Convexity of Spherical Bernstein-B ezier Patches and Circular Bernstein-B ezier Curves

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1 Illiois Wesleya Uiversity From the SelectedWorks of Tia-Xiao He 2015 Covexity of Spherical Berstei-B ezier Patches ad Circular Berstei-B ezier Curves Tia-Xiao He Ram Mohapatray Available at:

2 1 Covexity of Spherical Berstei-Bézier Patches ad Circular Berstei-Bézier Curves T. X. He ad Ram Mohapatra Abstract This paper discusses the criteria of covexity of spherical Berstei-Bézier patches, circular Berstei-Bézier curves, ad homogeeous Berstei-Bézier polyomials. keywords. spherical Berstei-Bézier patch, spherical Berstei-Bézier polyomial, circular Berstei-Bézier curve, homogeeous Berstei-Bézier polyomial, Bézier coefficiet, covexity. 1. Itroductio Let S be the uit sphere i R 3 with ceter at the origi. T = {v S : v = b 1 v 1 + b 2 v 2 + b 3 v 3, b i 0} is the spherical triagle geerated by the three uit vectors v 1, v 2, v 3 S. Here the boudary of T, three circular arcs, lie o great circles. Let v be a poit o S 1. The (spherical barycetric coordiates of v relative to T are the uique real umbers b 1, b 2, ad b 3 such that v = b 1 v 1 + b 2 v 2 + b 3 v 3. (1 It is clear from (1 that the spherical barycetric coordiates of a poit v o the sphere S 1 are exactly the same as the trihedral coordiates of v with respect to the trihedro geerated by {v 1, v 2, v 3 }. This implies that they have the followig properties (cf. [1]: (i. At the vertices v j, j=1, 2, 3, of T, b i (v j = δ ij, i=1,2,3.

3 2 (ii. For all v i the iterior of T, b i (v > 0. (iii. I cotrast to the usual barycetric coordiates o plaar triagles (which always sum up to 1, b 1 (v + b 2 (v + b 3 (v > 1 if v T ad v v 1, v 2, v 3. For the set f={f i : i = (i 1, i 2, i 3, i 1, i 2, i 3 0, i = i 1 + i 2 + i 3 = }, a th degree fuctioal spherical Berstei-Bézier (SBB polyomial is defied o the spherical triagle T as follows ([1]. p (v = B [f; b] = f i φ i (b, (2 where v 1, v 2, v 3 are three vertices of T, b=(b 1, b 2, b 3, v=b 1 v 1 + b 2 v 2 + b 3 v 3, ad φ i (b =! i! bi =! i!i 2!i 3! bi 1 i b i 2 2 b i 3 3, i = i 1 + i 2 + i 3 =. (3 f={f i } is called the set of Bézier coefficiets of the polyomial (2. If we do ot restrict {v 1, v 2, v 3 } to be o the uit sphere S, the the p (v show i (2 is called a homogeeous Berstei-Bézier (HBB polyomial of degree o the trihedro ˆT := {v R 3 : v = b 1 v 1 + b 2 v 2 + b 3 v 3, b i 0} geerated by {v 1, v 2, v 3 } ([1]. I may applicatios, the Bézier represetatio is used to form parametric surface patches by usig vector-valued coefficiets f={f i }. This will be idicated by usig the boldface otatio p (v = B [f; b] = f i φ i (b. (4 {(v i, f i } is called the Bézier et of p (v. Here v i = 1 i 3 i lv l. I [1], the spherical Berstei-Bézier (SBB patch was defied as the surface {p (vv : v T }. Usig the otatio E l mc i = c i+me l, where e l is the l th coordiate vector i R 3, we ca rewrite p (vv as p (vv = (i 1v 1 E i 2 v 2 E i 3 v 3 E 3 1c i φ +1 i (b. (5

4 3 Clearly, from (5, p (vv is also a parametric surface patch p +1 (v with f i = 1 i (i iv i E i 2 v 2 E i 3 v 3 E 3 1c i, i = + 1. (6 For this reaso we also called (4 the SBB patch of degree defied o the spherical triagle T. I [2], a theory of the circular Berstei-Bézier (CBB polyomials is developed. I additio to their itrisic iterest, the CBB polyomials are also useful for describig the behavior of SBB polyomials o the circular arcs makig up the edges of spherical triagles. Let C be the uit circle i R 2 with ceter at the origi, ad let A be a circular arc o C with legth less tha π ad vertices v 1 v 2. Let v be a poit o C. The the (circular barycetric coordiates of v relative to A are the uique pair of real umbers b 1, b 2 such that v = b 1 v 1 + b 2 v 2. (7 Circular barycetric coordiates have a very simple form if we express poits o C i polar coordiates. Suppose v 1 = (cos θ 1, si θ 1 T, v 2 = (cos θ 2, si θ 2 T, (8 with 0 < θ 2 θ 1 < π. let v C be expressed i polar coordiates as v = (cos θ, si θ T. The circular barycetric coordiates of v relative to circular arc A are b 1 (v = si(θ 2 θ si(θ 2 θ 1, b 2(v = si(θ θ 1 si(θ 2 θ 1. Similarly, for a give iteger > 0, the Berstei basis polyomial of degree o the circular arc A is φ i (θ := ( b 1 (θ i b 2 (θ i, i = 0, 1,,. i

5 We call p(θ := 4 c i φ i (θ (9 a circular Berstei-Bézier (CBB polyomial of degree o the circular arc A. Give a CBB polyomial p defied o a circular arc A, we defie a associated CBB curve by ( cos θ P (θ = p(θ. (10 si θ The aim of this paper is to study the covexity properties of SBB patches ad obtai some covexity criteria for HBB polyomials. The paper is orgaized as follows. I Sectio 2, we will discuss the covexity of SBB patches p (v. Some properties of spherical barycetric coordiates will also be give i this sectio. I Sectio 3, we will discuss the covexity of CBB curves. Some covexity criteria for HBB polyomials will be show i Sectio Covexity criteria of SBB patches I order to discuss the covexity of SBB patches, we eed the followig lemmas about the relatios amog the spherical barycetric coordiates of v that are defied i equatio (1. Lemma 1. Let T be a spherical triagle with vertices v i = (x i, y i, z i, i = 1, 2, 3, let v = (x, y, z be a poit o S, the uit sphere i R 3 with the ceter at the origi, ad let the vector b = (b 1, b 2, b 3 be the spherical barycetric coordiates of v relative to T. The 1 i j 3 b i b j v i, v j = 1. (11

6 5 Proof. From equatio (1, we have v, v = b 1 v 1 + b 2 v 2 + b 3 v 3, b 1 v 1 + b 2 v 2 + b 3 v 3 = Sice v, v = 1, we obtai equatio (11. 1 i j 3 b i b j v i, v j. Lemma 2. Let T be a spherical triagle with vertices v i = (x i, y i, z i, i = 1, 2, 3, let v = (x, y, z be a poit o S, the uit sphere i R 3 with the ceter at the origi, ad let the vector b = (b 1, b 2, b 3 be the spherical barycetric coordiates of v relative to T. The b 3 ca be cosidered as a fuctio of b 1 ad b 2, ad b 3 = β i(b, i = 1, 2, (12 b i where β l = 3 k=1 b k v l, v k, l = 1, 2, 3, ad v l, V k is the ier product of v l ad v k. Proof. It is sufficiet to prove the expressio of b 3 b 1. Takig derivative i terms of b 1 o both sides of equatio (11, we have b 1 1 i,j 3 b i b j v i, v j = 0. Expadig the left had side of the above equatio ad trasposig all terms with b 3 b 1 to the right had side, we obtai b 1 v 1, v 1 + b 2 v 1, v 2 + b 3 v 1, v 3 = b 3 b 1 [b 1 v 3, v 1 + b 2 v 3, v 2 + b 3 v 3, v 3 ]. Thus the lemma is proved. Lemma 3. Let the SBB patch of f be the B [f, b] defied i (4. The for l = 1, 2, B [f; b] = 1 b l ψ(v, E, v 3 E1 l v l E1 f 3 i φ i (b, (13

7 where β 3 = 3 k=1 b k v 3, v k, ψ(v, E= i 1 v 1 E 1 1+i 2 v 2 E 2 1+i 3 v 3 E 3 1, av α E l m, bv β E q = ab v α, v β E l me q, ad the shift operator E l mc i = c i+me l. Here e l is the l th coordiate vector i R 3. Proof. It is sufficiet to prove the expressio we have B [f; b] = b 1 [! f i (i 1 1!i 2!i 3! bi b i 2 2 b i b 1 B [f; b]. Notig equatio (12,! i 1!i 2!(i 3 1! bi 1 1 b i 2 2 b i = 1! (b 1 v 3, v 1 + b 2 v 3, v 2 + b 3 v 3, v 3 f i (i 1 1!i 2!i 3! bi b i 2 2 b i 3 3! (b 1 v 1, v 1 + b 2 v 1, v 2 + b 1 v 3, v 3 f i i 1!i 2!(i 3 1! bi 1 1 b i 2 2 b i = 1 [i 1 v 3, v 1 E 1E i 2 v 3, v 2 E1E i 3 v 3, v 3 E1E i 1 v 1, v 1 E 1E i 2 v 1, v 2 E 1E i 3 v 1, v 3 E1E 3 1]f 3 i φ i (b = 1 (i 1 v 1 E i 2 v 2 E i 3 v 3 E 3 1 (v 3 E1 1 v 1 E1f 3 i φ i (b. Thus equatio (13 has bee proved. Similarly, we obtai ] b 3 b 1 ad 2 b 2 l B [f; b] = 2 b 1 b 2 B [f; b] = 1 [] 2 + b l B [f; b] b l 1 [] 2 [ ψ(v, E, v 3 E l 1 v l E 3 1 ] 2 f i φ i (b ( 1, l = 1, 2, (14 [ 2 ] ψ(v, E, v 3 E1 l v l E1 3 f i φ i (b + B [f; b] b 2 b 1 ( 1,

8 = 1 [] 2 [ 2 ] ψ(v, E, v 3 E1 l v l E1 3 f i φ i (b + B [f; b] b 1 b 2 7 ( 1. (15 We ow deote p i = ψ(v, E, V 3 E1 1 V 1 E1 f 3 i, (16 q i = ψ(v, E, v 3 E1 2 v 2 E1 f 3 i, (17 U i = ψ(v, E, v 1 E1 3 v 3 E1 v 1 3 E1 2 v 3 E1 1 + (v 1 v 2 E1, 3 ψ(v, E f i, (18 V i = ψ(v, E, v 2 E1 3 v 3 E1 v 2 3 E1 1 v 3 E1 2 + (v 2 v 1 E1, 3 ψ(v, E f i, (19 ad W i = ψ(v, E, v 3 E 1 1 v 1 E 3 1 v 3 E 2 1 v 2 E 3 1, ψ(v, E f i. (20 Remark: By chagig f i to f i i (13-(20, we obtai the correspodig partial derivatives of B [f; b] ad the correspodig p i, q i, U i, V i, ad W i. ad Obviously, we have B [f; b] = 1 b 1 B [f; b] = 1 b 2 2 b b B [f; b] = B [f; b] = b 2 1 b B [f; b] = 2 1 [] 2 1 [] 2 1 [] 2 p i φ i (b, q i φ i (b, (U i + W i φ i (b + B b 1 b 1 (V i + W i φ i (b + B b 2 b 2 W i φ i (b + B b 1 b 2 ( 1, ( 1, ( 1,

9 = 1 [] 2 W i φ i (b + B b 2 b 1 ( 1. It is well kow that if the Gaussia curvature of a compact surface π i R 3 is positive everywhere, the surface π is covex (lies o oe side of each taget plae([9]. I additio, if π is defied by r = r(u, v, a parametric vector-valued fuctio i C 2, the the Gaussia curvature of π is k = (LN M 2 /(EG F 2, where E, G, F ad L, M, N are respectively the first fudametal form ad the secod fudametal form of π. It is well kow that L = 1 D (r u, r v, r uu, M = 1 D (r u, r v, r uv, N = 1 D (r u, r v, r vv, ad D 2 = EG F 2 > 0. Here (a, b, c is the scalar product of a, b, c ad is defied by (a, b, c = a b, c. Therefore the Gaussia curvature K of r = r(u, v ad LN M 2, or (r u, r v, r uu, (r u, r v, r vv (r u, r v, r uv 2 have the same sig. I particular, for the surface r =r(b 1, b 2 = B [f; b], we have r b1 = B [f; b], r b2 = B [f; b], b 1 b 2 r b1 b 1 = 2 b 2 B [f; b], 1 etc. Thus, if (r b1, r b2, r b1 b 1 (r b1, r b2, r b2 b 2 (r b1, r b2, r b1 b 2 2 > 0, the K > 0. By usig the otatio i (16-(20, from Lemma 3, we have (r b1, r b2, r b1 b 1 (r b1, r b2, r b2 b 2 (r b1, r b2, r b1 b 2 2 = (r p, r q, r U (r p, r q, r V + (r p, r q, r U (r p, r q, r W + (r p, r q, r V (r p, r q, r W,(21 where r p = 1 p iφ i (b, rq = 1 q i φ i (b, ru = 1 [] 2 U i φ i (b, rv = 1 [] 2 V i φ i (b, ad rw = 1 [] 2 W i φ i (b. Deotig (1 i,j,k = (p i, q j, U k, (2 i,j,k = (p i, q j, V k, ad (3 i,j,k = (p i, q j, W k, from (21 we have the followig theorem. 8 Theorem 1. If 1 α<β 3 j =3 s =2 r =2 t = k = (α,β j,s,r,t,k (i > 0 (22

10 9 holds for all i Z 3 +, i = 6, where ( s ( r ( j i j ( i (α,β j,s,r,t,k (i = (α t,s t,j s (β t k s( r j k,r k,i j r ( 2 2 ( 3 2 ( 6, 2 3 the B [f; b] is covex over the spherical triagle T. Proof. To prove the theorem, we eed the followig two represetatios. f i φ i (b i =m q i φ m i (b = where h i = j = (f j q i j ( ( i j / +m, ad f i φ i (b i =m g i φ m i (b = where h i = j = (f jg i j ( ( i j / +m. I fact, j = = i =m j = = f j φ j (b i =m j = = i =m j = = i =m j = = i =m j = = i =m j = i =m g i φ m i (b (f j g i φ j (bφ m i (b (f j g i ( j ( m i b i+j +m +m ( ( ( m fj g i j j i b i j ( fj g i j! j! m! (i j! bi ( i! ( + m! fj g i j (i j!j! i! ( ( i fj g i j j φ +m i (b/ h i φ +m i (b, (23 h i φ +m i (b, (24 ( + m!m! ( + m! bi Similarly, we may obtai equatio (24 by usig the same argumet..

11 10 Therefore, (r p, r q, r U =(r p r q r U = 1 p t φ t (b 1 Similarly, t = = 1 [] 4 s =2 = 1 [] 4 = 1 [] 4 (r p, r q, r V =(r p r q r V = 1 Thus, t = i =3 s =2 t = i =3 s =2 t = k = = 1 [] 4 s = p t q s t q s φ s (b ( s ( t 2 φ 2 s (b 1 [] 2 U i φ i (b ( ( i s (p t q s t U i s s t ( ( ( (1 i s 2 t,s t,i s φ 3 i (b/ s t p k φ k(b 1 i =3 r =2 k = (r p, r q, r U (r p, r q, r V = 1 [] 4 i =3 s =2 t = 1 [] 4 = 1 [] 8 i =3 r =2 k = (1 t,s t,i s r = (2 k,r k,i r ( i s (2 k,r k,i r q r φ r (b ( i r ( s t ( i r i =6 j =3 s =2 r =2 t = k = ( r ( j i j ( i k s( r j i ( s (1 t,s t,i s (2 t k,r k,i j r ( 2 2 ( 3 2 ( r k ( r k φ 3 i (b 2 ( 6 φ 6 (b. 3 φ 3 i (b/ U i φ i (b ( 2. ( [] 2 φ 3 i (b/ φ 3 i (b/ ( 2 ( 3 2 V i φ i (b ( ( ( 3 2 2

12 By usig the above two represetatios ad expressio (17, we immediately 11 have (r b1, r b2, r b1 b 1 (r b1, r b2, r b2 b 2 (r b1, r b2, r b1 b 2 2 = 1 [] 4 Thus Theorem 1 is proved. B [f; b]. i =6 j =3 s =2 r =2 t = k = 1 α<β 3 (α,β j,s,r,t,k (iφ6 i (b. (25 From coditio (25, we also have the followig criterio for the covexity of Theorem 2. If oe of the followig coditio is satisfied for all i, j, k Z 3 +, i = j = k =, the B [f; b] is covex over the spherical triagle T. (i (u i,j,k + (w i,j,k > 0, (ii (1 i,j,k + (3 i,j,k > (3 (u i,j,k (v i,j,k + (v i,j,k (w i,j,k + (w i,j,k (u i,j,k (iii 1 α<β 3 (α i,j,k (β i,j,k > 0; (iv (α i,j,k > 0, α = 1, 2, 3, ad (2 i,j,k + (3 where (u, v, w is a permutatio of (1, 2, 3. i,j,k > (3 i,j,k ; i,j,k > 0; Proof. Deote (l i,j,k, l = 1, 2, 3, by a 1, b 1, c 1, respectively ad (l i,j,k, l = 1, 2, 3, by a 2, b 2, c 2, respectively. It is obvious that iequality (22 holds if a 1 b 2 + b 1 c 2 + c 1 a 2 + a 2 b 1 + b 2 c 1 + c 2 a 1 > 0 (26 for all i, j, k Z 3 +, where i = j = k = ad i = j = k =. We ca prove that iequality (26 holds if the followig iequalities a l + c l > 0, ad a l b l + b l c l + c l a l > 0, l = 1, 2, hold. I fact, if the above iequalities hold, the we have (a 1 + c 1 b 1 > a 1 c 1, (a 2 + c 2 b 2 > a 2 c 2,

13 12 ad (a 1 + c 1 (a 2 + c 2 (a 1 b 2 + b 1 c 2 + c 1 a 2 + a 2 b 1 + b 2 c 1 + c 2 a 1 =(a 1 + c 1 (a 2 + c 2 [b 1 (a 2 + c 2 + b 2 (a 1 + c 1 + c 1 a 2 + a 1 c 2 ] =(a 2 + c 2 2 (a 1 + c 1 b 1 + (a 1 + c 1 2 (a 2 + c 2 b 2 + c 1 a 2 + a 1 c 2 > a 1 c 1 (a 2 + c 2 2 a 2 c 2 (a 1 + c c 1 a 2 + a 1 c 2 =(a 1 c 2 a 2 c Thus iequality (26 holds. Cosequetly, iequality (22 holds if (1 i,j,k + (3 i,j,k > 0, ad (1 i,j,k (2 i,j,k + (2 i,j,k (3 i,j,k + (3 i,j,k (1 i,j,k > 0, which is equivalet to that the followig matrix is positive defiite. [ ] (1 i,j,k + (3 i,j,k (1 i,j,k (1 i,j,k (1 i,j,k +. (2 i,j,k Obviously, the above matrix is positive defiite if it is strictly strogly diagoally domiat, that is, (1 i,j,k + (3 i,j,k > (1 i,j,k ad (1 Furthermore, this coditio ca be implied by Thus, Theorem 2 is proved. i,j,k + (2 (α i,j,k > 0, α = 1, 2, 3. i,j,k > (1 Remark 5. Equatio (23 ad iequality (26 were give i [12] without ay proof. i,j,k. 3. Covexity criteria of CBB curvess

14 13 I this sectio, we will give the covexity criteria for CBB curves. Obviously, a CBB curve is covex if ad oly if its curvature k 0; i.e., the curve lies o oly oe side of each taget lie. The followig lemma gives the curvature k = k(θ of P (θ at θ, which is defied as equatio (10. Lemma 4. Let the P (θ defied as (10 be a CBB curve ad p(θ be the associated CBB polyomial defied i equatio (9. The CBB curve is covex if ad oly if (p(θ 2 + 2(p (θ 2 p(θp (θ 0. (27 Proof. It is sufficiet to prove that the sig of the curvature of the CBB curve P (θ at ay θ is Sig[k(θ] = Sig[(p(θ 2 + 2(p (θ 2 p(θp (θ]. (28 Obviously, the curvature of the parametric curve P (θ = (x(θ, y(θ is k(θ = y (θx (θ x (θy (θ [(x (θ 2 + (y (θ 2 ] 3/2. Sice x(θ = p(θ cos θ ad y(θ = p(θ si θ, we obtai y (θx (θ x (θy (θ =[p (θ si θ + 2p (θ cos θ p(θ si θ][p (θ cos θ p(θ si θ] [p (θ cos θ 2p (θ si θ p(θ cos θ][p (θ si θ + p(θ cos θ] =(p(θ 2 (si 2 θ + cos 2 θ + 2(p (θ 2 (si 2 θ + cos 2 θ p(θp (θ(si 2 θ + cos 2 θ =(p(θ 2 + 2(p (θ 2 p(θp (θ. Thus the lemma is proved.

15 We ow derive p (θ ad p (θ. Sice p(θ = c i( i b1 (θ i b 2 (θ i, where b 1 (θ = si(θ 2 θ/ si(θ 2 θ 1 ad b 2 (θ = si(θ θ 1 / si(θ 2 θ 1, we have Thus 1 p! (θ = c i ( i 1!i! b 1(θ i 1 b 2 (θ i cos(θ 2 θ si(θ 2 θ 1 + i=1 c i! ( i!(i 1! b 1(θ i b 2 (θ i 1 cos(θ θ 1 si(θ 2 θ 1 1 ( 1 = c i b 1 (θ i 1 b 2 (θ i cos(θ 2 θ i si(θ 2 θ 1 1 ( 1 + c i+1 b 1 (θ i 1 b 2 (θ i cos(θ θ 1 i si(θ 2 θ 1 1 = [cos(θ θ 1 c i+1 cos(θ 2 θc i ] φ 1 i (θ. si(θ 2 θ 1 2 p ( 1 (θ = si 2 [(cos(θ θ 1 c i+2 cos(θ 2 θc i+1 cos(θ θ 1 (θ 2 θ 1 (cos(θ θ 1 c i+1 cos(θ 2 θc i cos(θ 2 θ] φ 2 i (θ 1 [si(θ θ 1 c i+1 + si(θ 2 θc i ] φ 1 i (θ si(θ 2 θ 1 2 ( 1 [ = cos 2 si 2 (θ θ 1 c i+2 2 cos(θ 2 θ cos(θ θ 1 c i+1 (θ 2 θ 1 + cos 2 ] (θ 2 θc i φ 2 i (θ 1 [b 2 (θc i+1 + b 1 (θc i ] ( 1! i!( i 1! b 1(θ i 1 b 2 (θ i 2 ( 1 [ = cos 2 si 2 (θ θ 1 c i+2 2 cos(θ 2 θ cos(θ θ 1 c i+1 (θ 2 θ 1 + cos 2 ] (θ 2 θc i φ 2 i (θ 1 1 (i + 1c i+1 φ i+1(θ ( ic i φ i (θ 14

16 15 2 ( 1 [ = cos 2 si 2 (θ θ 1 c i+2 2 cos(θ 2 θ cos(θ θ 1 c i+1 (θ 2 θ 1 + cos 2 ] (θ 2 θc i φ 2 i (θ 1 ic i φ i (θ ( ic i φ i (θ i=1 2 ( 1 [ = cos 2 si 2 (θ θ 1 c i+2 2 cos(θ 2 θ cos(θ θ 1 c i+1 (θ 2 θ 1 + cos 2 ] (θ 2 θc i φ 2 i (θ ic i φ i (θ ( ic i φ i (θ 2 ( 1 [ = cos 2 si 2 (θ θ 1 c i+2 2 cos(θ 2 θ cos(θ θ 1 c i+1 (θ 2 θ 1 + cos 2 ] (θ 2 θc i φ 2 i (θ c i φ i (θ Notig the trigoometric idetities cos 2 (θ 2 θ = 1 si 2 (θ 2 θ, cos 2 (θ θ 1 = 1 si 2 (θ θ 1, ad cos(θ 2 θ cos(θ θ 1 = cos(θ 2 θ 1 + si(θ 2 θ si(θ θ 1, we may re-write p (θ ito 2 p ( 1 (θ = si 2 [c i+2 2 cos(θ 2 θ 1 c i+1 + c i ] φ 2 i (θ (θ 2 θ 1 2 ( 1 b 2 (θ 2 c i+2 φ 2 2 ( 1 b 1 (θ 2 c i φ 2 i (θ 2 i (θ 2( 1 b 2 (θb 1 (θc i+1 φ 2 i (θ c i φ i (θ 2 ( 1 = si 2 [c i+2 2 cos(θ 2 θ 1 c i+1 + c i ] φ 2 i (θ (θ 2 θ 1 2 ( 2! ( 1 c i+2 ( i 2!i! b 1(θ i 2 b 2 (θ i+2 2 ( 2! 2( 1 c i+1 ( i 2!i! b 1(θ i 1 b 2 (θ i+1

17 2 ( 2! ( 1 c i ( i 2!i! b 1(θ i b 2 (θ i c i φ i (θ 2 ( 1 = si 2 [c i+2 2 cos(θ 2 θ 1 c i+1 + c i ] φ 2 i (θ (θ 2 θ 1 2! (i + 2(i + 1c i+2 ( i 2!(i + 2! b 1(θ i 2 b 2 (θ i+2 2! 2 (i + 1( i 1c i+1 ( i 1!(i + 1! b 1(θ i 1 b 2 (θ i+1 2! ( i( i 1c i ( i!i! b 1(θ i b 2 (θ i c i φ i (θ 2 ( 1 = si 2 [c i+2 2 cos(θ 2 θ 1 c i+1 + c i ] φ 2 i (θ (θ 2 θ 1 i(i 1c i φ i (θ 2 i( ic i φ i (θ ( i( i 1c i φ i (θ c i φ i (θ 2 ( 1 = si 2 [c i+2 2 cos(θ 2 θ 1 c i+1 + c i ] φ 2 i (θ (θ 2 θ 1 [i(i 1 + 2i( i + ( i( i 1 + ] c i φ i (θ 2 ( 1 = si 2 [c i+2 2 cos(θ 2 θ 1 c i+1 + c i ] φ 2 i (θ (θ 2 θ 1 2 c i φ i (θ (29 16 By usig equatio (24 for the case of two dimesio: m f i φ i (θ g j φ m j (θ j=0

18 +m = m j=0 ( + m i f i j g j φ +m i (θ/ m j ( + m m 17, (30 we obtai where [p(θ] 2 = 1 ( 2 d i = j=0 2 d i φ 2 i (θ, ( ( i 2 i c j c i j j j ad 2 2 p(θp ( 1 (θ = si 2 c j [c i j+2 2 cos(θ 2 θ 1 c i j+1 (θ 2 θ 1 j=0 ( ( ( i 2 i c i j ] φ 2 2 i (θ/ j j 2 ( 2 2 d i φ 2 i (θ/ 2 2 2(2 1 = ( 2 si 2 c j [c i j+2 2 cos(θ 2 θ 1 c i j+1 (θ 2 θ 1 j=0 ( ( i 2 i 2 +c i j ] φ 2 2 i (θ j j 2 ( ( ( 2 i 2 i 2 c j c i j φ 2 i (θ. j j j=0 Here, c i j = 0 if i < j or i > j +. For the sake of coveiece, we will use the relatio [(p(θ 2 ] = 2(p (θ 2 + 2p(θp (θ to derive the expressio for (p (θ 2. Similar to the process of derivig equatio (29, we ca obtai [(p(θ 2 ] = ( 2 2(2 1 si 2 (θ 2 θ [d i+2 2 cos(θ 2 θ 1 d i+1 + d i ] φ 2 2 i (θ

19 18 Therefore, (22 ( 2 2 d i φ i (θ 2 2 2(2 1 [ ( ( i i 2 = ( 2 si 2 c j c i j+2 (θ 2 θ 1 j j j=0 ( ( i i 1 2 cos(θ 2 θ 1 c i j+1 j j ( ( ] i 2 i +c i j φ 2 2 i (θ j j 2 ( ( ( 42 i 2 i 2 c j c i j φ 2 i (θ. j j j=0 (p(θ 2 + 2(p (θ 2 p(θp (θ =(p(θ 2 + [(p(θ 2 ] 3p(θp (θ 2 2 2(2 1 [ (( ( ( ( i i 2 i 2 i 2 = ( 2 si 2 c j c i j+2 3 (θ 2 θ 1 j j j j j=0 (( ( ( ( i i 1 i 2 i 2 2 cos(θ 2 θ 1 c i j+1 3 j j j j (( ( ( ( ] i 2 i i 2 i 2 +c i j 3 φ 2 2 i (θ j j j j ( ( ( 2 i 2 i 2 c j c i j φ 2 i (θ (31 j j j=0 We will use the followig degree-raisig formula which was give i [2], to raise the degree of the first summatio i the above expressio of (p(θ 2 + 2(p (θ 2 p(θp (θ to 2. where p(θ = d d+2 e i φ d i (θ = ē i φ d+2 i (θ, 1 ē i (θ = (d + 2(d + 1 [i(i 1e i cos(θ 2 θ 1 i(d i + 2e i 1

20 19 for i = 0, 1,, d (d i + 2(d i + 1e i ], Thus, from equatio (31 ad the degree-raisig formula, we obtai (p(θ 2 + 2(p (θ 2 p(θp (θ 1 2 [ ( (( ( 2 i i i 2 = ( 2 si 2 c j [i(i 1 c i j 3 (θ 2 θ 1 j j j j=0 (( ( ( ( i 1 2 i + 1 i 2 2 i 2 cos(θ 2 θ 1 c i j 1 3 j j j j ( (( ( ] i 2 2 i i +c i j 2 3 j j j [ ( (( ( 2 i 1 i + 1 i cos(θ 2 θ 1 i(2 i c i j+1 3 j j j (( ( ( ( i 2 i i 1 2 i 1 2 cos(θ 2 θ 1 c i j 3 j j j j ( (( ( ] i 1 2 i i 1 +c i j 1 3 j j j [ ( (( ( 2 i 2 i + 2 i + (2 i(2 i 1 c i j+2 3 j j j (( ( ( ( i i 1 i 2 i 2 2 cos(θ 2 θ 1 c i j+1 3 j j j j ( (( ( ] i 2 i 2 i 2 +c i j 3 φ 2 i (θ j j j ( ( ( 2 i 2 i 2 c j c i j φ 2 i (θ j j = where ( 2 a i = j=0 1 si 2 (θ 2 θ 1 2 a i φ 2 i (θ, (32 ( ( i 2 i [ c j b (2 (i, jc i j+2 + b (1 (i, jc i j+1 + b (0 (i, jc i j j j j=0 ] +b ( 1 (i, jc i j 1 + b ( 2 (i, jc i j 2, (33

21 c k = 0 for k = 1, 2,, 2 or k = + 1, + 2,, 2 + 2, ad b (l i j+l ca be foud from equatio (32 by usig the combiatio formulas: ( = ( 1 k k k or ( 1 k = k (. k I fact, for i < j or j < i, we have b (l (i, j = 0; for i j i, we obtai ( (( ( ( ( 2 i 2 i + 2 i i 2 i b (2 (i, j =(2 i(2 i 1 3 / j j j j j ( (i + 2(i + 1 =( i + j( i + j 1 (i j + 2(i j + 1 3, (34 [ ( (( ( 2 i 1 i + 1 i 1 b (1 (i, j =2 cos(θ 2 θ 1 i(2 i 3 j j j (( ( i i 1 (2 i(2 i 1 j j ( ( ] ( ( i 2 i 2 i 2 i 3 / j j j j [ (2i 2 + 1(i + 1 =2( i + j cos(θ 2 θ 1 i j ( 2i + 2j 1], (35 [ ( (( ( 2 i i i 2 b (0 (i, j = i(i 1 3 j j j (( ( ( ( i 2 i i 1 2 i 1 4i(2 i cos 2 (θ 2 θ 1 3 j j j j ( (( ( i 2 i 2 i 2 + (2 i(2 i 1 3 j j j ( ( ] ( ( i 2 i i 2 i +(1 2 si 2 (θ 2 θ 1 / j j j j =i(i 1 3(i j(i j 1 4i(2 i cos 2 (θ 2 θ (i j( i + j cos 2 (θ 2 θ 1 + (2 i(2 i 1 3( i + j( i + j 1 + (1 2 si 2 (θ 2 θ 1 =i(i 1 + (2 i(2 i

22 21 3 [(i j(i j 1 + ( i + j( i + j 1] + cos 2 (θ 2 θ 1 [ i(2 i 12(i j( i + j ], (36 [ (( ( ( ( i 1 2 i + 1 i 2 2 i b ( 1 (i, j =2 cos(θ 2 θ 1 i(i 1 3 j j j j ( (( ( ] ( ( i 1 2 i i 1 i 2 i +i(2 i 3 / j j j j j [ (2 2i + 1(2 i + 1 =2(i j cos(θ 2 θ 1 i + j + 1 3( 2i + 2j + 1], (37 ad ( (( ( ( ( i 2 2 i i i 2 i b ( 2 (i, j =i(i 1 3 / j j j j j [ ] (2 i + 2(2 i + 1 =(i j(i j 1 ( i + j + 2( i + j (38 From [7], we have the followig positivity criterio for CBB polyomial p(θ = a iφ i (θ. If ad a 0 + ( 1! a + ( 1! ( i=1 a i <0 ( i=1 a i <0 i i!( i! a i 0 ( i i!( i! a i 0, the p(θ 0. Therefore, from Lemma 4 ad equatio (32 we obtai the followig covexity criterio for p(θ. Theorem 3. Let the P (θ defied i (10 be a CBB curve ad p(θ be the associated CBB polyomial defied i equatio (9. If ( 2 a 0 + (2 1! 2 ad i=1 a i <0 i 2 i!(2 i! a i 0

23 22 ( (2 i 2 a 2 + (2 1! 2 i!(2 i! a i 0, i=1 a i <0 where a i is defied by (33-(38, the the CBB curve P (θ is covex. If we use the positivity criterio give i [11] (if ad a 0 + ( 1! a + ( 1! 1 i=1 a i <0 1 i=1 a i <0 i i!( i! a i 0 i i!( i! a i 0, the p(θ 0, we have aother covexity criterio for p(θ, which is as follows. Theorem 4. Let the P (θ defied i (10 be a CBB curve ad p(θ be the associated CBB polyomial defied i equatio (9. If ad a 0 + (2 1! a 2 + (2 1! 2 1 i=1 a i <0 2 1 i=1 a i <0 i i!(2 i! a i 0 2 i i!(2 i! a i 0, where a i is defied by (33-(38, the the CBB curve P (θ is covex. 4. Covexity criteria of HBB polyomials I Sectio 1, we have show that if ˆT is a trihedro geerated by {v 1, v 2, v 3 } ad if b 1 (v, b 2 (v, b 3 (v deote the trihedro coordiates, i.e., ˆT = {v R 3 : v = b 1 v 1 + b 2 v 2 + b 3 v 3, b i 0},

24 23 the the HBB polyomials of degree ca be writte as p (v = a i φ i (b. (39 Here φ i was defied i equatio (3 i Sectio 1. I this sectio, we will discuss the covexity criteria of HBB polyomial (39. Sice a positively homogeeous covex fuctio is called a gauge fuctio, these covexity criteria ca be also cosidered as coditios for makig a HBB polyomial a gauge fuctio (cf. [4]. I the followig, we will use the otatio D γ = γ 1 x + γ 2 y + γ 3 z, where γ = (γ 1, γ 2, γ 3. For γ = v l, l = 1, 2, 3, we deote D l = D γ = D vl. If we defie that E l a i = a i+e l, where e l deotes the l th coordiate vector i R 3, we have D l p = 1 E l a i φ 1 i (b. (40 For ay directio V, there exists a vector c V = (c 1, c 2, c 3 such that Thus, from (41, we have where c V = (c 1, c 2, c 3 T ad for i = 2. V = D 2 V p = ( 1 s c l v l. (41 2 c T V Q i,a c V φ 2 i (b, (42 Q i,a := (E u E w a i 3,3 u,w=1 (43 Obviously, p (v is covex o ˆT if ad oly if D 2 V p (v 0 for ay directioal vector V ad at ay poit v ˆT. Deotig q i,a (c V = c T V Q i,ac V, we have D 2 V p (v = ( 1 2 q i,a (c V φ 2 i (b. (44

25 24 We defie a fuctio c V associated with q i,f (c as follows: w i,a (c V = { 0, if qi,a (c V 0, 1, if q i,a (c V < 0, (45 where i = 2 ad i l 2, l = 1, 2, 3. If i = 2 ad i l = 2 for l = 1, 2, 3, the w i,a (c V = 1. The followig two iequalities about φ i (b, (46 ad (47, were obtaied i [7] ad [11], respectively, by usig iequalities from [6, p. 17]. 0 φ i (b 0 φ i (b ( 1! 3 i! 1 ( i l b l, (46 ( 1! i! 3 i l b l. (47 We ow give some covexity criteria for the homogeeous Berstei-Bézier polyomials over triagle ˆT. Theorem 5. Let r i {0, 1} for i = 2 ad i ( 2e l, l = 1, 2, 3, ad r i = 1 for i = ( 2e l, l = 1, 2, 3. The Berstei-Bézier polyomial p (v show i (39 is covex o ˆT if for all u {1, 2, 3} its Bézier coefficiets satisfy either or 2 2 ( s l=0 ( s l=0 i 2 l i 2 l r i i! E ue u a i w=1,2,3 w u r i i! 2 E ue u a i ( s l=0 i 2 l w=1,2,3 w u 2 2 r i i! E ue w a i (48 E u E w a i 0. (49

26 Proof. It is sufficiet to prove iequality (48, sice iequality (49 is implied by iequality (48. Notig iequality (46, we have 1 ( 1 D2 V p (v = q i,a (c V φ 2 i (b = = q i,a (c V w i,a (c V φ 2 i (b ( 3 2 ( 3! q i,a (c V w i,a (c V i!( 2 3 i l b l ( 3 ( 3! q i,a (c V w i,a (c V i!( 2 3 ( ( 3! ( 2 3 b 2 l c T V ( ( 3! ( 2 3 b 2 l c T V 2 2 ( 3 ( 3 i 2 l i 2 l i 2 l ( 3 b 2 l w i,a (c V i! w i,a (c V i! 2 2 Q i,a c V E u E w a i 3,3 u,w=1 Obviously, if the last symmetric matrix is strogly diagoally domiat, i.e., for all u = 1, 2, 3, ( 3 2 the the matrix i 2 l w i,a (c V E u E u a i i! ( s 2 l=0 w=1,2,3 w u ( 3 i 2 l 2 2 w i,a (c V i! i 2 l 2 2 w i,a (c V i! E u E w a i 3,3 u,w=1 E u E w a i, 25 c V.

27 26 is semi-positive defiite. Thus DV 2 p (v 0 ad p (v is covex. Obviously, the above coditio is implied by iequality (48. Thus, Theorem 5 is proved. Similarly, we may use iequality (30 to obtai the followig result. Theorem 6. Let r i {0, 1} for i = 2 ad i ( 2e l, l = 1, 2, 3, ad r i = 1 for i = ( 2e l, l = 1, 2, 3. The Berstei-Bézier polyomial p (v show i (35 is covex o ˆT if for l = 1, 2, 3 ad u = 1, 2, 3, its Bézier coefficiets satisfy either or 2 2 i l i! r ie u E u a i w=1,2,3 w u 2 i l i! r i E ue u a i i l i! r ie u E w a i w=1,2,3 w u (50 E u E w a i 0. (51 Proof. It is sufficiet to prove iequality (50, sice iequality (51 is implied by iequality (50. Notig iequality (47, we have 1 ( 1 D2 V p (v = q i,a (c V φ 2 i (b = ( 3! q i,a (c V w i,a (c V φ 2 i (b ( 3! q i,a (c V w i,a (c V i! 3 b 2 l c T V 2 i l 3 i l b 2 l i! w i,a(c V Q i,a c V

28 = ( 3! 3 b 2 l c T V 2 i l i! w i,a(c V E u E w a i 3,3 u,w=1 c V 27 Obviously, if the last symmetric matrix is strogly diagoally domiat, i.e., for all u = 1, 2, 3, 2 i l i! w i,a(c V E u E u a i w=1,2,3 w u 2 i! i! w i,a(c V E u E w a i, the the matrix i! i! w i,a(c V E u E w a i 2 3,3 u,w=1 is semi-positive defiite. Thus D 2 V p (v 0 ad p (v is covex. Obviously, the above coditio is implied by iequality (50. Thus, Theorem 6 is proved. Remark 4. A stroger covexity coditio is implied by iequalities (49 ad (51 as follows: E u E u a i w=1,2,3 E u E w a i. Here, u = 1, 2, 3. w u Remark 5. The coditios give i Theorem 5 ad Theorem 6 are idepedet (see [7] ad [8]. Remark 6. There is aother approach for fidig covexity criteria from the positivity criteria ad is show i [8] for plae BB polyomials. This approach ca also be applied here for HBB polyomials. Ackowledgmet The first author would like to thak Larry L. Schumaker s valuable suggestios ad commets for the draft of this paper. REFERENCES

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