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
Demonstration 3: Osculating Circles (demo under construction) 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 EvoluteEnter 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. |