Triangular Bézier Approximations to Constant Mean Curvature Surfaces

Transcription

1 riagular Bézier Approximatios to Costat Mea Curvature Surfaces A. Aral 1,A.Lluch 1, ad J. Moterde 2 1 Dep. de Matemàtiques, Uiversitat Jaume Castelló, Spai paral@mat.uji.es,lluch@mat.uji.es 2 Dep. de Geometria i opologia, Uiversitat de Valècia, Burjassot (Valècia, Spai moterde@uv.es Abstract. We give a method to geerate polyomial approximatios to costat mea curvature surfaces with prescribed boudary. We address this problem by fidig triagular Bézier extremals of the CMCfuctioal amog all polyomial surfaces with a prescribed boudary. Moreover, we aalyze the C 1 problem, we give a procedure to obtai solutios oce the taget plaes for the boudary curves are also give. 1 troductio Surfaces with costat mea curvature (CMC-surfaces are the mathematical abstractio of physical soap films ad soap bubbles, ad ca be see as the critical poits of area for those variatios that left the eclosed volume ivariable. he study of these surfaces is actually relevat sice there is a wide rage of practical applicatios ivolvig surface curvatures, ragig from rederig problems to real settigs i automotive idustry as measuremet ad calibratio problems, for istace. geeral, the characterizatio of area miimizig uder volume costrait is o loger true from a global poit of view, sice they could have self-itersectios ad exted to ifiity. But locally, every small eighborhood of a poit is still area miimizig while fixig the volume which is eclosed by the coe defied by the eighborhood s boudary ad the origi. A exhaustive discussio of the existece of surfaces of prescribed costat mea curvature spaig a Jorda curve i R 3 ca be foud i [2]. Give H R the fuctioal D H is defied as follows D H ( x =D( x +2HV ( x = 1 ( x u 2 + x v 2 dudv + 2H < x u x v, x > dudv, 2 3 where <, > ad deote the scalar ad the cross product respectively. f a isothermal patch is a extremal of the fuctioal D H, the it is a CMCsurface. he volume term, V ( x, measures the algebraic volume eclosed i M. Bubak et al. (Eds.: CCS 2008, Part, LNCS 5102, pp , c Spriger-Verlag Berli Heidelberg 2008

2 riagular Bézier Approximatios to Costat Mea Curvature Surfaces 97 the coe segmet cosistig of all lies joiig poits x (u, v othesurface with the origi. he first term, D( x, is the Dirichlet fuctioal. We will give a method to geerate Bézier extremals of D H, for prescribed boudary curves ad costat mea curvature. Our method lets to obtai approximatios to CMC-surfaces, sice we have cosidered the problem of miimizig this fuctioal restricted to the space of polyomials. Moreover, we will cosider the C 1 problem, that is, we give a way to geerate a polyomial approximatio to CMC-surface oce the boudary curves ad the taget plaes alog them have bee prescribed. 2 Existece of riagular Bézier Surfaces of Prescribed Costat Mea Curvature Here, we are ot workig with parametrizatios, we are workig istead with triagular cotrol ets. So, our aim is to fid the miimum of the real fuctio P D H ( x P, x P beig the triagular Bézier patch associated to the cotrol et P. he Dirichlet fuctioal, D, has a miimum i the Bézier case due to the followig facts: First, it ca be cosidered as a cotiuous real fuctio defied o R 3( 1( 2 2, sice there are ( 1( 2 2 iterior cotrol poits which belog to R 3. Secod, the fuctioal is bouded from below. hird, the ifima is attaied: whe lookig for a miimum, we ca restrict this fuctio to a suitable compact subset. O the other had, the fuctio assigig the value V ( x P to each cotrol et, P, with fixed boudary cotrol poits, has o global miimum. f that miimum existed, sice spatial traslatios do ot affect the curvature of the surface, we could suppose that the origi is located far eough away from the surface so that the cotrol et is eclosed i a half-space passig through the origi. Let us move a iterior cotrol poit, P 0, toward the origi. he, a wellkow property of Bézier surfaces states that all the poits of x (u, v chage i a parallel directio with itesity B 0 (u, v. he, sice the ew coe segmet is totally icluded i the iitial oe, its volume decreases. As we said, the fuctio, P D( x P, for cotrol ets with fixed boudary always has a miimum ad, as we have just see, the fuctio P V ( x P, ever has a miimum. herefore, by usig the costat H to balace both fuctios we ca say that the fuctio, P D H ( x P, will have a miimum oly for H [a, a] forsomecostata R. t should be oted that whe H =0, D H is reduced to the Dirichlet fuctioal, D, ad the there is a miimum, whereas whe H is too big, the mai term i D H is V, ad therefore the miimum does ot exist. he value of a depeds o the boudary cotrol poits ad the symmetry of the iterval, [a, a], is acosequeceofthe factthatreversigthe orietatioofa surface meas a chage i the sig of the mea curvature. A detailed explaatio

3 98 A. Aral, A. Lluch, ad J. Moterde about the existece coditios of CMC-surfaces suited to a boudary ad this depedecy ca be foud i [2]. 3 he CMC-Fuctioal Bézier Form he followig propositio gives a characterizatio of a isothermal CMC-surface. Propositio 1. [2] A isothermal patch, x, is a CMC-surface if ad oly if Δ x =2H x u x v. (1 Expressio (1 is the Euler-Lagrage equatio of the fuctioal D H.Moreover, a isothermal patch satisfies the PDE i (1 if ad oly if it is a extremal of D H. [1], it was proved that a extremal of the Dirichlet fuctioal amog all Bézier triagular surfaces with a prescribed boudary always exists ad it is the solutio of a liear system. Now we fid two qualitative differeces, the existece of the extremal of D H ca oly be esured with certaity whe H a, for a certai costat, a, depedig o the boudary cofiguratio, ad they are computed as solutios of a quadratic system. Moreover, sice the Euler-Lagrage equatio of the fuctioal D H, i Equatio (1, is ot liear we caot determie abézier solutio as a solutio of a liear system of equatios i terms of the cotrol poits. Here we will give a expressio of the CMC-fuctioal i terms of the cotrol poits of a triagular Bézier surface, which implies that the restrictio of the fuctioal to the Bézier case ca be see as a fuctio istead of as a fuctioal. he followig two results will simplify the way to obtai the formula i terms of cotrol poits of the fuctioal D H. Propositio 2. he Dirichlet fuctioal, D( x, ofatriagularbézier surface, x, associated to the cotrol et, P = {P } =, ca be expressed i terms of the cotrol poits, P =(x 1,x2,x3, with = {1, 2, 3 } = by the formula where ad D( x = a=1 0 = 1 = C 0 1 x a 0 x a 1 (2 ( ( C 0 1 = 0 ( 1 2 (a 1 + a 2 +2a 3 b 13 b 23 ( { 0 r =0, a r = r 0 r (0 r+r (0 r+r 1 r > 0 b rs = r 0 s + s 0 r ( r 0 + r ( s 0 + s. (4 Proof. he Dirichlet fuctioal is a secod-order fuctioal, therefore we compute its secod derivative i order to obtai the coefficiets C 0 1.

4 riagular Bézier Approximatios to Costat Mea Curvature Surfaces 99 he first derivative with respect to the coordiates of a iterior cotrol poit P 0 = ( x 1 0,x 2 0,x 3 0 where 0 =(0 1,2 0,3 0 for ay a {1, 2, 3}, aday 0 =, with0 1,0 2,0 3 0,is D( x x a = (< x u 0 x a, x u > + < x v 0 x a, x v > du dv, 0 ad the secod derivative 2 D( x (( ( x a 0 x a = B 0 u B 1 u + ( B ( 0 v B 1 v <ea,e a >dudv 1 ( 2 ( 2(2 1 = 0 ( 1 2( (a 1 + a 2 +2a 3 b 13 b 23, + 0 where we took ito accout the formula for the product of the Berstei polyomials ad the value of its itegral. herefore ( ( C 0 1 = 0 ( 1 2 (a 1 + a 2 +2a 3 b 13 b 23, + 0 where a 1,a 2,a 3,b 13,b 23 were defied i Equatio (4. Now, we will work the volume term of the CMC-fuctioal. Propositio 3. Let x be the triagular Bézier surface associated to the cotrol et, P = {P } =, the the volume, V ( x, ca be expressed i terms of the cotrol poits, P =(x 1,x2,x3, with =, bytheformula V ( x = 0 = 1 = 2 = C x 1 0 x 2 1 x 3 2 where C = ( ( ( 0 1 ( 2 3 (d d d ( with d JK rs = r J s J r s ( r + J r + K r ( s + J s + K s. (6 Proof. he term V ( x, is a cubical polyomial of the cotrol poits, so i order to compute the coefficiets C we will compute its third derivative. he derivative with respect to a first coordiate x 1 0 of a arbitrary iterior poit P 0 = ( x 1 0,x 2 0,x 3 0, where 0 = ad 0 1,0 2,0 3 0, is give by

5 100 A. Aral, A. Lluch, ad J. Moterde V ( x x 1 = 1 ( < ( B u e1 x v, x>+ < x u ( B 0 v e1, x > + < x u x v, ( B 0 e 1 > du dv = <B 0 e 1 x v, x> u <B 0 e 1 x vu, x>+ < x u B 0 e 1, x > v < x uv B 0 e 1, x>+ < x u x v,b 0 e 1 >dudv. After computig the derivative with respect to a arbitrary first coordiate, we applied the itegratio by parts formula. Now, bearig i mid that <B 0 e 1 x v, x> u = < x u B 0 e 1, x> v =0, sice B 0 (1 v, v =B 0 (0,v=B 0 (u, 0 = B 0 (u, 1 u =0for 0 = with 0 1,2 0,3 0 0, ad the properties of the cross ad the scalar triple product, we obtai that V ( x x 1 = 1 < x u x v,b e 1 >. (7 Now we must compute the derivative with respect to a secod coordiate, x 2 1, of a arbitrary iterior poit, such that, as before, 1 = with 1 1,2 1, Usig the same process as before we have: Z Z 2 V ( x = 1 x 1 0 x = < (B 1 u e 2 x v,b 0 e 1 > + < x u (B 1 v e 2,B 0 e 1 >dudv (B 0 u (B 1 v (B 0 v (B 1 u <e 1 e 2, x > dudv. Fially we compute the derivative with respect to a arbitrary third coordiate x 3 2 with 2 = adsuchthat2 1,2 2,2 3 0,thatis, C x 1 0 x 2 x 3 = 2 V ( x 1 2 x 1 0 x 2 1 x 3 = ((B 0 u (B 1 v (B 0 v (B 1 u B 2 dudv 2 ( ( ( = 0 1 ( 2 3 (d d d where we have achieved the last formula after computig the itegral of the Berstei polyomials ad performig some simplificatios like the followig: Z Z 1 B 1 0 e 1 B 1 1 e 2 B 2 dudv = 0 e e 2 2 B e 1 e 2 dudv = e 1 e ( ( (

6 riagular Bézier Approximatios to Costat Mea Curvature Surfaces 101 Lemma 1. he coefficiets C JK verify the followig symmetry relatios C JK = C JK = C JK. Proof. he symmetry of the coefficiets C s is a direct cosequece of the symmetry of d s: d JK rs = d JK rs, which is immediate from its defiitio i Propositio 3, sice: d JK J r s r J s rs = ( r + J r + K r ( s + J s + K s. the followig propositio we give a formula for the CMC-fuctioal, D H ( x i terms of the cotrol et, P = {P } =, of the Bézier triagular surface, x. Propositio 4. Let x be the triagular Bézier surface associated to the cotrol et, P = {P } =,wherep =(x 1,x2,x3 with = {1, 2, 3 } =. he CMC-fuctioal, D H, ca be expressed by the formula D H ( x = a=1 0 = 1 = C 0 1 x a 0 x a 1 +2H 0 = 1 = 2 = where ( ( C 0 1 = 0 ( 1 2 (a 1 + a 2 +2a 3 b 13 b with a r ad b rs defied i Equatio (4 ad ( ( ( C = 0 1 ( 2 3 (d d d with d JK rs defied i Equatio (6. 4 Bézier Approximatios to CMC-Surfaces C x 1 0 x 2 1 x 3 2 We have just see i Propositio 4 that the CMC-fuctioal, is a fuctio of the cotrol poits, so let us ow compute its gradiet with respect to the coordiates of a arbitrary cotrol poit. his will let us to give a characterizatio of the cotrol et of the triagular Bézier extremals of D H, which are Bézier approximatios to CMC-surfaces. he gradiet of the first added, correspodig to the Dirichlet fuctioal, with respect to the coordiates of a cotrol poit P 0 = ( x 1 0,x 2 0,x 3 0 = C 0Jx 1 J 0 D( x P C 0Jx 2 J, C 0Jx 3 J = C 0JP J (8 J = J = J = J =

7 102 A. Aral, A. Lluch, ad J. Moterde So, let us cosider the volume expressio V ( x =, J, K = C JKx 1 x2 J x3 K, ad compute its gradiet with respect to the coordiates of a cotrol poit P 0. V ( x P 0 = J, K = = J, K = C 0JK(x 2 J x3 K, x1 J x3 K,x1 J x2 K (9 C 0JK C 0KJ (x 2 J 2 x3 K, x1 J x3 K,x1 J x2 K = 1 2 = 1 2 C 0JK(x 2 J x3 K x2 K x3 J,x1 K x3 J x1 J x3 K,x1 J x2 K x1 K x2 J J, K = J, K = C 0JK P J P K. (10 Now we ca characterize the triagular cotrol et of a extremal of the CMCfuctioal amog all triagular Bézier patches costraied by a give boudary. Propositio 5. A triagular cotrol et, P = {P } =, is a extremal of the CMC-fuctioal, D H, amog all triagular cotrol ets with a prescribed boudary if ad oly if: 0= C 0JP J + H C 0JK P J P K (11 J = J, K = for all 0 =( 1 0,2 0,3 0 = with 1 0,2 0,3 0 0,wherethecoefficietsC 0J ad C 0JK are defied i Equatio (3 ad Equatio (5 respectively. helastresultletsustoobtaibézier approximatios to CMC-surfaces sice we compute solutios to a restricted problem, that is, we fid extremals of the fuctioal D H amog all polyomial patches with prescribed border. he followig propositio characterizes the extremals of this restricted problem: x is a extremal of the fuctioal D H amog all triagular Bézier patches with a prescribed boudary if ad oly if a weak versio of the coditio i Equatio (1 is fulfilled. Propositio 6. AtriagularBézier patch x is a extremal of the CMC-fuctioal, D H, amog all patches with a prescribed boudary if ad oly if: 0= (Δ x 2H x u x v B 0 dudv for all 0 =(0 1,0 2,0 3 = (12 with 1 0, 2 0,

8 riagular Bézier Approximatios to Costat Mea Curvature Surfaces 103 Proof. We simply compute the gradiet of the CMC-fuctioal with respect to a arbitrary cotrol poit. he boudary curves of our example i Fig. 1 describe a approximatio to a circle. herefore we obtai approximatios to spheres. Fig. 1 top, we have asked the iterior cotrol poits to fulfill a symmetry coditio: 4π3 4π3 2π3 2π3 P 112 = a cos,asi,b P 121 = a cos,asi,b P 211 =(a, 0,b ad we show three differet approximatios to CMC-surfaces. he three surfaces at the bottom are obtaied as a solutio of the system of quadratic equatios described i Equatio (11. Here we do t ask for ay kid of symmetry. Fig. 1. hese surfaces are approximatios to CMC-surfaces with curvatures H = 1.5, H = 1 adh = 0.5 respectively Fig. 2 we preset two more examples. he boudary curves i the first are built i such a way that ay associated patch would be isothermal at the corer poits ad i the bottom surfaces i Fig. 2 the boudaries are approximatios to three circular arcs, ad therefore our results look like pieces of a sphere. he resultig plots are pleasat ad moreover they ca be cotiuously deformed by the parameter H, thus allowig the desiger to choose of the shape which best fits the objective. We maitai the good shapes we got with the Dirichlet results i [1], but ow the choice of the curvature gives the desiger aother degree of freedom, although the surfaces are obtaied as a solutio of a quadratic system of the cotrol poits.

9 104 A. Aral, A. Lluch, ad J. Moterde Fig. 2. hese surfaces are approximatios to CMC-surfaces with curvatures H = 1, H =0adH = 1 at the top ad H = 2, H = 1.5 adh = 1 respectivelyatthe bottom 5 he C 1 Problem this sectio we will cosider the prescriptio of ot oly the boudary but also the taget plaes alog the boudary curves, the C 1 problem. Now, the boudary ad the ext to the boudary cotrol poits are fixed, but agai the extremals of the CMC-fuctioal, where the other iterior cotrol poits are cosidered as variables, ca be computed. Here we show a example. We prescribe the border cotrol poits alog a plaar equilateral triagle ad three more lies of cotrol poits as it is show i Fig. 3. Fig. 3. he border cotrol poits ad their eighborig lies of cotrol poits are prescribed he followig figures show approximatios to CMC-surfaces obtaied as a solutio of the quadratic system of the cotrol poits i Equatio (11, but ow for all 0 =( 1 0, 2 0, 3 0 = with 1 0, 2 0, 3 0 > 1.hefreepoitsaretheiterior cotrol poits outside the boudary ad its ext lie of cotrol poits.

10 riagular Bézier Approximatios to Costat Mea Curvature Surfaces 105 Fig. 4. hese surfaces are approximatios to CMC-surfaces with curvatures H = 2, H = 1.5 adh = 1 respectively 6 Coclusios A isothermal patch has costat mea curvature H if ad oly if it is a extremal of the fuctioal D H ( x =D( x +2HV ( x. We have geerated approximatios to CMC-surfaces, sice we have cosidered the problem of miimizig this fuctioal restricted to the space of polyomials. We have obtaied a expressio of D H i terms of the cotrol poits of a triagular Bézier surface. After that, we deduced the coditio that a triagular cotrol et must fulfill i order to be a extremal of D H amog all Bézier triagles with a prescribed boudary. his characterizatio of the Bézier extremals of D H allowed us to compute them as a solutio of a quadratic system of the cotrol poits. he surfaces that are obtaied have regular shapes ad have the advatage of allowig prescriptio of the desired curvature i additio to the boudary. his makes it possible to esure, for a give boudary, the existece of a family of polyomial approximatios to CMC-surfaces with this boudary ad curvatures withi a particular iterval. herefore, the prescriptio of the curvature i this method ca be see as aother degree of freedom i compariso with the Dirichlet surface geeratio method i [1]. Fially, i the last sectio, we cosider the C 1 problem, that is, oce the boudary curves ad the taget plaes alog them have bee prescribed we give a way to geerate a polyomial approximatio to CMC-surface associated to this iitial iformatio. Refereces 1. Aral, A., Lluch, A., Moterde, J.: riagular Bézier Surfaces of Miimal Area. : Kumar, V., Gavrilova, M.L., a, C.J.K., L Ecuyer, P. (eds. CCSA LNCS, vol. 2669, pp Spriger, Heidelberg ( Struwe, M.: Plateau s problem ad the calculus of variatios. Mathematical Notes. Priceto Uiversity Press, Priceto (1988