“Evolute”的概念、定义、习题解题方法及翻译-数学辞典

单词

Evolute

释义

Evolute

An evolute is the locus of centers of curvature (the envelope) of a plane curve's normals. The original curve is thensaid to be the Involute of its evolute. Given a plane curve represented parametrically by ,the equation of the evolute is given by

			(1)
			(2)

where

are the coordinates of the running point,

is the Radius of Curvature

(3)

and

is the angle between the unit Tangent Vector

(4)

and the x-Axis,

			(5)
			(6)

Combining gives

			(7)
			(8)

The definition of the evolute of a curve is independent of parameterization for any differentiable function (Gray 1993). If

is the evolute of a curve

, then

is said to be the Involute of

. The centers of the Osculating Circles to a curve form the evolute to that curve (Gray 1993, p. 90).

The following table lists the evolutes of some common curves.

Curve	Evolute
Astroid	Astroid 2 times as large
Cardioid	Cardioid 1/3 as large
Cayley's Sextic	Nephroid
Circle	point (0, 0)
Cycloid	equal Cycloid
Deltoid	Deltoid 3 times as large
Ellipse	Lamé Curve
Epicycloid	enlarged Epicycloid
Hypocycloid	similar Hypocycloid
Limaçon	Circle Catacaustic for a point source
Logarithmic Spiral	equal Logarithmic Spiral
Nephroid	Nephroid 1/2 as large
Parabola	Neile's Parabola
Tractrix	Catenary

See also Involute, Osculating Circle

References

Cayley, A. ``On Evolutes of Parallel Curves.'' Quart. J. Pure Appl. Math. 11, 183-199, 1871.

Dixon, R. ``String Drawings.'' Ch. 2 in Mathographics. New York: Dover, pp. 75-78, 1991.

Gray, A. ``Evolutes.'' §5.1 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 76-80, 1993.

Jeffrey, H. M. ``On the Evolutes of Cubic Curves.'' Quart. J. Pure Appl. Math. 11, 78-81 and 145-155, 1871.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 40 and 202, 1972.

Lee, X. ``Evolute.''http://www.best.com/~xah/SpecialPlaneCurves_dir/Evolute_dir/evolute.html.

Lockwood, E. H. ``Evolutes and Involutes.'' Ch. 21 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 166-171, 1967.

Yates, R. C. ``Evolutes.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 86-92, 1952.

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