Graphics Reference
In-Depth Information
sented by a sequence of points. This is not going to work here, although we will want
finite descriptions since everything is going to be represented in a computer. The two
standard representations are by parameterizations or implicitly via equations. Rela-
tively few objects have a convenient implicit representation, with the conics being the
main exception. Our emphasis will be on parametric curves and surfaces.
How are curves and surfaces defined in practice? This depends on what one is
trying to do. There are three aspects to this problem though—the definition, the imple-
mentation, and the user interface. The underlying mathematical theory is obviously
important but often the actual formulas have been known for a long time and the real
problem is to make it easy for a user to specify them. In other words, it is often the
user interface that is the driving force. Users would like to be able to create and manip-
ulate objects easily in terms of properties relevant to their task and ones they can
understand. One way to think of this is in terms of a black box that has some dials,
one for each property that the user wants to adjust. A large percentage of the papers
on curves and surfaces have to do with finding more intuitive and convenient ways
to define and manipulate the
same
underlying mathematical curve. For example, one
may want to define a piecewise cubic curve by simply specifying some points that
control its shape, or by some tangent vectors, or by conditions on its curvature. It is
the mathematician's task to make this possible
and
efficient.
As one looks over how curves and surfaces are used in geometric modeling one
finds that the subject develops in two directions. Are we trying to model a very precise
object, so that accuracy is paramount, or are we designing shapes in a more rough
outline manner? For example, in the design of the wing of an airplane or the blades
of a turbine one is dealing with analytical models that must be reproduced faithfully
within strict tolerance limits. On the other hand, when designing an automobile body,
this is more intuitive and involves aesthetics. Here the tolerances are not so strict.
Many definitions of curves and surfaces are derived from data-fitting-type problems
and in one sense their study deals with special cases of the following:
The general approximation problem:
Given a fixed collection of functions j
1
, j
2
,..., j
k
,
find coefficients c
i
such that
k
Â
()
=
()
g
x
c
ii
j
x
(11.1)
i
=
1
is an approximation to some “theoretical” function f(
x
). The functions j
i
are often called
primitive
or
basis
functions.
For example, if f(x) is a real-valued function of a real variable, the function j
i
(x) could
be the monomial x
i
, in which case we are simply looking for the polynomial g(x) that
best approximates f(x).
Figure 11.2 depicts the environment in which we are operating. The function f
is thought of as a given function in some large function space
X
over some domain
D
. We are looking for a function g that comes from a certain special linear subspace
A
of
X
defined by the j
i
. The domain
D
could be quite general, so that the variable
x
is not necessarily a real number. Often
D
consists of points in
R
n
. For example, the
functions could be defined on a surface and then the
x
s would be elements of
R
2
. The
function g should also be a “good” approximation. Desirable properties are:
