This document discusses Hermite bicubic surface patches, which are a type of synthetic surface. Hermite bicubic surface patches connect four corner data points using a bicubic equation involving the corner points, corner tangent vectors, and corner twist vectors. The document provides the parametric equation for a Hermite bicubic surface patch and explains how boundary conditions and geometry are defined. It also discusses how Hermite bicubic surface patches provide C1 continuity between patches while maintaining C2 continuity within each patch when blending patches.

1. 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

2. 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?
•Corner points=4, corner tangent vectors=4×2=8, corner twist
vectors=4.
2
∑∑= =
≤≤≤≤=
3
0
3
0
10,10,),(
i j
ji
ij vuvuCvuP
•The parametric equation of Hermite bicubic surface patch is

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6. Influencing factors
on position and
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6
on position and
tangent vectors
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8. [ ] [ ]
[ ][ ]
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zero.are(u)Fofrest1,(u)F1,uwithedgeon the,
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Proof
8
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vF
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Proof

9. [ ]
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9
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10. [ ] [ ]
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11. Continuity of Blent Hermite Bicubic Patches
• Similar to the Hermite cubic splines, the Bicubic
Hermite patches give maximum C1 continuity from
one patch to the next, though they give C2 continuity
inside each of the patches.
• While blending two Bicubic Hermite patches, the
necessary conditions are:
11
– Same curves (C0 continuity) at the common edge
– Same direction of tangent vectors (C1 continuity) at the
common edge
– The magnitudes of the tangent vectors do not have to be
the same

12. Blending Two Hermite Patches along u edges
• [P(0,v)]patch2=[P(1,v)]patch1, C0 continuity
• [Pu(0,v)]patch2=K[Pu(1,v)]patch 1, C1 continuity
The B matrix for the
adjacent patches are
shown. Only these
elements need to be
constrained, rest could
be arbitrary for C1
12
be arbitrary for C1
continuity

13. 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.
13121110
03
30
02
20
01
10
00
00
3
0
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0
),(
10,10,),(
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i j
ji
ij
++++
++++=
≤≤≤≤= ∑∑= =
13
3
3
2
210
33
33
32
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13
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22
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HCC.toreducesitcases,1vand0vofeachFor
uCuCuCCvuP
vuCvuCvuCvuC
vuCvuCvuCvuC
vuCvuCvuCvuC
+++=
==
+++
++++
++++

14. • The second way to prove is:
2) Let u edges coincide. P00 coincides with P10, and P01 coincides
with P11. Pv00=Pv10 and Pv01=Pv11. All four twist vectors will be
zero. Pu00=Pu10= Pu01=Pu11=0.
[ ]
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14
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014003012001
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

15. ===
++=+++
++++
++++
++++
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matrix,[B]resultantThezero.toequalareallrest
ˆ;ˆ;
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33
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sLCrLCPC
Hence
svLruLPvuCvuCvuCvuC
vuCvuCvuCvuC
vuCvuCvuCvuC
vuCvuCvuCvuC
svLruLPvuCvuP
ij
vu
vu
vu
i j
ji
ij By equivalence, find the bicubic
planar surface patch.
15
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[B]
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][]][][[][),(
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00
00
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Can you prove it?