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).