Continuity of Bçezier patches. Jana Pçlnikovça, Jaroslav Plaçcek, Juraj ç Sofranko. Faculty of Mathematics and Physics. Comenius University

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1 Continuity of Bezier patches Jana Plnikova, Jaroslav Placek, Juraj Sofranko Faculty of Mathematics and Physics Comenius University Bratislava Abstract The paper is concerned about the question of smooth glueing of triangular Bezier patches. In the begining polar forms are brieæy explained. After that they are applied on parametric continuity. We'll obtain a geometric interpretation of C and C 2 smoothly joined Bezier patches. In the next part we'll deal in geometric continuity. We will show relations between derivatives of two maps which are forced by the condition of their geometric continuity. In the place of shape parameters matrices occur. The rest of the paper is devoted to investigating of this matrices and reparametrization functions. It seems that the matrices are not arbitrary, but very strong conditions are put on them. Basic deænitions. Bezier patches Let points t ;t ;t 2 2 E 2 are non-colinear. Then for each u 2 E 2 there exist unique triplet of numbers èuè; èuè; 2 èuè2 R, satisfying u = èuèt + èuèt + 2 èuèt 2 ; èuè+ èuè+ 2 èuè=: The numbers èuè; èuè; 2 èuè are called barycentric coordinates of the point uwith respect to the triangle t t t 2. In next let's suppose that the points t ;t ;t 2 are æxly given. Polynomial B 4 ijk : E2! R deæned B 4 ijk èuè = n ijk i èuè j èuè k 2èuè; i + j + k = n is called Bernsten polynomial. Notice, that all Bernstein polynomials of degree n èi.e. i + j + k = nè create basis of the space of polynomials of degree at most n.

2 Let B : E 2! E d is deæned : Bèuè = X i+j+k=n B 4 ijk èuèb ijk; b ijk are points from E d The restriction of this map to 4t t t 2 is called triangular Bezier patch of degree n. Points b ijk are control points a create a control net..2 Polar forms A map f : M! N is linear iæ it preserves linear combinations, i. e. fè P æ i u i è= P æ i fèu i è; u i 2 M; æ i 2 R f : M! N is aæne iæ it preserves aæne combinations of points : fè P æ i j i è= P æ i fèu i è; u i 2 M; P æ i =; æ i 2 R f :èmè n!n is multilinear èmultiaæneè iæ f is linear èaæneè in each argument when the others are æxed. f :èmè n!n is symmetric iæ its value doesn't depends on the order of arguments. It means fèu ;u 2 ;:::u n è=fèu ;u 2 ;:::u n è where è ; 2 ;::: n èisapermutation of the set è; 2;:::nè. Now we can approach a deænition of polar forms. Let F is polynomial E 2! E d of degree n. Then there exists unique map f :èe 2 è n!e d which ismultiaæne, symmetric and has a diagonal property, i.e. fèu;:::uè=fèuè. This map is called èaæneè blossom èpolar formè of F. For our cogitations linear blossoms are more usefull. For this purpose we must present this two special insertions E 2 to E 3 : u =èu ;u 2 è è ^u=èu ;u 2 ;è u =èu ;u 2 è è u=èu ;u 2 ;è Linear blossom èpolar formè of the polynomial F : E 2! E 3 is a map fæ : èe 3 è n! E d, which ismultilinear, symmetric and has a diagonal property, i.e. fæèu; u;::: uè = Fèuè. Such map always exist and is unique. Linear and aæne blossoms of the same polynomial F satisfy fæèu ; u 2 ;:::u n è=fèu ;u 2 ;:::u n è; u i 2 E 2

3 2 Parametric continuity Maps F; G : E 2! E d are parametric continuous of the order q in u 2 E 2 if their directional derivatives in u up to order q are the same. The theory of polar forms can be applied on parametric continuity of surfaces. But ærst let's mention their two important properties: Let F beabezier patch of degree n with control points b ijk for i + j + k = n with respect to 4t t t 2 and let f be its polar form. Then b ijk = fèt i tj tk 2è; i + j + k = n Furthermore if we denote D 2:::qF èuè q-th directional derivative off in u with respect to vectors ::: q, then the linear blossom of F : E 2! E d satisfy relation D 2:::q F èuè = n! èn,qè! f æèu n,q ^ ^2 ::: ^ q è Proofs of this properties you can ænd for example in ë3ë. We started to use a simpler and lucider multiplicative notation: Instead of fèu ;u 2 ;:::u n èwe are going to write fèu u 2 :::u n è. By the entry fèu n,k u u 2 :::u k èwewant tosay that argument u occurs èn, kè times there. Therefore, if we want to ænd out a value of a derivative of a polynomial, we don't need to derivate, but it is enough to look at a certain value of its blossom. From the previous theorem we get conditions for parametric continuity of two patches: Let F; G : E 2! E d are triangular Bezier patches and let u be a point from E 2. Then according to èè F and G are continuous in u of q-th order iæ èè f æ èu n,i ^ :::^ i è=g æ èu n,i ^ :::^ i è; i =;:::q 3 Geometric interpretation t o e t 2 6 e t e = t, t e e = t 2, t e 2 e 2 = e t, t t Figure : Mapped space

4 Let's denote barycentric coordinates of e t with respect to 4t t t 2 as ; ; 2 : e t = t + t + 2 t 2 where é. Let Bezier patches F; G map this point coæguration in this way: F : 4t t t 2! E 2 G : 4e t t 2 t! E 2 Further let b ijk are control points of the patch F and e b ijk are control points of the patch G. Let's investigate a continuity of this two patches along the common edge t t 2.Ifwe suppose C continuity, itisobvious, that derivatives in u 2 t t 2 with respect to e are the same í it's the same Bezier curve. So that it is enough to ænd only one more direction already, which is lineary independent with e, with respect to which the maps F; G poses the same derivative. Then the C continuity is satisæed because of the fact that D F èuè is linear in for a æxed u and so the other directional derivatives can be composed from the mentioned two. We can write : e t, t = èt, t è+ 2 èt 2,t è e 2 = e + 2 e D e2 Fèuè= D e Fèuè+ 2 D e Fèuè D e2 Gèuè=D e2 Fèuè D e2 Gèuè= D e Fèuè+ 2 D e Fèuè g æ èu n,^e 2 è= f æ èu n,^e è+ 2 f æ èu n,^e è ::: gèe t t i tj 2è= fèt t i tj 2è+ fèt i+ t j 2è+ 2 fèt i tj+ 2 è e bij = b ij + b i+j + 2 b ij+ ; i + j = n, ; u2 t t 2 And thanks to the blossoming we have simply obtained well-known geometric interpretation of the C continuity: Bezier patches F : 4t t t 2! E d ;G:4t e t t 2 are C smoothly joined along the common boundary iæ the control points b ij ;b i+j ; e b ij ;b ij+ lie in the same plane and create an aæne image of the quadrangle t t e t t 2 èlook at æg. 2è. After similar cogitations we can obtain not so known geometric interpretation of C 2 continuity: Bezier patches F : 4t t t 2! E d ;G : 4t e t t 2 are C 2 smoothly joined along the common boundary iæ moreover there exist points d i ; i =;:::n, such that verteces b 2ij ;b i+j ;d i ;b ij+ lie in the same plane and create an aæne image of the quadrangle t t e t t 2 and also points d i ; e b i+j ; e b 2jk ; e b ij+ are coplanar and create an aæne image of the quadrangle t t e t t 2 too èæg. 3è. 4 Geometric continuity Let F; G : E 2! E d are maps. Let u 2 E 2. Then F; G are in u geometric continuous of q-th order iæ there exist such parametrization of surfaces F; G, that

5 Figure 2: C continuity Figure 3: C 2 continuity they are parametric continuos of q-th order. It means: F = F èx; yè i.e. is a function of parameters x; y G = Gès; tè : Let's try to reparametrize the map F and express it by s; t. Soxand y will be written as functions: x = xès; tè; y = yès; tè. Let both maps are now expressed in the way, that they are parametric continuous in u =ès ;t è:

6 Continuity of ærst degree: F s èuè =G s èuè ^ F t èuè=g t èuè For continuity of second degree moreover F ss èuè =G ss èuè ^ F st èuè =G st èuè ^ F tt èuè =G tt èuè But F has to be derivated as a compound function. Let's denote a ij èuè = b ij èuè j u 2 E2 Then the relation between derivatives of F and G with respect to the original parameters can be expressed by connection G ss G st G tt A a 2 b 2 a b a 2 b 2 A F x G s = G t F y + a b a b F x F a2 2a b b 2 a a a b + a b b b a 2 2a b b 2 F xx Let G is the matrix of ærst derivatives of G, G is the matrix of its second derivatives and so on. Similary let F is the matrix of ærst derivatives of F, F is the matrix of its second derivatives and so on. Then the relations è2è, è3è and other can be written in a simpler way: G =æ F F xy F yy A è2è è3è G =æ 2 F +æ 22 F è4è G =æ 3 F +æ 23 F +æ 33 F ::: Surfaces F matrices and G are in u geometric continuous of q-th degree iæ there exist æ ij ; j =:::q; i=:::j satisfying the mentioned relations. We observe that the relations for the geometric continuity of surfaces are very similar to the ones for the geometric continuity of curves. But as shape parameters there are matrices æ ij instead of real numbers.

7 5 Geometric continuity of Bezier patches Now we are going to apply the results of the previous section on the Bezier patches. Let Bezier patches F; G are deæned on the quadrangle t t e t t 2 like this: F : 4t t t 2! E 2 G : 4e t t 2 t! E 2 Let F and G are regularly aparametrized in this way: F = F èr;sè G= Gès; tè where parameter r raises in the direction e, s raises in the direction e and t in the direction e 2 èlook æg.è. We suppose C continuity :Fè;sè=Gès; è: In the begining we require geometric continuity of the ærst order of the patches F and G along the common boundary t t 2. It means that for all u from this edge there exist a matrix æ, satisfying è2è. In this special case it can be written: De G D e2 G where a ij ;b ij are functions of the point u. = a b De F a b D e F Thanks to the fact that parameter s is common for both F and G, it holds D e Gèuè =D e Fèuè for all u 2 t t 2 and so a = and b = constantly. Let a = a b = b for simplicity. Then : æ = a b ; a = aèuè b = bèuè The result can be geometricaly interpreted : Bezier patches are in u geometric continuous of ærst degree iæ they have in uthe same tangent plane. Let's continue by investigating of geometric continuity of second degree of two Bezier patches. Besides the matrix æ, with the speciæed form, the matrices æ 22 and æ 2 must exist, such that G =æ 2 F +æ 22 F If we look at è3è, we can notice, that all elements of the matrix æ 22 are already known, because it is composed only of coeæcients occured in æ. And as for æ 2, it's possible to proceed like in the case of æ. Because the same parameter and the derivative of the same curve is considered, it holds D e e Gèuè =D e e Fèuè è a 2 = b 2 =

8 It will be a bit more complicated for the combined derivative: D e e 2 Gèuè =D e èd e2 Gèuèè = D e èad e F èuè+bd e F èuèè = ad e èd e F èuèè + bd e èd e F èuèè = ad e e F èuè+bd e e F èuè and after comparing with è3è we obtain a = b = constantly for all u 2 t t 2. The result of this observations is the next equation D e e G D e e 2 GA A D e F + D a b F D e2 e 2 G c d a 2 2ab b 2 D e e F D e e FA D e e F We could continue similary. After the small cogitation above the equation for the q-th derivative G èqè = æ q F +æ 2q F +æææ+æ qq F èqè è5è we realize that if the previous derivatives has been investigated, new coeæcients appear only in matix æ q í the q-th derivatives of the reparametrization functions are only there. Let's look at it more closely. Let the reparametrization functions x and y are polynomials of degree m èlook ë2ëè : X X xèu; vè = æ ij u i v j ^ yèu; vè= æ ij u i v j i+jm i+jm From the matrix æ 2 it follows x uv = a = and also y uv = b =. It means that nor xèu; vè neither yèu; vè contains combined member. Furthermore from æ it is x u = a =,i.e.xdoesn't depend on parameter u. In the case of y it is a bit diæerent: y u = b = constantly along the whole boundary t t 2. It means that yèu; vè contains u only as a linear member witch coeæcient. The result can be written: xèu; vè=xèvè= yèu; vè=u+ nx i= nx i= And then we obtain more speciæed form of æ j : æ q = B@ a q ::: b q a q, b q, CA = a q b q æ i v i æ i v i B@ ::: a q b q CA

9 Only last two members of æ q can get an arbitrary value. Let's look at the result. It seems that if the conditions of GC q, smooth joining of the patches F and G has been formulated and we want to reach smoothnes of q-th order, it can be determined by only two shape parameters in the matrix æ q. There's no possibility to manipulate with other matrices occured in the relation between G èqè and F èqè èlook è5èè. So the geometric continuity of curves and surfaces are very similar not only because of the form of the equation system è4è but also because of the constraints for the work with shape parameters. References ëë J. M. Hahn. Geometric continuos patch complexes. Computer Aided Geometric Design, è6è:55í67, 989. ë2ë J. Hoschek and L. Dieter. Fundamentals of Computer Aided Geometric Design, pages 33í37. A K Peters, Ltd, 993. ë3ë H. P. Seidel. Polar forms and triangular B-spline surfaces. Computing in Euclidean Geometry, pages 235í286, 992.