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. See for example osculating circle and osculating orbit.
Contact forms are particular differential forms of degree 1 on odd-dimensional manifolds; see contact geometry. Contact transformations are related changes of co-ordinates, of importance in classical mechanics. See also Legendre transformation.
Contact between manifolds is often studied in singularity theory, where the type of contact are classified, these include the A series (A0: crossing, A1: tangent, A2: osculating, ...) and the umbilic or D-series where there is a high degree of contact with the sphere.
For a smooth curve S in the plane then for each point, S(t) on the curve then there is always exactly one osculating circle which has radius 1/κ(t) where κ(t) is the curvature of the curve at t. If the curve has zero curvature (i.e. an inflection point on the curve) then the osculating circle will be a straight line. The set of the centers of all the osculating circles form the evolute of the curve.
If the derivative of curvature κ'(t) is zero, then the osculating circle will have 4-point contact and the curve is said to have a vertex. The evolute will have a cusp at the center of the circle. The sign of the second derivative of curvature determines whether the curve has a local minimum or maximum of curvature. All closed curves will have at least four vertices, two minima and two maxima (the four-vertex theorem).
In general a curve will not have 5-point with any circle. However, 5-point contact can occur generically in a 1-parameter family of curves, where two vertices (one maximum and one minimum) come together and annihilate. At such points the second derivative of curvature will be zero.