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| {{redirect|Kissing circles|Descartes' theorem on mutually tangent (kissing) circles|Descartes' theorem}}
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| [[File:Osculating circle.svg|thumb|right|An osculating circle]]
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| 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 [[infinitesimal]]ly close to ''p''. Its center lies on the inner [[Normal (geometry)|normal line]], and its [[curvature]] is the same as that of the given curve at that point. This circle, which is the one among all '''[[tangent circles]]''' at the given point that approaches the curve most tightly, was named ''circulus osculans'' (Latin for "kissing circle") by [[Gottfried Wilhelm Leibniz|Leibniz]].
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| The center and radius of the osculating circle at a given point are called '''center of curvature''' and '''[[radius of curvature (mathematics)|radius of curvature]]''' of the curve at that point. A geometric construction was described by [[Isaac Newton]] in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'':
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| {{Quotation|There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre.|Isaac Newton, ''Principia''; PROPOSITION V. PROBLEM I.}}
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| ==Description in lay terms==
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| 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]] 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.
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| ==Mathematical description==
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| Let ''γ''(''s'') be a [[regular parametric curve|regular parametric plane curve]], where ''s'' is the [[arc length]], or natural parameter. This determines the unit tangent vector ''T'', the unit normal vector ''N'', the [[Curvature#Precise definition|signed curvature]] ''k''(''s'') and the radius of curvature at each point:
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| : <math> T(s)=\gamma'(s),\quad T'(s)=k(s)N(s),\quad R(s)=\frac{1}{\left|k(s)\right|}.</math>
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| 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''.
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| 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''.
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| The plane curve can also be given in a different regular parametrization
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| <math> \gamma(t)\,= \, \begin{pmatrix} x_1(t) \\
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| x_2(t) \end{pmatrix}\, </math>
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| where regular means that <math>\gamma'(t)\ne 0</math> for all <math>t</math>. 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 cicle are
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| : <math>k(t) = \frac{x_1'(t) \cdot x_2''(t) - x_1''(t) \cdot x_2'(t)}{\Big( x_1'(t)^2+x_2'(t)^2 \Big)^{\frac{3}{2}}} \qquad\qquad \qquad\qquad\qquad N(t)\,=\,\frac{1}{|| \gamma'(t)||}\cdot\begin{pmatrix} -x_2'(t) \\ x_1'(t) \end{pmatrix}</math>,
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| : <math>R(t) = \left| \frac{\Big( x_1'(t)^2+x_2'(t)^2 \Big)^{\frac{3}{2}}}{x_1'(t) \cdot x_2''(t) - x_1''(t) \cdot x_2'(t)} \right|\qquad\qquad \mathrm{and} \qquad\qquad Q(t)\,=\,\gamma(t)\,+ \, \frac{1}{k(t)\cdot|| \gamma'(t)||}\cdot\begin{pmatrix} -x_2'(t) \\ x_1'(t) \end{pmatrix}\,. </math>
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| ==Properties==
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| 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''.<ref name=Lamb>Actually, point ''P'' plus two additional points, one on either side of ''P'' will do. See Lamb (on line): {{cite book |title=An Elementary Course of Infinitesimal Calculus |author=Horace Lamb |page=406 |url=http://books.google.com/books?id=eDM6AAAAMAAJ&pg=PA406&dq=%22osculating+circle%22&lr=&as_brr=0 |publisher=University Press |year=1897}}</ref> 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''.
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| The osculating circle ''S'' to a plane curve ''C'' at a regular point ''P'' can be characterized by the following properties:
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| * The circle ''S'' passes through ''P''.
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| * The circle ''S'' and the curve ''C'' have the [[tangent lines to circles|common tangent]] line at ''P'', and therefore the common normal line.
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| * 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.
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| This is usually expressed as "the curve and its osculating circle have the third or higher order contact" at ''P''. Loosely speaking, the vector functions representing ''C'' and ''S'' agree together with their first and second derivatives at ''P''.
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| 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 four. 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.
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| 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.
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| ==Examples==
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| ===Parabola===
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| [[File:Krümmungskreis-Parabel.png|thumb|250px|The osculating circle of the parabola at its vertex has radius 0.5 and fourth order contact.]]
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| For the parabola
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| :<math>\gamma(t) = \begin{pmatrix} t\\t^2 \end{pmatrix}</math>
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| the radius of curvature is
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| :<math>R(t)= \left| \frac{ \left(1+4\cdot t^2 \right)^{\frac{3}{2}}}{2} \right| </math>
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| At the vertex <math>\gamma(0) = \begin{pmatrix} 0\\0 \end{pmatrix}</math> the radius of curvature equals ''R(0)=0.5'' (see figure). The parabola has fourth order contact with its osculating circle there. For large ''t'' the radius of curvature increases ~ ''t<sup>3</sup>'', that is, the curve straightens more and more.
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| ===Lissajous curve===
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| [[File:Lissajous-Curve+OsculatingCircle+3vectors animated.gif|thumb|250px|Animation of the osculating circle to a Lissajous curve]]
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| A [[Lissajous curve]] with ratio of frequencies (3:2) can be parametrized as follows
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| :<math> \gamma(t)\,= \, \begin{pmatrix} \cos(3t) \\
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| \sin(2t) \end{pmatrix}\,. </math>
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| It has signed curvature ''k''(''t''), normal unit vector ''N''(''t'') and radius of curvature ''R''(''t'') given by
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| : <math>k(t) = \frac{6\cos(t)(8\cos(t)^4-10\cos(t)^2+5)}{(232\cos(t)^4-97\cos(t)^2+13-144\cos(t)^6)^{3/2}}\,, </math>
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| : <math>N(t)\,=\,\frac{1}{|| \gamma'(t)||}\cdot\begin{pmatrix} -2\cos(2t) \\ -3\sin(3t)
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| \end{pmatrix}</math>
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| and
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| : <math>R(t) = \left| \frac{(232\cos(t)^4-97\cos(t)^2+13-144\cos(t)^6)^{3/2}}{6\cos(t)(8\cos(t)^4-10\cos(t)^2+5)} \right|\,.</math>
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| See the figure for an animation. There the "acceleration vector" is the second derivative <math>\frac{\mathrm{d}^2\gamma(s)}{\mathrm{d}s^2}</math> with respect to the [[arc length]] <math>s</math>.
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| ==See also==
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| *[[Circle packing theorem]]
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| *[[Contact (mathematics)]]
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| *[[Osculating curve]]
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| ==Notes==
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| <references/>
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| ==Further reading==
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| For some historical notes on the study of curvature, see
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| *{{cite book |title=From the Calculus to Set Theory 1630-1910: An Introductory History |author=Grattan-Guinness & H. J. M. Bos |url=http://books.google.com/books?id=OLNeNIbD3jUC&pg=PA72&vq=curvature&dq=Leibniz+Calculus+Bos&lr=&source=gbs_search_s&sig=ACfU3U1ksyW8pE2DkRUdMQ6-Dw5jo4WdJA
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| |page=72 |isbn=0-691-07082-2 |year=2000 |publisher=Princeton University Press}}
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| *{{cite book |url=http://books.google.com/books?id=KDSqLsOHc9UC&pg=PA313&dq=motion+%22center+of+curvature%22&lr=&as_brr=0&sig=ACfU3U1xXmwMgWfk0Is_CToKzTbY591kXQ#PPA313,M1
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| |page=313 |author=Roy Porter, editor |isbn=0-521-57243-6 |year=2003 |publisher=Cambridge University Press |title=The Cambridge History of Science: v4 - Eighteenth Century Science}}
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| For application to maneuvering vehicles see
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| *JC Alexander and JH Maddocks: [https://drum.umd.edu/dspace/bitstream/1903/4630/1/TR_87-122.pdf ''On the maneuvering of vehicles'']
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| *{{cite book |title= Problems in Applied Mathematics: selections from SIAM review |author=Murray S. Klamkin |page=1
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| |url=http://books.google.com/books?id=-b2hQ7_ARocC&pg=PA1&dq=motion+%22center+of+curvature%22&lr=&as_brr=0&sig=ACfU3U0e0jN1k65IAWtXyKbYwYxtdeGlRw#PPA1,M1
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| |isbn=0-89871-259-9 |publisher=Society for Industrial and Applied Mathematics |year=1990}}
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| ==External links==
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| {{Commons category|Illustrations for curvature and torsion of curves|Graphical illustrations of curvature and osculating circles}}
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| *[http://www.math.uni-muenster.de/u/urs.hartl/gifs/CurvatureAndTorsionOfCurves.mw Create your own animated illustrations of osculating circles] ([[Maple (software)|Maple]]-Worksheet)
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| * {{MathWorld | urlname= OsculatingCircle | title= Osculating Circle }}
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| *[http://math.fullerton.edu/mathews/n2003/CurvatureMod.html Module for Curvature]
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| [[Category:Circles]]
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| [[Category:Differential geometry]]
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| [[Category:Curves]]
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