# Compute the curvature of the evolute

1. Sep 26, 2007

### buzzmath

1. The problem statement, all variables and given/known data
Let ?(t):I?R^2 be a C^4 curve with nonvanishing curvature. Show that is evolute ?(t)=?(t)+r(t)N(t) is a regular curve which also has nonvanishing curvature. where r(t) = 1/k(t) is the radues of curvature of ?

2. Relevant equations
k(t)=|T'|/|?'| T=?'/|?'| N(t)=T'/|T'| and the curvature of a circle is 1/radius

3. The attempt at a solution

I started by seeing if the evulote just traced the curve itself just on a different scale but found this to be wrong. I think the way to solve this is to show that the evolute doesn't have any line segments. So you could assume that it did have a line segment and then consider the three cases of curvature for? it's either constant, increasing, or decreasing. if it's constant then the radius of the osculating circles will be the same and if the evolute makes a line segment that that would mean that ? is a line at that point which is a contradiction. I'm not sure exactly where to go next. I tried also just to compute the curvature of the evolute but I kept messing up with the manipulations and it got real messy and I couldn't get it to work.