^_^physicist
Jan30-07, 12:12 AM
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.
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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).
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).