Involute and Evolute Proof (Differential Geometry )

[^_^physicist]

Homework Statement

Suppose that a(s) is a unit speed curve.

A) If B(s) is an involute of a (not necessarily unit speed), prove that B(s)=
a(s)+ (c-s)*T(s), where c is a constant and T= a'

B) Under what conditions is a(s) + (c-s)T(s) a regular curve and hence an involute of "a."

Homework Equations

T is the tangent vector field to a(s), such that, T= d(a)/ds, and for this relation to hold "a" must be a function of arc length.

Regular curve: In R^3, is a function a:(a,b)->R^3 which is of class C^k for some k (greater than or equal to) 1, and for which da/dt does not equal zero for all t in (a,b).

<a,b>= dot product of a and b

The Attempt at a Solution

Ok for part A)

Let B = a(s) +(c-s)T(s), where T(s)=a'. Than B is the involute of a(s) by defination, if <B,T>= 0.

<a(s) + (c-s)*T(s), T(s)>=0
Let s=c

Than===> <a(c), T(c)>=0.

Since a(c) and T(c), are by defination perpendicular to one another, thus, the statement holds.

--------

However, I am pretty sure this is wrong. In fact I am positive that I must have made a mistake in my proof, because after the problem the text mentons that I should have found from the proof that |B-a| is a measure of arclength. Which I need for the next problem. So somewhere I am getting this all screwed up.

Also, for part B), I put down that B(s) must have s= c for B to be the involute of "a." Which I have a feeling is incorrect (I couldn't even come up with a reason as to why).

 

[mighty2000]

Hello! I have a few suggestions for your proof and some ideas for part B).

For part A), instead of using the dot product, try using the fact that the tangent vector T(s) is perpendicular to the curve a(s) (since a(s) is a unit speed curve). Then you can use the Pythagorean theorem to show that |B-a| is a measure of arclength.

For part B), instead of setting s=c, try considering the conditions under which the curve B(s) is regular. Remember that a regular curve is defined as a curve whose tangent vector is never equal to 0. Can you think of a way to show that B(s) is regular for certain values of c?