Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Involute and Evolute Proof (Differential Geometry please help)

  1. Jan 29, 2007 #1
    1. The problem statement, all variables and given/known data
    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."

    2. Relevant 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
    3. 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).
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?