Contact (mathematics)

In mathematics, two functions have a contact of order k if, at a point P, they have the same value and their first k derivatives are equal.

One speaks also of curves and geometric objects having k-th order contact at a point: this is also called osculation (i.e. kissing), generalising the property of being tangent.

Contact transformations are related changes of coordinates, of importance in classical mechanics.

If the derivative of curvature κ'(t) is zero, then the osculating circle will have 3rd-order contact and the curve is said to have a vertex.

Ccircles which have two-point contact with two points S(t1), S(t2) on a curve are bi-tangent circles.