Graphics Reference
In-Depth Information
Some people tend to see “bands” near such discontinuities; the effect is called
Mach banding
(see Section 1.7).
If we use piecewise linear interpolation in animation, having computed the
positions of objects at certain “key” times, then between these times objects move
with constant velocities, and hence zero acceleration; all the acceleration is con-
centrated in the key moments. This can be very distracting.
(a)
If we have a polyhedral shape with nontriangular faces, we can triangulate each
face to get a triangle mesh. Then we can, as before, interpolate function values at
the vertices over the triangular faces. But the results of this interpolation can vary
wildly depending on the particular triangulation. It's easiest to see this with a very
simple example (see Figure 9.9) in which a function defined on the corners of a
square is extended to the interior in two different ways. The results are evidently
triangulation-dependent.
(b)
As we hinted above, taking function values at the vertices of a mesh and trying
to find
smoothly
interpolated values over the interior of the mesh is a difficult
task. Part of the difficulty arises in defining what it means to be a smooth function
on a mesh. If the mesh happens to lie in the
xy
-plane, it's easy enough: We can
use the ordinary definition of smoothness (existence of various derivatives) on the
plane. But when the mesh is simply a polyhedral surface in 3-space (e.g., like a
dodecahedron), it's no longer clear how to measure smoothness.
Of course, if we replace the dodecahedron with the sphere that passes through
its vertices, then defining smoothness is once again relatively easy. Each point
of the dodecahedron corresponds to a point on the surrounding sphere (e.g., by
radial projection), and we can declare a function that's smooth on the sphere to
be smooth on the dodecahedron as well. Unfortunately, finding a smooth shape
that passes through the vertices of a polyhedron is itself an instance of the exten-
sion problem: We have a function (the
xyz
-coordinates of a point) defined at
each vertex of the mesh; we'd like a function (the
xyz
-coordinates of the smooth-
surface points) that's defined on the interiors of triangles. Such a function is what
a solution to the smooth interpolation problem would give us. Thus, in suggest-
ing that we use a smooth approximating shape, we haven't really simplified the
problem at all.
A partial solution to this is provided by creating a sequence of meshes through
a process called
subdivision
of the original surface. These subdivided meshes
converge, in the limit, to a fairly smooth surface. We'll discuss this further in
Chapter 22.
(c)
Figure 9.9: (a) A square with
heights assigned at the four cor-
ners; (b) one piecewise linear
interpolation of these values; and
(c) a different interpolation of the
same values.
The piecewise linear extension technique works when the values at the vertices
are real numbers; it's easy to extend this to tuples of real numbers (just do the
extension on one coordinate at a time). It's also easy to apply it to other spaces in
which convex combinations, that is,
