Curvature

From AoPSWiki

Curvature is a a number associated with every point on each smooth curve that describes "how curvy" the curve is at that point. In particular, the "least curvy" curve is a line, and fittingly lines have zero curvature. For a circle of radius , the curvature at every point is . Intuitively, this grows smaller as grows larger because one must turn much more sharply to follow the path of a circle of small radius than to follow the path of a circle with large radius.

Given a twice-differentiable function , the curvature of the graph of the function at the point is given by the formula

For a curve given in parametric form by the pair , the curvature at a point is This expression is invariant under positive-velocity reparametrizations, that is the curvature is a property of the curve and not the way in which you traverse it.

Curvature of surfaces

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