Evolute of an Ellipse

The evolute of a curve is the locus of all its centers of curvature. For an ellipse, the radius of curvature changes as you move along the curve, being smallest at the major vertices and largest at the minor vertices. This variation traces out an astroid-like shape known as the evolute.

Try the evolute experiment

Open the interactive Evolute of an Ellipse module to see the osculating circle and its center change dynamically as you trace the ellipse.

1. Set Horizontal semiaxis a to 3 and Vertical semiaxis b to 2. Compare the ellipse and its four-cusp evolute. A center of curvature can lie outside the ellipse; at Angle theta / pi = 0.5 the center is (0, -2.5).
2. Change Angle theta / pi from 0 to 0.5. At θ = 0 (major axis vertex), the osculating circle is small and fits snugly inside the narrow end. At Angle theta / pi = 0.5, or θ = π/2 (minor axis vertex), the circle is much larger, matching the flatter top edge of the ellipse.
3. Observe how the center C traces the rose-colored evolute curve. As θ moves from 0 to 0.5, C travels from the cusp on the positive x-axis to the cusp on the negative y-axis.
4. Set Horizontal semiaxis a to 2 and Vertical semiaxis b to 2. The ellipse becomes a circle. The radius of curvature is constantly 2, and the center of curvature C collapses to a single point at the origin (0, 0).