Orientation preserving reparameterization

  • #1
This is from O'Neil's differential geometry. I'm having trouble parsing through the problem/hint.

Given any curve a that does not pass through the origin has an orientation-preserving reparameterization in the polar form:

a(t) = (r(t)*Cos(V(t)),r(r)*Sin(V(t)))

where r(t)= ||a(t)||

The hint has us reference an older problem. I was wondering if someone could give me a push on how to start proving this.
 

Answers and Replies

  • #2
336
0
Its just switching to polar form with r(t)= sqrt(x^2 + y^2).
 
  • #3
http://tinypic.com/r/xqf446/7
^^This is the hint I mentioned.

http://tinypic.com/r/24101at/7
^^This is the problem its referencing.

I'm having trouble figuring out what it wants me to do for part (a).

When it mentions denoting a f and g as follows:

f = U_1.(a/||a||) = (1,0).(a/||a||) = Cos(V(t))
g = U_2.(a/||a||) = ... = Sin(V(t))

I'm not sure where I should go after that.
 

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