Geometry Expressions

Center of Curvature

We generate code for the center of curvature of a curve.

First create a parametric curve with X=f(T), Y=g(T).

Picture 1

Now create a tangent to the curve at parametric location t.

Picture 2

Create the normal.

Construct the envelope of the normal as t varies. This is the evolute curve, the locus of the centers of curvature.

Picture 3

Put a point on the evolute at parametric location t.

Picture 4

Create the circle of curvature.

Picture 5

View the coordinates of the center of curvature.

Picture 6

Copy as Java

Picture 7

And paste into a code editor.

Picture 8

Note we have placeholder generic functions f() and g() and numerical approximations to their first and second deivatives, which appear in the formula for the center of curvature.