![Osculating circles of the Archimedean spiral](https://upload.wikimedia.org/wikipedia/commons/0/0f/Osculating_circles_of_the_Archimedean_spiral.svg)
In
differential geometry of curves
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.
Many specific curves have been thoroughly investigated using the ...
, the osculating circle of a sufficiently smooth plane
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
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
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
ly close to ''p''. Its center lies on the inner
normal line
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at t ...
, and its
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
defines the curvature of the given curve at that point. This circle, which is the one among all
tangent circles
In geometry, tangent circles (also known as kissing circles) are circles in a common plane that intersect in a single point. There are two types of tangency: internal and external. Many problems and constructions in geometry are related to tange ...
at the given point that approaches the curve most tightly, was named ''circulus osculans'' (Latin for "kissing circle") by
Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of ma ...
.
The center and radius of the osculating circle at a given point are called
center of curvature
In geometry, the center of curvature of a curve is found at a point that is at a distance from the curve equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circl ...
and
radius of curvature
In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
of the curve at that point. A geometric construction was described by
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
in his ''
Principia'':
Nontechnical description
Imagine a car moving along a curved road on a vast flat plane. Suddenly, at one point along the road, the steering wheel locks in its present position. Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
of the circle is equal to that of the road at that point. That circle is the osculating circle of the road curve at that point.
Mathematical description
Let be a
regular parametric plane curve, where is the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
(the
natural parameter
In probability and statistics, an exponential family is a parametric set of probability distributions of a certain form, specified below. This special form is chosen for mathematical convenience, including the enabling of the user to calculate ...
). This determines the ''unit tangent vector'' , the ''unit normal vector'' , the
signed curvature and the ''radius of curvature'' at each point for which is composed:
Suppose that ''P'' is a point on ''γ'' where . The corresponding center of curvature is the point ''Q'' at distance ''R'' along ''N'', in the same direction if ''k'' is positive and in the opposite direction if ''k'' is negative. The circle with center at ''Q'' and with radius ''R'' is called the osculating circle to the curve ''γ'' at the point ''P''.
If ''C'' is a regular space curve then the osculating circle is defined in a similar way, using the
principal normal vector
Principal may refer to:
Title or rank
* Principal (academia), the chief executive of a university
** Principal (education), the office holder/ or boss in any school
* Principal (civil service) or principal officer, the senior management level ...
''N''. It lies in the ''
osculating plane {{Unreferenced, date=May 2019, bot=noref (GreenC bot)
In mathematics, particularly in differential geometry, an osculating plane is a plane in a Euclidean space or affine space which meets a submanifold at a point in such a way as to have a second ...
'', the plane spanned by the tangent and principal normal vectors ''T'' and ''N'' at the point ''P''.
The plane curve can also be given in a different regular parametrization
where regular means that
for all
. Then the formulas for the signed curvature ''k''(''t''), the normal unit vector ''N''(''t''), the radius of curvature ''R''(''t''), and the center ''Q''(''t'') of the osculating circle are
Cartesian coordinates
We can obtain the center of the osculating circle in Cartesian coordinates if we substitute and for some function ''f''. If we do the calculations the results for the X and Y coordinates of the center of the osculating circle are:
Direct geometrical derivation
Consider three points
,
and
, where
. To find the center of the circle that passes through these points, we have first to find the segment bisectors of
and
and then the point
where these lines cross. Therefore, the coordinates of
are obtained through solving a linear system of two equations:
where
,
for
.
Consider now the curve
and set
,
and
. To the second order in
, we have
and a similar expression for
and
where the sign of
is reversed. Developing the equation for
and grouping the terms in
and
, we obtain
Denoting
, the first equation means that
is orthogonal to the unit tangent vector at
:
The second relation means that
where
is the curvature vector. In plane geometry,
is orthogonal to
because
Therefore
and the radius of the osculating circle is precisely the inverse of the curvature.
Solving the equation for the coordinates of
, we find
Osculating circle as a minimization problem
Consider a curve
defined intrinsically by the equation
which we can envision as the section of the surface
by the plane
.
The normal
to the curve at a point
is the gradient at this point
Therefore, the centers of the tangent circles
are given by
where
is parameter. For a given
the radius
of
is
We wish to find, among all possible circles
, the one that ''matches'' best the curve.
The coordinates of a point
can be written as
where for
,
, i.e.
Consider now a point
close to
, where its "angle" is
. Developing the trigonometric functions to the second order in
and using the above relations, coordinates of
are
We can now evaluate the function
at the point
and its variation
.
The variation is zero to the first order in
by construction (to the first order in
,
is on the tangent line to the curve
). The variation proportional to
is
and this variation is zero if we choose
Therefore the radius of the osculating circle is
For an explicit function
, we find the results of the preceding section.
Properties
For a curve ''C'' given by a sufficiently smooth parametric equations (twice continuously differentiable), the osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on ''C'' as these points approach ''P''.
[Actually, point ''P'' plus two additional points, one on either side of ''P'' will do. See Lamb (on line): ] This is entirely analogous to the construction of the
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. More ...
to a curve as a limit of the secant lines through pairs of distinct points on ''C'' approaching ''P''.
The osculating circle ''S'' to a plane curve ''C'' at a regular point ''P'' can be characterized by the following properties:
* The circle ''S'' passes through ''P''.
* The circle ''S'' and the curve ''C'' have the
common tangent line at ''P'', and therefore the common normal line.
* Close to ''P'', the distance between the points of the curve ''C'' and the circle ''S'' in the normal direction decays as the cube or a higher power of the distance to ''P'' in the tangential direction.
This is usually expressed as "the curve and its osculating circle have the second or higher order
contact
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
" at ''P''. Loosely speaking, the vector functions representing ''C'' and ''S'' agree together with their first and second derivatives at ''P''.
If the derivative of the curvature with respect to ''s'' is nonzero at ''P'' then the osculating circle crosses the curve ''C'' at ''P''. Points ''P'' at which the derivative of the curvature is zero are called
vertices. If ''P'' is a vertex then ''C'' and its osculating circle have contact of order at least three. If, moreover, the curvature has a non-zero
local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ra ...
or minimum at ''P'' then the osculating circle touches the curve ''C'' at ''P'' but does not cross it.
The curve ''C'' may be obtained as the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a sh ...
of the one-parameter family of its osculating circles. Their centers, i.e. the centers of curvature, form another curve, called 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 ''C''. Vertices of ''C'' correspond to singular points on its evolute.
Within any arc of a curve ''C'' within which the curvature is monotonic (that is, away from any
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 ...
of the curve), the osculating circles are all disjoint and nested within each other. This result is known as the
Tait-Kneser theorem.
Examples
Parabola
![Parabola circle](https://upload.wikimedia.org/wikipedia/commons/a/a7/Parabola_circle.svg)
For the parabola
the radius of curvature is
At the vertex
the radius of curvature equals (see figure). The parabola has fourth order contact with its osculating circle there. For large the radius of curvature increases ~ , that is, the curve straightens more and more.
Lissajous curve
![Lissajous-Curve+OsculatingCircle+3vectors animated](https://upload.wikimedia.org/wikipedia/commons/6/63/Lissajous-Curve%2BOsculatingCircle%2B3vectors_animated.gif)
A
Lissajous curve
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
: x=A\sin(at+\delta),\quad y=B\sin(bt),
which describe the superposition of two perpendicular oscillations in x and y dire ...
with ratio of frequencies (3:2) can be parametrized as follows
:
It has signed curvature , normal unit vector and radius of curvature given by
and
See the figure for an animation. There the "acceleration vector" is the second derivative
with respect to the
arc length
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
.
Cycloid
![Cycloid osculating circle evolute 2](https://upload.wikimedia.org/wikipedia/commons/7/7f/Cycloid_osculating_circle_evolute_2.gif)
A
cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...
with radius can be parametrized as follows:
Its curvature is given by the following formula:
which gives:
See also
*
Circle packing theorem
The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in gen ...
*
Osculating curve
*
Osculating sphere
Figure of the Earth is a term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A sphere is a well-known historical approxima ...
Notes
Further reading
For some historical notes on the study of curvature, see
*
*
For application to maneuvering vehicles see
*JC Alexander and JH Maddocks (1988): ''On the maneuvering of vehicles''
*
External links
* {{MathWorld , urlname= OsculatingCircle , title= Osculating Circle
math3d : osculating_circle
Circles
Differential geometry
Curves