3.6: Curvature and Osculating Circles

In section 3.3 we defined the osculating plane at X(t0) to be the plane spanned by the tangent and principal normal vectors, T(t0) and P(t0). We now construct the osculating circle as the circle lying osculating plane that is tangent to X(t) and has radius (1/κ(t)). Locally in the osculating plane, the curve passes just inside of the circle on one side of the point of contact and just outside on the other. The curve swept out by the centers of the osculating circles is the evolute curve to X(t), which can be written as:

E(t) = X(t) + (1/κ(t))P(t)


This demonstration graphs a curve X(t) (in red) and its evolute curve E(t) (in yellow. The curve window also shows the osculating circle at X(t0). Use the tapedeck for t0 in the control panel to show the centers of the osculating circles sweeping out the evolute curve.
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