Proving Geodesic Curvature of Curve in Surface is Equal to Projection

[murmillo]

Homework Statement

Show that the geodesic curvature of an oriented curve C in S at a point p in C is equal to the curvature of the plane curve obtained by projecting C onto the tangent plane along the normal to the surface at p.

Homework Equations

Meusnier's theorem, and k^2 = (k_g)^2 + (k_n)^2

The Attempt at a Solution

The proposition makes sense. It's basically saying that the geodesic curvature is what's left after you take out the effect of how the surface is curved in the ambient space. But how to prove it?
I used Meusnier's theorem to get that the normal curvatures of both curves are the same. But I don't know what to do from there. Any help?

 

[mighty2000]

Thank you for your question. To prove the given proposition, we can use the fact that the geodesic curvature is defined as the curvature of the curve obtained by projecting C onto the tangent plane along the normal to the surface at p. So, we can rewrite the proposition as follows:

The geodesic curvature of C at p is equal to the curvature of the projected curve on the tangent plane at p.

Now, let us consider a point p on C and its projection onto the tangent plane. By Meusnier's theorem, we know that the normal curvatures of both curves are the same. This means that the curvature of the projected curve on the tangent plane is equal to the curvature of the original curve at p.

But, we also know that the geodesic curvature is the curvature of the projected curve on the tangent plane. Therefore, we can conclude that the geodesic curvature of C at p is equal to the curvature of the projected curve on the tangent plane at p. This proves the given proposition.

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