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In differential geometry of curves, 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 (geo ...
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, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
ly close to ''p''. Its center lies on the inner normal line, 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 (mathematics), surface deviates from being a plane (ge ...
defines the curvature of the given curve at that point. This circle, which is the one among all
tangent circles
tangent circles
at the given point that approaches the curve most tightly, was named ''circulus osculans'' (Latin for "kissing circle") by
Leibniz Gottfried Wilhelm (von) Leibniz ; see inscription of the engraving depicted in the "#1666–1676, 1666–1676" section. (; or ; – 14 November 1716) was a prominent Germany, German polymath and one of the most important Logic, logicians, Math ...

Leibniz
. The center and radius of the osculating circle at a given point are called center of curvature and
radius of curvature
radius of curvature
of the curve at that point. A geometric construction was described by
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March Old Style and New Style dates, 1726/27) was an English mathematician, physicist, astronomer, theologian, and author (described in his time as a "natural philosophy, natural philosopher") w ...

Isaac Newton
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 (mathematics), surface deviates from being a plane (ge ...
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 ''γ''(''s'') be a regular parametric plane curve, where ''s'' is the
arc length
arc length
(the natural parameter). This determines the ''unit tangent vector'' T(''s''), the ''unit normal vector'' N(''s''), the signed curvature ''k''(''s'') and the ''radius of curvature'' ''R''(''s'') at each point for which ''s'' is composed: : T(s)=\gamma'(s),\quad T'(s)=k(s)N(s),\quad R(s)=\frac. Suppose that ''P'' is a point on ''γ'' where ''k'' ≠ 0. 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 ''N''. It lies in the '' osculating plane'', 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 \gamma(t) = \begin x_1(t) \\ x_2(t) \end where regular means that \gamma'(t)\ne 0 for all t. 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 : k(t) = \frac, \qquad N(t) = \frac\cdot\begin -x_2'(t) \\ x_1'(t) \end : R(t) = \left, \frac \\qquad \text \qquad Q(t) = \gamma(t) + \frac\cdot\begin -x_2'(t) \\ x_1'(t) \end\,.


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: : x_c = x - f'\frac \quad\text\quad y_c = f + \frac


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 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 tangent lines to circles, 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 (mathematics), contact" 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 vertex (curve), 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 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 (mathematics), envelope of the one-parameter family of its osculating circles. Their centers, i.e. the centers of curvature, form another curve, called the ''evolute'' 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 (curve), vertex 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

For the parabola :\gamma(t) = \begin t\\t^2 \end the radius of curvature is :R(t)= \left, \frac \ At the vertex \gamma(0) = \begin 0\\0 \end the radius of curvature equals (see figure). The parabola has fourth order contact with its osculating circle there. For large ''t'' the radius of curvature increases ~ ''t''3, that is, the curve straightens more and more.


Lissajous curve

A Lissajous curve with ratio of frequencies (3:2) can be parametrized as follows : \gamma(t) = \begin \cos(3t) \\ \sin(2t) \end. It has signed curvature ''k''(''t''), normal unit vector ''N''(''t'') and radius of curvature ''R''(''t'') given by : k(t) = \frac\,, : N(t) = \frac \cdot \begin -2\cos(2t) \\ -3\sin(3t) \end and : R(t) = \left, \frac \. See the figure for an animation. There the "acceleration vector" is the second derivative \frac with respect to the
arc length
arc length
.


Cycloid

A cycloid with radius ''r'' can be parametrized as follows: : \gamma(t) = \begin r\left(t - \sin t\right) \\ r\left(1 - \cos t\right) \end Its curvature is given by the following formula: :\kappa(t) = - \frac which gives: :R(t) = \frac


See also

*Circle packing theorem *Osculating curve *Osculating sphere


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