Example finding the osculating circle, Multivariable Calculus

For a parametric curve r(t) with given vector information, we find the radius, the center, and a parametric description for the osculating circle at a given point on the circle. This is this circle most tangent to the curve at that point, with radius 1/curvature. This is the sequel to this video:    • Example finding the osculating plane ...  , although you do not have to watch that one first if you want to just start with the given information.

We visualize the geometry in MATLAB, and then discuss in general terms why the parametric description gives us the correct circle. Throughout, I like to emphasize how the unit length tangent, normal, and binormal vectors (TNB) create an orthogonal "frame" of vectors which is designed for the geometry of the curve. This is the Frenet frame.

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