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DIFFERENTIAL GEOMETRY COURSE

8. *EVOLVENT AND EVOLUTE*

*Definition 76:* Let *C* be a curve. Let *K* be a curve such that the tangents to *C* are normals of *K*.
Then we call *K* an evolvent of *C*, and *C* an evolute of *K*.

*Proposition 77:* Every curve has ∞^{1} evolvents and ∞^{1} evolutes.

*Proof :* Use a parametrisation of *C* according to arc length: __x__(s). Then an evolvent *K* has parametrisation
__y__(s)=__x__(s)+λ(s)__t__(s)
(s is not arc length of *K*).

Now __t__(s) must be perpendicular to __y__ '(s), where
__y__ '(s) = __x__^{ .}(s) + λ^{ .}(s)__t__(s) + λ(s)κ(s)__n__(s) =
(1 + λ^{ .}(s))__t__(s) + λ(s)κ(s)__n__(s).

So 1 + λ^{ .}(s) = 0 and s + λ(s) = c. Then we find for each constant c an evolvent *K*_{c} with parametrisation
__y__(s) = __x__(s) + (c-s)__t__(s).

On the other hand, use a parametrisation *K* according to arc length: __x__(s). A normal vector has the form __n__(s)cos(φ(s)) + __b__(s)sin(φ(s)).

Then a corresponding evolute *C* has the following parametrisation (where s is not arc length):
__y__(s)=__x__(s)+λ(s)(__n__(s)cos(φ(s)) + __b__(s)sin(φ(s))).

Now __y__ '(s) must be a scalar multiple of __n__(s)cos(φ(s)) + __b__(s)sin(φ(s)), where

__y__ '(s) = __t__ +
λ^{ .}(__n__ cos(φ) + __b__ sin(φ)) + λ(__n__^{ .}cos(φ) + __b__^{ .}sin(φ)
- __n__ sin(φ)φ^{ .} + __b__ cos(φ)φ^{ .}) =

__t__(1-λκ cos(φ)) + __n__(λ^{ .}cos(φ) - λτ sin(φ) - λ sin(φ)φ^{ .}) +
__b__(λ^{ .}sin(φ) + λτ cos(φ) + λ cos(φ)φ^{ .}).

So the following must hold: first, λ = (κ cos(φ))^{-1}, and, second,

Hence φ^{ .} = - τ, so φ = φ_{o} - ∫_{0}^{s} τ(u) du.

With each choice of φ_{o} we find an evolute *C*.

*Problem 78:*

Show that we get the normals of a curve *K* corresponding to an evolute *C* from the normals of another evolute *C'* by rotating each of them
in its normal plane over a fixed angle.

Furthermore, show that the contact point __y__(s) of the evolute *C* lies on the curvature axis of the point __x__(s) of *K*.

Finally show that the principal normals of a curve *K* are the tangents to a curve *C* (so that *C* is the "envelope" of these tangents) if and only if τ=0, so if *K* is planar.
Then *C* is the locus of the centers of curvature of *K*.

*Problem 79:* Show that the tangents to a circular helix intersect each plane perpendicular to the axis in the points of an evolvent of the circle that is the intersection of this plane
and the cilinder on which the helix is lying.

*Problem 80:* Given a cycloid (see 18, 28), determine its evolute, and show this is also a cycloid.

answers

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