Graphics Reference
In-Depth Information
9.11
Envelopes of Surfaces
Although the concept of envelopes of surfaces was often used by geometers it is
actually tricky to define carefully. We shall simply extend the definition given in the
case of curves.
Let a t : [0,1] Æ R 3 be a one-parameter family of surfaces defined by
Definition.
(
) =
(
)
a
t uv
,
a
uvt
, ,
for some C function
[] ¥ [] ¥ [ Æ R
3
a :
01
,
01
,
01
,
.
An envelope of this family is defined to be a surface p(u,v) that is not a member of
this family but that is tangent to some member of the family at every point.
Spivak ([Spiv75]) discusses the envelope of a family of planes. He shows that “in
general” a one-parameter family of planes has an envelope that is either a generalized
cylinder, a generalized cone, or the tangent surface of a curve. He also explains how
this led to a definition of parallel translation along a Riemannian manifold.
9.12
Canal Surfaces
This section is on a special type of envelope surface that is relevant to CAGD.
Definition. A canal surface is the envelope of a one-parameter family of spheres S(t).
If g(t) is the center of the sphere S(t), then the curve g(t) is called the center curve of
the canal surface. The function r(t), where r(t) is the radius of the sphere S(t), is called
the radius function of the canal surface. A canal surface whose radius function is con-
stant is called a tube surface .
Canal surfaces were first defined and studied by Gaspard Monge. If the center
curve for a canal surface is a straight line, then we get a surface of revolution. In
general, canal surfaces are a type of “sweep” surface. They are the boundary of the
solid that one gets by sweeping a sphere along a curve.
9.12.1. Lemma. If S(t) is the one-parameter family of spheres that defines a canal
surface S , then S(t) « S is a circle for every t.
Proof. This follows from the fact that S(t) « S is the limit of the intersections
S(t -d) « S(t +d) as d approaches 0.
Definition. The circles S(t) « S in Lemma 9.12.1 are called the characteristic circles
of the canal surface.
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