In mathematics, contact of order k of functions is an equivalence relation, corresponding to having the same value at a point P and also the same derivatives there, up to order k. The equivalence classes are generally called jets. The point of osculation is also called the double cusp.

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...

Page done by Foringoleite

Source: Wikipedia.org