# Dictionary Definition

osculation

### Noun

1 (mathematics) a contact of two curves (or two
surfaces) at which they have a common tangent

# User Contributed Dictionary

## English

### Etymology

From etyl la osculatio from osculor.### Pronunciation

### Noun

#### Related terms

#### Translations

a contact between curves or surfaces

- Swedish: oskulation

# Extensive Definition

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. Equivalence classes are generally called jets.

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.

## Contact between curves

Two curves in the plane intersecting at a point p are said to have:- 1-point contact if the curves have a simple crossing (not tangent).
- 2-point contact if the two curves are tangent.
- 3-point contact if the curvatures of the curves are equal. Such curves are said to be osculating.
- 4-point contact if the derivatives of the curvature are equal.
- 5-point 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 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.

### Bi-tangents

It is also possible to consider circles which have two point contact with two points S(t1), S(t2) on the curve. Such circles are bi-tangent circles. The centers of all bi-tangent 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

- Curves and Singularities

osculation in French: Contact (géométrie)

osculation in Chinese: 切点