Graphics Reference
In-Depth Information
the other hand,
C
3
and
C
4
have cusps and
C
4
has self-intersections on top of that. The
offset curve may also intersect the original curve. Equation (14.29) points out part of
the problem. Even though p(u) may be a regular curve, p
d
(u) will not be at those points
where
()
=
1
+
k ud
0
.
(14.31)
There may be cusps at those points. This is only a symptom of a bigger problem,
which is that a curve and its offset curve usually have different analytic types. For
example, the offsets of rational curves are typically not themselves rational curves.
One often wants to talk about offset curves even when the curve is not differen-
tiable everywhere and the normal n(u) may not be defined. See Figure 14.35(a) or
14.36(a), where we are dealing with a piecewise differentiable curve. One can deal
with such problems in the case of simple isolated singularities like cusps or corners.
After this litany of problems with offset curves, let us look at approaches to dealing
with them. The idea is to subdivide the domain of the curve into segments over which
it and the offset behave nicely. The breakpoints of the subdivision correspond to
special points on the curve for which there are tests. The points to look for are:
(1) Ordinary cusp of offset:
Defined by equation (14.31) and the condition
k¢(u) π 0
(2) Extraordinary cusp of offset:
Defined by equation (14.29) and the conditions
k¢(u) = k≤(u) = 0
(3) Turning point:
A point where p¢(u) is either vertical or
horizontal, that is, either x¢(u) = 0, y¢(u) π 0,
or x¢(u) π 0, y¢(u) = 0.
(4) Inflection point:
A point where k(t) = 0, that is, the curve is
locally flat.
(5) Vertex:
A point where dk/ds = 0, that is, k has a local
extremum.
The planar evolute of the curve is relevant here because one can show that the cusps
of an offset lie on the evolute of the original curve (Theorem 9.7.1 in [AgoM05]). The
planar evolute is the locus of the centers of curvature and is defined by
1
()
=
()
+
()
()
qu
pu
nu
(14.32)
k
u
The special points defined by (3-5) above (called
characteristic points
in [FarN90a])
are “intrinsic” properties of curves because they involve solving for zeros of p¢(u), k(u),
k¢(u), k≤(u), etc. These points are interesting on
both
curves, but one can show that
the turning points, inflection points, and vertices of the offset curve correspond to
those of the original curve, except that if k(t) =-1/d at a turning point or vertex of
p(u), then the corresponding point on p
d
(u) is a cusp or an extraordinary cusp. See
Theorem 9.7.2 in [AgoM05].
After the special points (1-5) above have been found one has a segmentation of
the curve and its offset. The curve can then be approximated over each primitive
