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单词 CircleOfCurvature
释义

circle of curvature


Let a given plane curveMathworldPlanetmath γ have a definitecurvaturePlanetmathPlanetmath (http://planetmath.org/CurvaturePlaneCurve) κ ina point P of γ.  The circle of curvatureMathworldPlanetmath of γ in thepoint P is the circle which has the radius 1|κ|and which has with the curve the common tangent in P and whichin a neighbourhood of P is on the same side as the curve.

The radius ϱ of the circle of curvature is theradius of curvature of γ in P.  The center of thecircle of curvature is the center of curvature of γ in P.

When the curve γ is given in the parametric form

x=x(t),y´=y(t),

the coordinates of the center of curvature belonging to the point  (x,y) of the curve are

ξ=x-(x2+y2)yxy′′-x′′y,η=y+(x2+y2)xxy′′-x′′y.

Example.  Since the curvature of the parabolaMathworldPlanetmathPlanetmathy=x2 in the origin is  -2,  the corresponding radius of curvature is12 and the center of curvature  (0,12).

Furthermore, it is possible to define the circle of curvature withoutfirst knowing about curvature of the curve. (In fact, using this definition,one could reverse the procedure and define curvature as the radius ofthe circle of curvature.) We may define the circle of curvaturea point P of γ as the unique circle passing through P whichmakes a second-order contact with γ at P.

Titlecircle of curvature
Canonical nameCircleOfCurvature
Date of creation2013-03-22 16:59:46
Last modified on2013-03-22 16:59:46
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id13
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 53A04
Synonymosculating circle
Related topicCurvatureOfaCircle
Related topicOsculatingCurve
Definesradius of curvature
Definescenter of curvature
Definescentre of curvature
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更新时间:2025/5/4 16:09:24