In mathematics, two

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...

s have a contact of order ''k'' if, at a point ''P'', they have the same value and ''k'' equal

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

s. This is an equivalence relation, whose equivalence classes are generally called jets. The point of osculation is also called the double cusp. Contact is a geometric notion; it can be defined algebraically as a valuation. One speaks also of

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s 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 In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

. (Here the derivatives are considered with respect to arc length.) An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a

tangent line In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...

is an osculating curve from the family of lines, and has first-order contact with the given curve; an

osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve i ...

is an osculating curve from the family of

circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...

s, and has second-order contact (same tangent angle and curvature), etc..

Applications

Contact forms are particular differential forms of degree 1 on odd-dimensional manifolds; see contact geometry. Contact transformations are related changes of coordinates, of importance in

classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...

. See also

Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...

. Contact between manifolds is often studied in

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

, where the type of contact are classified, these include the ''A'' series (''A''₀: crossing, ''A''₁: tangent, ''A''₂: osculating, ...) and the

umbilic In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...

or ''D''-series where there is a high degree of contact with the sphere.

Contact between curves

Two curves in the plane intersecting at a point ''p'' are said to have: *0th-order contact if the curves have a simple crossing (not tangent). *1st-order contact if the two curves are

. *2nd-order contact if the curvatures of the curves are equal. Such curves are said to be osculating. *3rd-order contact if the derivatives of the curvature are equal. *4th-order contact if the second derivatives of the curvature are equal.

Contact between a curve and a circle

For each point ''S''(''t'') on a

smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...

plane curve ''S'', there is exactly one

, whose radius is the reciprocal of κ(''t''), the curvature of ''S'' at ''t''. Where curvature is zero (at an

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case ...

on the curve), the osculating circle is a straight line. The

locus Locus (plural loci) is Latin for "place". It may refer to: Entertainment * Locus (comics), a Marvel Comics mutant villainess, a member of the Mutant Liberation Front * ''Locus'' (magazine), science fiction and fantasy magazine ** ''Locus Award' ...

of the centers of all the osculating circles (also called "centers of curvature") is the

evolute In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that cur ...

of the curve. 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 Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet * Vertex (computer graphics), a data structure that describes the positio ...

. 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 The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives fro ...

). In general a curve will not have 4th-order contact with any circle. However, 4th-order contact can occur generically in a 1-parameter family of curves, at a curve in the family where (as the parameter varies) two vertices (one maximum and one minimum) come together and annihilate. At such points the second derivative of curvature will be zero.

Bi-tangents in econometrics

econometrics Econometrics is the application of statistical methods to economic data in order to give empirical content to economic relationships. M. Hashem Pesaran (1987). "Econometrics," '' The New Palgrave: A Dictionary of Economics'', v. 2, p. 8 p. 8 ...

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 ''bi-tangent'' circles. The centers of all bi-tangent circles form the

symmetry set In geometry, the symmetry set is a method for representing the local symmetries of a curve, and can be used as a method for representing the shape of objects by finding the topological skeleton. The medial axis The medial axis of an object is ...

. The

medial axis The medial axis of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the topological skeleton, it was introduced in 1967 by Harry Blum as a tool for biological shape recogn ...

is a subset of the symmetry set. These sets have been used as a method of characterising the shapes of biological objects by Mario Henrique Simonsen, Brazilian and English econometrist.

References

* * Ian R. Porteous (2001) ''Geometric Differentiation'', pp 152–7,

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pre ...

{{ISBN, 0-521-00264-8 . Multivariable calculus Differential geometry Singularity theory Contact geometry