Graphics Reference
In-Depth Information
Chapter 23
Surfaces
Both spline curves and subdivision curves can be generalized, creating spline and
subdivision
surfaces
. In this chapter, we show how to make a
Bézier patch:
a
small piece of surface, parameterized by
[
0, 1
]
×
[
0, 1
]
,forwhich
u
→
S
(
u
,
v
0
)
is
a Bézier curve for each value of
v
0
and
v
S
(
u
0
,
v
)
is a Bézier curve for each
value of
u
0
(i.e., “it's Bézier in both directions”). Just as Bézier curves can be
joined together into longer curves, Bézier
patches
can be assembled together into
a “quilted” surface, although making the patches meet up smoothly at the edges
and corners is more complex than in the curve case. The quilt, in this situation,
generally has the form of a grid: squares meeting four at a corner. The web mate-
rial for this chapter describes the creation of spline surfaces in more detail.
If we want to make a shape where adjacent patches meet three at a corner or
five at a corner, the conditions for continuity are much messier and the resultant
shape is not as controllable. One popular solution in this case is to shift to
sub-
division surfaces,
which start from an arbitrary polyhedron and, through repeated
subdivision, converge to a surface that's generally very smooth. In the subdivision
scheme we present in this chapter we can start from any polygonal mesh, but
after subdivision all faces of the mesh become quadrilateral, and after repeated
subdivision most vertices meet exactly four faces. Faces whose vertices all have
valence four can be shown to be the same as cubic spline surfaces, which meet
their neighbors with
C
2
smoothness. At the “exceptional” vertices, where three
or five or six or more faces meet, the surface is generally
C
1
smooth, but it may
have curvature discontinuities. The web material for this chapter describes various
further subdivision schemes and their implementation.
→
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