In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 and geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an osculating curve is an extension of the concept of tangent

Tangent

In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

. A tangent line to a curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

 is the straight line that shares the location

Location (geography)

The terms location and place in geography are used to identify a point or an area on the Earth's surface or elsewhere. The term 'location' generally implies a higher degree of can certainty than "place" which often has an ambiguous boundary relying more on human/social attributes of place identity...

 and direction

Direction (geometry, geography)

Direction is the information contained in the relative position of one point with respect to another point without the distance information. Directions may be either relative to some indicated reference , or absolute according to some previously agreed upon frame of reference Direction is the...

 of the curve, while an osculating circle

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 infinitesimally close to p...

 to the same curve shares the location, direction, and curvature

Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...

.

Two curves are said to be osculating at a particular point if they share the same osculating circle, just as they are said to be tangent if they share the same tangent line. The term derives from the Latinate root "osculate", to kiss

Kiss

A kiss is the act of pressing one's lips against the lips or other body parts of another person or of an object. Cultural connotations of kissing vary widely. Depending on the culture and context, a kiss can express sentiments of love, passion, affection, respect, greeting, friendship, and good...

, because the two curves contact one another in a more intimate way than simple tangency.

If two smooth curves are tangent at a point and also cross there, they are not only tangent but also osculating. The converse – osculating curves cross at the point of osculation – is not necessarily true, but holds in almost all

Almost all

In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....

cases.