Blaga P. Lectures on the differential geometry of - tiera.ru

90 Chapter 2. Plane curves
whence
while
Figure 2.4: An involute of a circle
Jρ ′ = −(c − s)k±[r](s) · r ′ (s),
ρ ′′ (s) · Jρ ′ (s) = (c − s) 2 · (k±[r](s)) 3 .
The conclusion follows now from the definition of the signed curvature. �
The following theorem provides a connection between the involute and the evolute.
In many textbooks this connection is taken, in fact, as the definition of the involute.
Theorem. Let (I, r = r(s)) be a naturally parameterized curve and ρ – its involute with
the origin at c ∈ I. The the evolute of ρ is r.
Proof. The evolute of ρ is given, as known, by the equation
ρ 1(s) = ρ(s) +
1
k±[ρ](s) · Jρ′ (s)
�ρ ′ (s)� .
Using the previous lemma to express the signed curvature of ρ as a function of the signed
curvature of r, we get
ρ 1(s) = r(s) + (c − s)r ′ (s) +
|c − s|
sgn(k±[r](s)) · (c − s)k±[r](s) · J2r ′ (s)
�(c − s)k±[r](s) · Jr ′ = r(s).
(s)�
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