Bezier Surface Computation Assignment Help

Assignment Help: >> Surface Modeling - Bezier Surface Computation

Example: A 3 × 3 Bezier Surface Computation:

 A cubic Bezier surface may be attained by substituting n = 3 and m = 3 in

Eqs. 19-26

128_Bezier Surface Computation.png

                                                                                                0 ≤ u ≤ 1, 0 ≤ v ≤ 1

This equation may be expanded to give

253_Bezier Surface Computation1.png

= B0,3 (u) [P00 B0, 3 (v) + P01 B1, 3(v) + P02 B2, 3 (v) + P03 B3, 3 (v)]

+ B1, 3 (u) [P10 B0, 3 (v) + P11 B1, 3 (v) + P12 B2, 3 (v) + P13 B3, 3 (v)]

+ B2, 3 (u) [P20 B0, 3 (v) + P21 B1, 3 (v) + P22 B2, 3 (v) + P23 B3, 3 (v)]

+ B3, 3 (u) [P30 B0, 3 (v) + P31 B1, 3 (v) + P32 B2, 3 (v) + P33 B3, 3 (v)]

 This equation may be written in a matrix form as

1662_Bezier Surface Computation6.png

or        

1110_Bezier Surface Computation2.png

P(u, v) = UT  [MB] [P] [MB]TV

where the subscript B indicate Bezier and

1879_Bezier Surface Computation3.png

and the U and V vectors are [u3   u2   u 1]T respectively. Note that [MB] is the same matrix for the cubic Bezier curve.

UT [MH] [B] [MH]TV = UT [MB] [P] [MB]TV

 [MH] [B] [MH] = [MB] [P] [MB]T

or Solving out  for [B] gives [MH]- 1. This equation may be reduced to give

2475_Bezier Surface Computation4.png

Comparing this equation along with Equation for the bicubic patch reveals that the tangent and twist vectors of the Bezier surface are expressed in terms of the vertices of its characteristic polyhedron.

2337_Bezier Surface Computation5.png

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