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 curve
at the point
(where
), take a point
in the plane then
is a vector in the obfuscating plane.
A tangent vector to the curve is given by
so at
Since
and
is normal ,
Hence
Sinceand
are in the plane,
is perpendicular to the plne so
which simplifies to
