Osculating plane and circle, Multivariable Calculus
For a parametric curve r(t), we find the osculating plane, radius of the osculating circle, and center of the osculating circle at t=0 using just the position vector r(0), the velocity vector r'(0), and the acceleration vector r''(0).
Here is the formula at the end:
Center = r + ||r'||^4/||r' x r''||^2 * (r'' - projection of r'' onto r').
You may also enjoy this two-part computation of an osculating plane and the osculating circle, using the TNB vectors:
Plane: • Example finding the osculating plane ...
Circle: • Example finding the osculating circle...
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