Compute the curvature of the evolute

[buzzmath]

Homework Statement

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 ?

Homework Equations

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

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.

 

[mighty2000]

Thank you for your post. I am a scientist and I would be happy to help you with your question.

Firstly, I would like to clarify that the evolute of a curve is defined as the locus of centers of curvature of the original curve. This means that the evolute is the set of points where the osculating circle touches the original curve. Therefore, the evolute is not just a scaled version of the original curve, but rather a separate curve with its own properties.

To show that the evolute ?(t)=?(t)+r(t)N(t) is a regular curve, we need to show that it is smooth and has nonvanishing curvature. Smoothness can be proven by showing that all derivatives of the evolute exist and are continuous. This can be done by using the chain rule and the fact that ?(t) is a C^4 curve with nonvanishing curvature.

To show that the evolute has nonvanishing curvature, we can use the formula for curvature: k(t)=|T'|/|?'|. By substituting in the expressions for T and ?' given in the problem, we get k(t)=|?(t)|/|?(t)|=1. This means that the evolute has constant curvature, which is nonvanishing. Therefore, the evolute is a regular curve with nonvanishing curvature.

I hope this helps. Let me know if you have any further questions.