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. 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.

osculating circle | tacnode | tangent curves