. SIGGRAPH'9, Computer Graphics, 6():79{88, July 99. [BIW93] C. Bajaj, I. Ihm, and J. Warren. Higher order interpolation and least squares approximation using implicit algebraic surfaces. ACM Transactions on Graphics, ():37{37, Oct [BX9] C. Bajaj and G. Xu. A-splines: Local Interpolation and Approximation Using C k Continuous Piecewise Real Algebraic Curves. Technical Report CSD-TR-9-095,CAPO- 9-, Computer Sciences Department, Purdue University, December 99. [Dah89] W. Dahmen. Smooth piecewise quadratic surfaces. In T. Lyche and L. Schumaker, editors, Mathematical Methods in Computer Aided Geometric Design, pages 8{93. Academic Press, Boston, 989. [DTS93] [Far90] [Guo9a] [Guo9b] W. Dahmen and T-M. Thamm-Schaar. Cubicoids: modeling and visualization. Computer Aided Geometric Design, 0():89{08, Apr G. Farin. Curves and Surfaces for Computer Aided Geometric Design: a practical guide. Academic Press, Boston, MA, nd edition, 990. B. Guo. Modeling Arbitrary Smooth Objects with Algebraic Surfaces. PhD thesis, Computer Science, Cornell University, 99. B. Guo. Surface generation using implicit cubics. In N.M. Patrikalakis, editor, Scientic Visualizaton of Physical Phenomena, pages 85{530. Springer-Verlag,Tokyo, 99. 3

24 [Guo93] [MW9] [Pet90] B. Guo. Non-splitting Macro Patches for Implicit Cubic Spline Surfaces. Computer Graphics Forum, (3):3 { 5, 993. D. Moore and J. Warren. Approximation of dense scattered data using algebraic surfaces. In Proc. of the th Hawaii Intl. Conference on System Sciences, volume, pages 68{690, Kauai, Hawaii, 99. IEEE, IEEE Computer Society Press, Los Alamitos, California. J. Peters. Local smooth surface interpolation: a classication. Computer Aided Geometric Design, 7(-):9{95, June 990. [Pra87] V. Pratt. Direct least squares tting of algebraic surfaces. Computer Graphics, (3):5{5, 987. [Sed90] T. Sederberg. Techniques for cubic algebraic surfaces, tutorial part ii. IEEE Computer Graphics and Applications, 0(5):{, Sept. 990.

25 A Determining Number 5 and 6 Weights of Quintic It follows from (.0) and (.) that b (i) = (i) a(i) + (i) a(i) 03 + (i) 3 a(i) 0 + (i) a(i) 0 (A:) b (i) 0 = (i) (i) a(i) 0 +(i) (i) a(i) 30 +(i) (i) 3 a(i) 0 +(i) (i) a(i) + (i) (i) a(i) 00 +(i) (i) 3 a(i) 030 +(i) (i) a(i) 03 + (i) 3 (i) 3 a(i) 030 +(i) 3 (i) a(i) 0 + (i) (i) a(i) 0 for i =;. (A.3) can be written briey as (A:3) where is the known terms in (A.3). Since b (i) 0 =(i) (i) a(i) + (i) (i) a(i) 0 + (A:) b () 0 = b () + b () (A:5) b () 0, b () = b () 0, b () (A:6) then by substituting (A.) into (A.5) and (A.6) and then eliminating b (i) 0 (A.5) we get three equations related to four unknowns which could be written as: " (),,, (), #" () 0 0 () h, ( (), ) ( () #" a () 0 a () 0 0 a () 0, ) i" a () # # =, " (),,, (),, h () ;, () i " a () a () # from (A.) and #" a () a () # + (A:7) = (A:8) where are known terms. Since the coecient matrix of (A.7) is nonsingular, by solving [a () 0 a() 0 ]T from (A.7) and then substituting it into (A.8), we get one equation relating to the unknowns a () ; a(). Let the equation be in the form a () + a() =! (A:9) Therefore, these unknowns form a closed chain around the vertex p in one side of the tangent plane at p. Let 3 6 A = (A:50) k! = 6 5!!...! k k 5 (A:5)

26 We observe that system Ax =! (A:5) is of rank k,. In other words, it has innite number of solutions. Wehave not come up with a proof of this property yet. Assuming this observation is true, we choose the solution that least-squared approximate the default values. B Determining Number 9 Weights of the Quintic Scheme For i =;, and b (i) 0 = (i) a(i) 0 + (i) a(i) (i) 3 a(i) 0 + (i) a(i) 003 (B:53) b (i) 0 = (i) (i) a(i) 0 +(i) (i) a(i) 30 +(i) (i) 3 a(i) +(i) (i) a(i) 0 + (i) (i) a(i) 00 +(i) (i) 3 a(i) 03 +(i) (i) a(i) (i) 3 (i) 3 a(i) 0 +(i) 3 (i) a(i) 0 + (i) (i) a(i) 003 Substitute (B.53) and (B.5) into (B.56), we have (B:5) b () 300 = b () 0 + b () 0 (B:55) b () 0, b () 0 = b () 0, b () 0 (B:56) () (, () )b() 003, () (, () )b() 003 = This is a system of the form A = k k whose determinant is Q k+r i= i,(,) k+r Q k+r i= i. This matrix is nonsingular in general if the points given are in the general position. Hence the system can be solved. However, if the surrounding tetrahedra at the same side of p are not closed, the matrix A is in the form of A = k k By choosing one unknown, say the l-th to be a free parameter, A can be written as A = with A = l, l, l ; A = l+ l+ l k k " # A 0 0 A 6

27 Hence the system of equations breakup into two smaller sub-systems. Each of them can be solved separately. We identify two kinds of degeneracies as follows. () () = (), but not equal to zero. and need to be set to some values other than 0:5 so that A is not singular. () () and () are all zero. In this case, changing, would not solve the problem. We perturbed the tops of the face simplexes to avoid singularity in A. C Enforcing positivity of quartic and cubic BB polynomials C. Enforcing positivity of a cubic polynomial For a cubic BB polynomial F 3() =b 30 B 3 30()+b B 3 ()+b B 3 ()+b 03 B 3 03() (C:57) where b 30 > 0, b are given, set b and b 03 so that F 3() > 0in[0; ]. We observe that, if F 3() > 0, then there must exist a subdivision of the BB polynomial where the weights of each piece are all positive. We subdivide the univariate cubic polynomial F 3() into two pieces, F 3 () = b () 30 B3 30()+b () B3 ()+b () B3 ()+b () 03 B3 03() (C.58) F 3 () = b () 30 B3 30()+b () B3 ()+b () B3 ()+b () 03 B3 03() (C.59) upon = t (See gure.). We now just need to choose t = t 0 < b 30 b30,b so that b () =(, t 0)b 30 + t 0 b > 0 (C:60) The other weights are positive function of b and b 03 and hence can be set to be positive. In particular, Hence if b () = (, t 0 )b () t 0((, t 0 )b + tb ) (C.6) > t 0 ((, t 0 )b + t 0 b ) (C.6) (, t 0 )b + t 0 b > 0or b b 30, b 0 > 0; (C.63) b () > 0. It is trivial to verify that if furthermore b 03 > 0, all other control points are positive. 7

28 C. Enforcing positivity of a quartic polynomial For a quartic BB polynomial, F () = b 0 B 0()+b 3 B 3()+b B ()+b 3 B 3()+b 0 B 0() (C.6) where b 0 >", b 3 and b are given, set b 3 and b 0 so that F () > 0in[0; ]. We subdivide the univariate quartic polynomial F () into two pieces, F () () = b () 0 B 0 + b () 3 B 3 + b () B + b () 3 B 3 + b () 0 B 0 (C.65) F () () = b () 0 B 0 + b() 3 B 3 + b() B + b() 3 B 3 + b() 0 B 0 (C.66) upon = t (see gure.6). By choosing t carefully, one makes sure that b () 3 = (, t)b 0 + tb 3 > 0 (C.67) b () = (, t) b 0 + (, t)tb 3 + t b > 0: (C.68) Specically, Let r =,t t, inequality (C.68) is equivalent to Solving this inequality in r>0, b 0 r +b 3 r + b > 0 q r>,b 3 + b 3, b 0b b 0 (C:69) (C:70) p Hence by choosing r = r 0 >,b 3+ b 3,b0b, b () b0 > 0. It is easy to verify that so is b(). 3 Then, in order to make b () 3 > 0, as or b () 3 =(, t)b() + t(b 3B 0 + b B + b 3B 0 ) > 0 (C:7) b 3 B 0(t 0 )+b B (t 0 )+b 3 B 0(t 0 ) > 0 (C:7) where t 0 = +r0,or b 3 r 0 + b r 0 + b 3 > 0 (C:73) or q 0 q,b 3 + b 3 b 3 >, b 3 + b 3, b 0b b3 +b A b 0 b 0 Furthermore, with b 0 > 0, every other weights are positive. (C:7) 8

29 C.3 Enforcing positivity of a quintic polynomial For a quintic BB polynomial, F 5() = b 50 B 5 50()+b B 5 ()+b 3 B 5 3()+b 3 B 5 3()+b B 5 ()+b 05 B 05() (C.75) where b 50 >", b, b 3 and b 3 are given, set b and b 05 so that F 5() > 0in[0; ]. We subdivide the univariate quintic polynomial F 5() into two pieces, F 5 () () = b () 50 B b () B5 + b () 3 B5 3 + b () 3 B5 3b () B5 + b () 05 B5 05 (C.76) F 5 () () = b () 50 B b() B5 + b() 3 B5 3 + b() 3 B5 3 b() B5 + b() 05 B5 05 (C.77) upon = t (see gure.5). By choosing t carefully, one makes sure that b () = (, t)b 50 + tb > 0 (C.78) b () 3 = (, t) b 50 + (, t)tb + t b 3 > 0 (C.79) b () 3 = (, t) 3 b (, t) tb + 3(, t)t b 3 + t 3 b 3 > 0 (C.80) Similar to the cubic and quartic case, we solve in 0 <t<, or its equivalence (, t) 3 b (, t) tb + 3(, t)t b 3 + t 3 b 3 > 0 (C:8) r 3 b 50 +3r b +3rb 3 + b 3 > 0 (C:8) with r =,t t. Let t 0 be what we choose for t. We next choose b so that or b () = (, t 0 )t () 3 t 0(b B30(t 3 0 )+b 3 B(t 3 0 )+b 3 B(t 3 0 )+b B03(t 3 0 )) (C.83) > t 0 (b B 3 30 (t 0)+b 3 B 3 (t 0)+b 3 B 3 (t 0)+b B 3 03 (t 0)) > 0 (C.8) Setting b 05 > 0 makes every other weights positive. b >,(b r b 3 r 0 +3b 3 r 0 ) (C.85) 9