Osculating curve

[1][2] The term derives from the Latinate root "osculate", to kiss, because the two curves contact one another in a more intimate way than simple tangency.

[3] Examples of osculating curves of different orders include: The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces.

This is as high an order as is possible in the general case.

[5] In one dimension, analytic curves are said to osculate at a point if they share the first three terms of their Taylor expansion about that point.

This concept can be generalized to superosculation, in which two curves share more than the first three terms of their Taylor expansion.

A curve C containing a point P where the radius of curvature equals r , together with the tangent line and the osculating circle touching C at P
Osculating ellipses – The spiral is not drawn: we see it as the locus of points where the ellipses are especially close to each other.