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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
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