 Contact (mathematics)

"Osculation" redirects here. For other meanings, see Kiss.
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 kth 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 odddimensional manifolds; see contact geometry. Contact transformations are related changes of coordinates, 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 (A_{0}: crossing, A_{1}: tangent, A_{2}: osculating, ...) and the umbilic or Dseries where there is a high degree of contact with the sphere.
Contents
Contact between curves
Two curves in the plane intersecting at a point p are said to have:
 1point contact if the curves have a simple crossing (not tangent).
 2point contact if the two curves are tangent.
 3point contact if the curvatures of the curves are equal. Such curves are said to be osculating.
 4point contact if the derivatives of the curvature are equal.
 5point contact if the second derivatives of the curvature are equal.
Contact between a curve and a circle
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 4point 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 fourvertex theorem).
In general a curve will not have 5point with any circle. However, 5point contact can occur generically^{[disambiguation needed ]} in a 1parameter 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.
Bitangents
It is also possible to consider circles which have two point contact with two points S(t_{1}), S(t_{2}) on the curve. Such circles are bitangent circles. The centers of all bitangent circles form the symmetry set. The medial axis is a sub set of the symmetry set. These sets have been used as a method of characterising the shapes of biological objects.
References
 Bruce, J. W.; P.J. Giblin (1992). Curves and Singularities. Cambridge. ISBN 0521429994.
 Ian R. Porteous (2001) Geometric Differentiation, pp 152–7, Cambridge University Press ISBN 0521002648 .
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