Plane curve with the greatest order of contact with another curve
"Osculation" redirects here. For other uses, see Osculate (disambiguation).
A curve C containing a point P where the radius of curvature equals r, together with the tangent line and the osculating circle touching C at P
In differential geometry, an osculating curve is a plane curve from a given family that has the highest possible order of contact with another curve. That is, if F is a family of smooth curves, 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 derivatives (in succession, from the first derivative) at P equal to the derivatives of C as possible.[1][2]
The term derives from the Latinate root "osculate", to kiss, because the two curves contact one another in a more intimate way than simple tangency.[3]
^Rutter, J. W. (2000), Geometry of Curves, CRC Press, pp. 174–175, ISBN 9781584881667.
^Williamson, Benjamin (1912), An elementary treatise on the differential calculus: containing the theory of plane curves, with numerous examples, Longmans, Green, p. 309.
^Max, Black (1954–1955), "Metaphor", Proceedings of the Aristotelian Society, New Series, 55: 273–294. Reprinted in Johnson, Mark, ed. (1981), Philosophical Perspectives on Metaphor, University of Minnesota Press, pp. 63–82, ISBN 9780816657971. P. 69: "Osculating curves don't kiss for long, and quickly revert to a more prosaic mathematical contact."
first-order contact with C. The osculating circle to C at p, the osculatingcurve from the family of circles. The osculating circle shares both its first...
An osculating circle is a circle that best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has...
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point of a differentiable curve is the curvature of its osculating circle — that is, the circle that best approximates the curve near this point. The curvature...
differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together...
the point. The word osculate is from the Latin osculatus which is a past participle of osculari, meaning to kiss. An osculating plane is thus a plane...
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curves "kiss". An osculating orbit and the object's position upon it can be fully described by the six standard Kepler orbital elements (osculating elements)...
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nesting of osculating circles. The normals of the given curve at points of nonzero curvature are tangents to the evolute, and the normals of the curve at points...
that, at each point, the plane tangent to the surface is an osculating plane of the curve. Asymptotic directions can only occur when the Gaussian curvature...
approximation to the ellipsoid in the vicinity of a given point is the Earth's osculating sphere. Its radius equals Earth's Gaussian radius of curvature, and its...
curve whose curvature changes linearly with its curve length (the curvature of a circular curve is equal to the reciprocal of the radius). This curve...
curvature. Because this is an osculating circle at the vertex of the curve, it has third-order contact with the curve. The curve has two inflection points...
It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the centre of curvature. Cauchy defined the center...
cyclides both sheets form curves. * For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single...
exist, the osculating circles to parallel curves at corresponding points are concentric. As for parallel lines, a normal line to a curve is also normal...
convex hull of a smooth curve on the sphere that is not a vertex of the curve, then at least four points of the curve have osculating planes passing through...
In geometry, a cardioid (from Greek καρδιά (kardiá) 'heart') is a plane curve traced by a point on the perimeter of a circle that is rolling around a...
osculating circle at a given point P on a curve is the limiting circle of a sequence of circles that pass through P and two other points on the curve...
meters and more. Base curve radius Bend radius Degree of curvature (civil engineering) Osculating circle Track transition curve Weisstien, Eric. "Radius...
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