How to find the standard format for curvature when given x=x_1(t)e_1 + x_2(t)e_2 as the curve?

Question

I have to show that, along the plane curve x=x_1(t)e_1 + x_2(t)e_2, that the curvature is

k = | x_1' x_2'' - x_2'x_1''| / [|(x_1')^2 + (x_2')^2|^(3/2) ]

The formula I've got for curvature is

||T’(t)|| / ||r’(t)||

But how do I get T(t) and r(t) with just x(t)?

Answer 1

See the source. It is all explained and worked out.

As for getting stuff from just x(t). You really have two parametric equations ( I am taking e_1 and e_2 to be vectors defining a two dimensional, orthogonal coordinate system). These are:

In e_1 direction: x1 = x_1(t)
In e_2 direction: x2 = x_2(t)

And the two axes are x1 (unit vector e_1) and x2 (unit vector e_2).

Answer 2

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