Curvature and the Shape Operator

The Osculating Circle

Now is your chance to play with the osculating circle. Below is the graph of a fourth-degree polynomial, a currently selected point, and the relevant osculating circle. Use the slider to move the current point along the whole range of the graph. Watch the size of the circle grow/shrink and note how the circle "switches" sides of the function, depending on whether the curvature is positive or negative.

Current curvature at x = :

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