Curvature
Implicit curve

For a curve defined by an implicit equation F (x, y) = 0 with partial derivatives denoted F, F, F, F, F,the curvature is given by

The signed curvature is not defined, as it depends on an orientation of the curve that is not provided by the implicit equation. Note that changing F into – F would not change the curve defined by F (x, y) = 0, but it would change the sign of the numerator if the absolute value were omitted in the preceding formula.

A point of the curve where F = F = 0 is a singular point, which means that the curve is not differentiable at this point, and thus that the curvature is not defined (most often, the point is either a crossing point or a cusp).

The above formula for the curvature can be derived from the expression of the curvature of the graph of a function by using the implicit function theorem and the fact that, on such a curve, one has
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