Graphics Reference
In-Depth Information
to find the vertices of a parallel curve we need a formula for the derivative of its cur-
vature function. Use equations (9.3) and (9.33) to get
d
ds
k
d
dt
k
dt
ds
1
d
dt
k
k
Ê
Ë
ˆ
¯
d
d
=
=
.
1
+
d
¢
()
pt
d
This and (9.32) leads to the equation
d
ds
k
1
d
ds
k
d
d
=
.
(9.34)
3
(
)
1
+
k
d
For the final special point, let p(t) = (x(t),y(t)) be a regular curve in the plane.
Definition.
A
turning point
for p(t) is a point where p¢(t) is either vertical or hori-
zontal, that is, either x¢(t) = 0, y¢(t) π 0, or x¢(t) π 0, y¢(t) = 0.
Turning points of a parallel curve are easily found from equation (9.32).
9.7.2. Theorem.
The turning points, inflection points, and vertices of the parallel
curve p
d
(t) correspond to those of the original curve p(t) (except when k(t) =-1/d at
a turning point or vertex of p(t), in which case the corresponding point on p
d
(t) is
then a cusp or an extraordinary point).
Proof.
The theorem follows from equations (9.32)-(9.34).
Turning points depend on the coordinate system. On the other hand, inflection
points and vertices are intrinsic properties of the curve. All three are invariant under
regular reparameterizations.
We finish this section by listing some facts about length and area for parallel
curves.
9.7.3. Theorem.
If p(t), t Π[0,1], is a regular plane curve of total length L, then the
total length L
d
of the nondegenerate parallel curve p
d
(t) is given by
LLd
d
=+Dq,
where Dq is the total angle of rotation of the principal normal n(t) of p(t). If the curve
corresponding to p(t) is closed and convex, then
LL d
d
=+2p
.
Proof.
See [FarN90a] for the first part.
9.7.4. Theorem.
If p(t), t Π[0,1], is a regular plane curve of total length L, then the
area A between p(t) and the nondegenerate parallel curve p
d
(t) is given by
1
2
(
)
ALLd
=+
,
d
