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
| y^((k))(x_0)=f^((k))(x_0) |
for k=0, 1, 2. The point of tangency is called a tacnode.
OsculatingCurves
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 CurvesExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Osculating Curves." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/OsculatingCurves.html