Inverting a Point in the Osculating Circles of a Curve

Just as the tangent to a curve at a point is the best linear fit there, so the osculating circle is the best circular fit. At that point, the tangents to the osculating circle and the curve coincide, and the curvatures of the curve and the osculating circle are equal.

Inverting a point in a circle amounts to reflecting it in the circle thought of as a mirror, with distortion inversely proportional to the radius. Thus, for a point , the inversion point with respect to a circle of radius centered at is defined by . Inversion maps the interior of a circle to its exterior and vice versa.

In this Demonstration, the red curve is the set of inversions of the locator point in the osculating circles of the blue curve (four choices). You can drag the locator.

You can vary a green osculating circle and the dark red point that is the inverse of the locator.