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Finding the Equation of the Osculating Plane to a Curve Given As Parametric Coordinates

The osculating plane at a point on a curve is the plane containing the tangent vector and the principal normal to the curve at that point.

To find the equation of the osculating plane to the curveat the point(where), take a pointin the plane thenis a vector in the obfuscating plane.

A tangent vector to the curve is given by

so at

Sinceandis normal ,

Hence

Sinceandare in the plane,is perpendicular to the plne so which simplifies to