Synthetic Surfaces 1) Hermite (Bicubic Surface) Patch 2) Bezier (Surface) Patch 3) B-Spline (Surface) Patch 4) Coons (Surface) Patch 5) Blending offset (Surface) Patch 6) Triangular (Surface) Patch 7) Sculptured (Surface) Patch 1 8) Rational surfaces (Surface) Patch All these surfaces are based on polynomial forms. Fourier series can also be used to approximate the surfaces instead. But they are not meant for general use. Because the facts are: (i) they can approximate any curve, not just periodic (ii) computations involved are high
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Synthetic Surfaces
1) Hermite (Bicubic Surface) Patch
2) Bezier (Surface) Patch
3) B-Spline (Surface) Patch
4) Coons (Surface) Patch
5) Blending offset (Surface) Patch
6) Triangular (Surface) Patch
7) Sculptured (Surface) Patch
1
8) Rational surfaces (Surface) Patch
All these surfaces are based on polynomial forms.
Fourier series can also be used to approximate the surfaces instead. But they are not meant for general use. Because the facts are:
(i) they can approximate any curve, not just periodic
(ii) computations involved are high
Hermite Bicubic Surface
•The parametric bicubic surface patch connects four corner data
points and utilizes a bicubic equation.
•Therefore, 16 vectors or 16×3=48 scalars are required to
determine the unknown coefficients in the equation. How?
HERMITE BICUBIC PATCH IS A “SIMPLE EXTENSION” OF THE HERMITE CUBIC CURVE
• There are two ways to prove it.
1) Substitute u=1 or v=1 in the parametric equation of the
Hermite patch, it degenerates to that of HCC.
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HCC. toreducesit cases, 1 vand 0 vofeach For
uCuCuCCvuP
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+++=
==
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• The second way to prove is:
2) Let u edges coincide. P00 coincides with P10, and P01 coincideswith P11. Pv00=Pv10 and Pv01=Pv11. All four twist vectors will bezero. Pu00=Pu10= Pu01=Pu11=0.
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matrix, [B]resultant The zero. toequal are allrest
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Hence
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ij
vu
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i j
ji
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planar surface patch.
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matrix, [B]resultant The zero. toequal are allrest