Curvature and Osculating Circle

Curvature measures how sharply a curve bends at a given point. The osculating circle (from the Latin "osculari" meaning "to kiss") is the circle that best approximates the curve at that point. It shares the same tangent, normal, and curvature as the curve itself. For a parametric curve r(t) = (t, a·t²), we can compute the curvature κ, the radius of curvature R = 1/κ, and the center of the osculating circle explicitly.

Experiment

Follow these steps to experiment with the module.

1. Set Parabola coefficient a to 1 and Curve parameter t to 0. Notice that the radius of curvature at the vertex is 0.5.
2. Change Curve parameter t to 1. Observe how the curvature decreases and the osculating circle grows larger as you move away from the vertex.
3. Set Parabola coefficient a to 0. Observe the straight line limit: curvature drops to zero, and the osculating circle is appropriately undefined as its radius approaches infinity.
4. Set Parabola coefficient a to 0.5 and Curve parameter t to 1. Check point (1, 0.5), radius 2√2 and center (−1, 2.5).
5. Set Parabola coefficient a to 1 and Curve parameter t to −0.5. The center reflects to (0.5, 1.25), while curvature and radius match t = 0.5.
6. Clear Parabola coefficient a. Its error is visible and the old circle/results disappear. Restore 1, pan, and choose Reset view; the input values remain unchanged.

You can try this interactive module yourself: Curvature and Osculating Circle