Graphics Reference
In-Depth Information
The next four sections discuss some of the more common tensor product
surfaces: the bicubic patch, Bézier surfaces, B-spline surfaces, and rational B-spline
surfaces.
12.9
The Bicubic Patch
Definition.
A parametric surface p(u,v) defined by
3
3
Â
Â
i
j
(
)
=
puv
,
uv
p
,
(12.34)
ij
i
=
0
j
=
0
where 0 £ u,v £ 1 is called the
general bicubic tensor product surface
or
bicubic patch
.
Similar to the case of curves, the points
p
i,j
are called the
algebraic coefficients
of the
bicubic patch.
It is easy to see that the bicubic patch can be interpreted as the tensor product of
two cubic curves defined by their algebraic coefficients as in equation (11.37). Just as
in the case of curves, the algebraic description (12.34) is usually not very convenient
for a user. One wants a more geometric way to specify the surface. One way to get
this geometric description is with the following observations. There are 16 (vector)
degrees of freedom. Some obvious geometric constraints are the four corner points
and the eight tangent vectors in the u and v direction at those points. That leaves four
degrees of freedom and we can use the mixed partials p
uv
(u,v) at the corner points.
They are called the
twist vectors
.
Definition.
The matrix
B
defined by
(
)
(
)
(
)
(
)
p
00
,
p
01
,
p
00
,
p
01
,
Ê
ˆ
v
v
Á
Á
Á
˜
˜
˜
()
()
()
()
p
10
,
p
11
,
p
10
,
p
11
,
v
v
B
=
(12.35)
(
)
(
)
(
)
(
)
p
00
,
p
01
,
p
00
,
p
01
,
u
u
uv
uv
Ë
¯
()
()
()
()
p
10
,
p
11
,
p
10
,
p
11
,
u
u
uv
uv
is called the
geometric matrix
for the bicubic patch. Its elements are called the
geometric coefficients
of the patch.
The geometric matrix determines the algebraic coefficients completely. Here is
how the geometric coefficients would determine the point p(u,v) on the patch by
repeatedly using the Hermite principle that the endpoints and tangents of a curve
determine the curve completely: See Figure 12.15.
(1) p(u,0) is determined from p(0,0), p(1,0), p
u
(0,0), and p
u
(1,0).
(2) Similarly, p(u,1) is determined from p(0,1), p(1,1), p
u
(0,1), and p
u
(1,1).
(3) Next, p
v
(u,0) is determined from p
v
(0,0), p
v
(1,0), p
uv
(0,0), and p
uv
(1,0).
(4) Similarly, p
v
(u,1) is determined from p
v
(0,1), p
v
(1,1), p
uv
(0,1), and p
uv
(1,1).
(5) Finally, p(u,v) is determined from p(u,0), p(u,1), p
v
(u,0), and p
v
(u,1).
