differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, an osculating curve is a

plane curve In mathematics, a plane curve is a curve in a plane that may be a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane c ...

from a given family that has the highest possible order of contact with another curve. That is, if is a family of smooth curves, is a smooth curve (not in general belonging to ), and is a point on , then an osculating curve from at is a curve from that passes through and has as many of its

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

s (in succession, from the first derivative) at equal to the derivatives of as possible... The term derives from the Latinate root "osculate", to

kiss A kiss is the touching or pressing of one's lips against another person, animal or object. Cultural connotations of kissing vary widely; depending on the culture and context, a kiss can express sentiments of love, passion, romance, sex ...

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

Examples

Examples of osculating curves of different orders include: *The

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

to a curve at a point , the osculating curve from the family of straight lines. The tangent line shares its first derivative (

slope In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of t ...

) with and therefore has first-order contact with .. *The

osculating circle An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that point. The osculating circle provides a way to unders ...

to at , the osculating curve from the family of

circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...

s. The osculating circle shares both its first and second derivatives (equivalently, its slope and

curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...

) with . *The osculating parabola to at , the osculating curve from the family of

parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

s, has third order contact with . *The osculating conic to at , the osculating curve from the family of

conic section A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...

s, has fourth order contact with .

Generalizations

The concept of osculation can be generalized to higher-dimensional spaces, and to objects that are not curves within those spaces. For instance an osculating plane to a

space 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 ...

is a plane that has second-order contact with the curve. This is as high an order as is possible in the general case.. In one dimension, analytic curves are said to osculate at a point if they share the first three terms of their

Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

about that point. This concept can be generalized to superosculation, in which two curves share more than the first three terms of their Taylor expansion.

References

{{reflist Curves

Examples

Generalizations

See also

References