TNB Frame, Osculating Planes and Circles

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TNB Frame

At any point of a regularly parametrized curve, the unit tangent vector, the unit normal vector and the unit binormal vector form a local coordinate frame called the TNB frame:

Try it yourself

Osculating plane and circle

We already know that a regularly parametrized curve can be locally approximated by a line called the tangent line. At points where the curvature is not zero, we can get even better approximation by a circle. The circle that provides the best approximation of the curve at a given point is called the osculating circle of the curve at that point:

An example of TNB frame and osculating circle:

The following video shows an animation which visualizes a TNB frame and osculating circle moving along a curve:

The following illustration shows the curve from the above video with several of the TNB frames: