Graphics Reference
In-Depth Information
Note that even if the original curve p(t) is a regular curve, the parallel curve p d (t)
may not be. Therefore because of their importance in CAGD we shall analyze their
singularities and other special points. First of all, there may be cusps at those points
where
() =
1
+
k td
0
.
Definition.
A point on the parallel curve p d (t), d π 0, is called a (ordinary) cusp if
() =- 1
k t
d
and k¢(t) π 0. A parallel curve without cusps is called nondegenerate .
Definition.
A point on the parallel curve p d (t) is called an extraordinary cusp if
() =- 1
k t
d
and k¢(t) = 0 and k≤(t) π 0.
One can show that the curvature of the parallel curve goes to infinity as we
approach an extraordinary cusp.
9.7.1. Theorem. The cusps of the parallel curve p d (t), d π 0, lie on the evolute of
p(t) and meet that curve orthogonally. The extraordinary cusps of the parallel curve
coincide with the cusps of the evolute.
Proof. See Figure 9.15. The first part is clear from the definition of the plane evolute.
See [FarN90a] for a proof of the second.
Now cusps and extraordinary cusps of a parallel curve correspond to differentia-
bility discontinuities that did not exist on the original curve. There are other special
points that do not correspond to discontinuities. One of these is an inflection point.
This is where the curvature vanishes. We can find such points for a parallel curve from
Equation (9.33). Another special point is a vertex of a curve. Recall that a vertex on
a curve is a place where the derivative of the curvature function vanishes. Therefore,
evolute
Figure 9.15.
Cusps of parallel curves lie on evolute.
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