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# Visual Continuity of Bezier Triangles

IP.com Disclosure Number: IPCOM000101039D
Original Publication Date: 1990-Jun-01
Included in the Prior Art Database: 2005-Mar-16
Document File: 4 page(s) / 131K

IBM

## Related People

Kuriyama, S: AUTHOR

## Abstract

Method of generating surfaces from polyhedrons by using Bezier triangles is proposed. Quartic triangular patch interpolation using the Clough-Tocher split method can generate visually smooth surfaces. We extend the Clough-Tocher split to the subdivision of an arbitrary N-gonal domain by using N triangular patches, and generate uniformly smooth surfaces in that domain.

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Visual Continuity of Bezier Triangles

Method of generating surfaces from polyhedrons by using
Bezier triangles is proposed. Quartic triangular patch interpolation
using the Clough-Tocher split method can generate visually smooth
surfaces. We extend the Clough-Tocher split to the subdivision of an
arbitrary N-gonal domain by using N triangular patches, and generate
uniformly smooth surfaces in that domain.

Visual Continuity of Bezier Triangles Quartic Bezier triangles
are represented by a linear combination of Bernstein polynomials with
barycentric coordinates (u,v,w):
Q(u,v,w) = V i,j,k B 4 i,j,k (u,v,w),
B n i,j,k (u,v,w) = n! / (i! j! k!) u i v j w
k, (i,j,k >= 0, i+j+k=4, 0 <= u,v,w <= 1, w = 1 - u
- v), where the shapes of surfaces are represented by Bezier points V
i,j,k.  The two adjacent patches represented by P i,j,k and Q i,j,k,
as shown in Fig. 1, have the same tangential plane along their common
edge, if the following equations are satisfied:
m1 R0 + n1 T0 = l1 S0, m2 R3 + n2 T3 = l2
S3, (1) 3 (m1 R1 + n1 T1) + (m2 R0 + n2 T0 )
= 3 l1 S1 + l2 S0, (2) (m2 R1 + n2 T1) + (m1
R2 + n1 T2 ) = l2 S1 + l1 S2, (3)
3 (m2 R2 + n2 T2) + (m1 R3 + n1 T3) = 3 l2
S2 + l1 S3, (4) where Ri = Q 3-i,i,1 - Q 4-i,i,0, Si =
P 3-i,i+1,0 - P 4-i,i,0, and Ti = P 3-i,i,1 - P 4-i,i,0 (i =
0,...,3), and mi, ni, li (i = 1,2) are scalar values. The values of
mi, ni, and li (i = 1,2) are computed so as to satisfy mi +
ni = 1 for normalization.

Surface Generation Algorithm from Multilateral Skeleton
N-lateral domains are subdivided into N triangular parts each
corresponding to N Bezier triangles defined by V n i,j,k (n =
1,...,N), as shown in Fig. 2. The points V 0,3,1 and V 0,1,3 are
calculated from the vertices of a given multilateral domain V 0,4,0
and V 0,0,4. Let Vd 0,0,4 be the projection point of V 0,0,4 on the
tangential plane at V 0,4,0, then V 0,3,1 is computed as a follows:
V 0,3,1 = V 0,4,0 + t   V 0,0,4 V 0,4,0 ( Vd 0,0,4 -
V 0,4,0 ) /   Vd 0,0,4 V 0,4,0
where the scalar t is a parameter that controls the shape, and V
0,1,3 is computed along with Vd 0,4,0. The point V n 1,3,0 is
computed from V n 0,3,1, V n-1 0,1,3, and V n 0,4,0:
V n 1,3,0 = V n 0,4,0 + 1 / (1 - cos ( 2
pi / N )) { 1 / 2 ( V n 0,3,1 + V n-1
0,1,3 ) - V n 0,4,0 } (5)
where V n 1,0,3  is similarly computed from V n 0,1,3, V n+1 0,3,1,
and V n 0,0,...