Graphics Reference
In-Depth Information
Figure 12.26.
Ensuring that triangular
Bézier patches meet
smoothly.
not enough. The relative size of their domain comes into play and one has to add the
condition that the points are in the same ratio (d-c):(e-d).
Next, consider triangular Bézier patches. Let p(u,v) and q(u,v) be two triangular
Bézier patches of degree d defined by triangular nets { b ijk } and { c ijk }, respectively, and
assume that they meet along the edge u = 0 and b 0,j,d-j = c 0,j,d-j . A necessary condition
for the patches to meet with G 1 continuity is that the triangles from the two patches
that meet in a common edge are coplanar. For C 1 continuity those triangles for each
patch must be the image of the same affine map. See Figure 12.26.
Finally, we look at bicubic patches as defined in Section 12.9. An interpolating
bicubic B-spline surface can be thought of as a collection of such patches, but we
want to go in the other direction. Given a rectangular grid of points p ij , 0 £ i £ m,
0 £ j £ n, we would like to find conditions on the geometric coefficients of the bicubic
patches defined by the individual rectangles of the grid that will guarantee a globally
smooth surface. We could transform a bicubic patch into a Bézier patch and use
what we know about Bézier patches, but here we want to approach the problem
directly.
We begin by analyzing the conditions under which two bicubic patches meet
smoothly. Let p(u,v) be a patch defined by points p 00 , p 10 , p 01 , and p 11 and q(u,v) a
patch defined by points q 00 , q 10 , q 01 , and q 11 . Assume that q 00 = p 10 , q 01 = p 11 , and
that the patches meet along the boundary curve
() =
() =
(
)
g vpvqv
1
,
0
,
.
See Figure 12.27, where we use the abbreviations
u
()
uv
()
p
=
p
i j
,
and
p
=
p
i j
,
u
uv
ij
ij
and similar abbreviations for q. To get C 1 continuity where the patches meet it must
be possible to find a change of coordinates of the function q(u,v) in the v direction,
so that the new function has the same derivatives along g as the function p(u,v). This
means that ∂q/∂u and ∂p/∂u must be multiples of each other along g. It follows that
having
u
u
uv
uv
u
u
uv
uv
q
=
a
pq
,
=
a
pq
,
=
a
p
,
and
q
=
a
p
,
00
10
00
10
01
11
01
11
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