Osculating Curves

A curve y(x) is osculating to f(x) at x_0 if it is tangent at x_0 and has the same curvature there. Osculating curves therefore satisfy


for k=0, 1, 2. The point of tangency is called a tacnode.


One of simplest examples of a pairs of osculating curves is x^2 and x^2-x^4, which osculate at the point x_0=0 since for k=0, 1, 2, y^((k))(0)=f^((k))(0) is equal to 0, 0, and 2.

See also

Osculating Circle, Tacnode, Tangent Curves

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Cite this as:

Weisstein, Eric W. "Osculating Curves." From MathWorld--A Wolfram Web Resource.

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