Graphics Reference
In-Depth Information
curves carries over to surfaces. Although parameterizations typically have rectangu-
lar domains, there are times when triangular domains are more convenient and
Section 12.12.3 discusses those. Rational B-spline and NURBS surfaces are defined
in Section 12.12.4 and efficient evaluation algorithms for both B-spline and NURBS
surfaces are discussed in Section 12.12.5. Section 12.12.6 is on interpolation using B-
spline surfaces. Section 12.13 defines the very special cyclide surfaces. Sections
12.14-12.16 revisit, in the context of surfaces, some of the topics we encountered with
curves in Sections 11.8-11.12. We discuss the subdivision of surfaces into smaller
patches, the addition of control points and knots, composite surfaces, and fairing sur-
faces. Next, in Section 12.17, we switch from smooth surfaces and describe the class
of polygonal surfaces defined by recursive subdivision. Section 12.18 gives a summary
of some of the main points to remember when it comes to curves and surfaces and
we finish with a few historical comments in Section 12.19.
12.2
Surfaces of Revolution
Surfaces of revolution are a frequently encountered type of surface. Spheres and cylin-
ders can be thought of as surfaces of revolution. In general, one gets an “object of
revolution” by revolving a set about some arbitrary axis. To analyze what this means
in more detail, consider the simplest object to revolve, namely, a point. In that case
one gets a circle in a plane orthogonal to the axis with center on the axis and radius
equal to the distance of the point to the axis. It follows that one can think of an object
of revolution as consisting of a union of circles centered on the axis, one for each
point of the object being revolved. This also suggests that a way to parameterize a
point
p
of an object gotten by revolving a curve about an axis is to use two parame-
ters. One parameter is the parameter of the point on the curve, which gave rise to
p
and the other is the angle through which it was rotated. For general objects of revo-
lution we would need k + 1 parameters, where k is the number of parameters needed
to parameterize the object being revolved.
Our actual definition of a surface of revolution, which will be in terms of a para-
meterization, will restrict itself to the case where a curve is being revolved about the
x-axis. This will simplify the definition. Besides, one can get surfaces of revolution
about an arbitrary axis from this using rigid motions. Another simplifying hypo-
thesis will be to assume that the curve lies in the x-y plane. Extending the definition
to allow arbitrary space curves is left as an exercise for the reader (see Exercise
12.2.1.). One should note however that although it is easy to define an “object” of
revolution, it is not easy to guarantee that the result will be a surface. We shall see
that even in the special cases we shall analyze it is not trivial to ensure that the result
will not have any singularities.
Definition.
Let g : [a,b] Æ
R
2
be a planar parametric curve and let g(t) = (g
1
(t),g
2
(t)).
Define a function
[
]
¥
[
]
Æ
R
3
p : a,b
cd
,
