Osculating curve

File:Osculating circle.svg

A curve Template:Mvar containing a point Template:Mvar where the radius of curvature equals Template:Mvar, together with the tangent line and the osculating circle touching Template:Mvar at Template:Mvar

In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve. That is, if Template:Mvar is a family of smooth curves, Template:Mvar is a smooth curve (not in general belonging to Template:Mvar), and Template:Mvar is a point on Template:Mvar, then an osculating curve from Template:Mvar at Template:Mvar is a curve from Template:Mvar that passes through Template:Mvar and has as many of its derivatives (in succession, from the first derivative) at Template:Mvar equal to the derivatives of Template:Mvar as possible.

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.

Examples

File:Nested Ellipses.svg

Osculating ellipses – The spiral is not drawn: we see it as the locus of points where the ellipses are especially close to each other.

Examples of osculating curves of different orders include:

• The tangent line to a curve Template:Mvar at a point Template:Mvar, the osculating curve from the family of straight lines. The tangent line shares its first derivative (slope) with Template:Mvar and therefore has first-order contact with Template:Mvar.
• The osculating circle to Template:Mvar at Template:Mvar, the osculating curve from the family of circles. The osculating circle shares both its first and second derivatives (equivalently, its slope and curvature) with Template:Mvar.
• The osculating parabola to Template:Mvar at Template:Mvar, the osculating curve from the family of parabolas, has third order contact with Template:Mvar.
• The osculating conic to Template:Mvar at Template:Mvar, the osculating curve from the family of conic sections, has fourth order contact with Template:Mvar.

Generalizations

The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an osculating plane to a space curve is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.

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.

See also

• Osculating orbit