Graphics Reference
In-Depth Information
dimensional examples of these operations. Clearly, shelling can be defined in terms of
standard offset operations. Forsyth describes how to define such offsetting operations
for the boundary representation of solids.
14.10
Envelopes
Envelopes are spaces that are a generalization of offset curves or surfaces. These
spaces arise from the boundary of regions swept out by moving parts of machinery.
Understanding the geometry of envelopes is therefore important in the design of
machinery and its operation, such as in the case of NC machines or robots. In par-
ticular, it is relevant in making sure that there is adequate clearance of these parts in
the work environment. Envelopes have mostly been studied when one is sweeping
circles, planes, and spheres. Even there the analysis can get very tricky. A discussion
of general envelopes can be found in [Brec92].
Here is a definition for envelopes in the plane.
Definition.
Let a
t
: [0,1] Æ
R
2
be a one-parameter family of curves in the plane
defined by a
t
(u) =a(u,t) for some C
•
function a: [0,1] ¥ [0,1] Æ
R
2
. An
envelope
of
this family is defined to be a curve p(u) that is
not
a member of this family but which
is tangent to some member of the family at every point.
Figure 14.43(a) shows a nicely structured envelope. Figure 14.43(b) shows the
envelope of normals to an ellipse whose ends are the centers of the osculating circles.
In other words, an envelope can have bad singularities even if we start with nice
functions.
One approach to studying the envelope p(u) is to think of p(u) as being the limit
as e approaches 0 of the intersections of p(u) and p(u+e). Although this has serious
problems in general, it seems to work in many cases of interest.
Figure 14.43.
Envelopes of curves and normals.
