Previous: Evolutes


Osculating Circles

Given a plane curve, we want to find the circle which best approximates the curve in a neighborhood of a point t{_0}. A circle is determined by three points, so we choose three points on the curve which are near the point in question. If the three points are not collinear, they determine a circle, with center of a circle given by the intersection of the perpendicular bisectors of any two pairs of them. If we take the limit as the points approach to, these lines perpendicular to chords of the circle will approach the normal vectors to the curve near the point. The circle we obtain as the limit will have its center a distance 1kg along the normal vector to the curve at to. This circle is the osculating circle, derived from the Latin for "kiss". Its radius is known as the radius of curvature and the inverse of the radius of curvature is the curvature.

Demonstration 3: Osculating Circles (demo under construction)
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Note that for any piece of curve which has monotonically increasing curvature (as the exponential spiral does), the osculating circles of lesser curvature contain the osculating circles of greater curvature.

This demo shows a series of osculating circles of a curve.

The centers of the osculating circles trace out the evolute of the curve as the parameter runs through its domain.

The Evolute

Enter a curve and its domain. Then hit play on the tape deck and watch the center of the osculating circles trace out the evolute. Note that the demo will draw the evolutes of some curves, such as X(t) = (t, t{^3}) with singularities. Sometimes they are removable. Try to interpolate.

This demo shows the evolute being traced out for any given curve.

    Type in a curve with an inflection point. Watch the osculating circle grow large on one side and then grow small again on the other side as the point is passed. Relate this phenomenon to the sign of the curvature.


Next: The Tangential Image