In
differential geometry, an osculating curve is a
plane curve
In mathematics, a plane curve is a curve in a plane that may be either 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 ...
from a given family that has the highest possible order of
contact with another curve.
That is, if ''F'' is a family of
smooth 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, ''C'' is a smooth curve (not in general belonging to ''F''), and ''p'' is a point on ''C'', then an osculating curve from ''F'' at ''p'' is a curve from ''F'' that passes through ''p'' and has as many of its
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 at ''p'' equal to the derivatives of ''C'' as possible.
[.][.]
The term derives from the Latinate root "osculate", to
kiss
A kiss is the touch or pressing of one's lips against another person or an object. Cultural connotations of kissing vary widely. Depending on the culture and context, a kiss can express sentiments of love, passion, romance, sexual attraction, ...
, 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 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 ...
to a curve ''C'' at a point ''p'', 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 is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
) with ''C'' and therefore has first-order contact with ''C''.
[.]
*The
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 ...
to ''C'' at ''p'', the 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. The osculating circle shares both its first and second derivatives (equivalently, its slope and
curvature) with ''C''.
*The osculating parabola to ''C'' at ''p'', 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 exact ...
s, has third order contact with ''C''.
*The osculating conic to ''C'' at ''p'', the osculating curve from the family of
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a spe ...
s, has fourth order contact with ''C''.
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 seri ...
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.
See also
*
Osculating orbit
In astronomy, and in particular in astrodynamics, the osculating orbit of an object in space at a given moment in time is the gravitational Kepler orbit (i.e. an elliptic or other conic one) that it would have around its central body if pertu ...
References
{{reflist
Curves