curvature:

en.wikipedia.org/wiki/Curvature

For parametric equations
K = x'y'' - y'x'' / (x'^2 + y'^2 ) ^(3/2)

This should not be too hard for the computer. It is only

derivatives, but usually messy for the thing that you want to do.

It is easy to graph the cycloid,
x = a(t - sin(t))
y = a(1 - cos(t))

The curvature is
- 1 / ( 4*a * sin(.5*t))

This simplified answer is not so easy to find, the situation will

be quite different (and interesting) for different radian

frequencies say
t1 = 0.5
t2 = 1.25

If there is a beta test on this topic, I would be interested to

know about it.

I would like to suggest so that this application is able:
1. To plot vector-valued functions in 2D & 3D, r(t) = [x(t), y(t), z(t)],
2. To plot vector fields in 2D & 3D, F(x,y,z) = [f(x,y,z), g(x,y,z), h(x,y,z)],
3. To show polar coordinates alternatively instead of Cartesian coordinates (for 2D),
4. To plot 3D functions which expressed in terms of (a) Cartesian (x,y,z), (b) cylindrical (rho, theta, z) and (c) spherical (r, phi, theta) coordinates,
5. To plot the functions of 2 or 3 variables, z = f(x,y), w = f(x,y,z),
6. To plot direction fields which the functions are involved derivatives,
7. To evaluate line integrals, surface integrals and volume integrals, and
8. To customize the areas (or regions, solids) between functions by colors (line color, filled color, gradient color).

These are my suggestions and wish that this application could be better and better.
Thank you very much.