| 释义 |
CardioidThe curve given by the Polar equation
 | (1) |
 | (2) |
 , the Cartesian equation
 | (3) |
The cardioid is a degenerate case of the Limaçon. It is also a 1-Cusped Epicycloid (with  ) and is the Caustic formed by rays originating at a point on the circumference ofa Circle and reflected by the Circle.
The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. ItsArc Length was found by La Hire in 1708. There are exactly three Parallel Tangents tothe cardioid with any given gradient. Also, the Tangents at the ends of any Chord through theCusp point are at Right Angles. The length of any Chord through the Cusp pointis  .
The cardioid may also be generated as follows. Draw a Circle  and fix a point  on it. Now draw a set ofCircles centered on the Circumference of and passing through . The Envelope ofthese Circles is then a cardioid (Pedoe 1995). Let the Circle be centered at the origin andhave Radius 1, and let the fixed point be  . Then the Radius of a Circle centered at anAngle  from (1, 0) is
The Arc Length, Curvature, and Tangential Angle are
As usual, care must be taken in the evaluation of  for  . Since (7) comes from an integralinvolving the Absolute Value of a function, it must be monotonic increasing. Each Quadrant can betreated correctly by defining
 | (10) |
 is the Floor Function, giving the formula
 | (11) |
The Perimeter of the curve is
The Area is
See also Circle, Cissoid, Conchoid, Equiangular Spiral, Lemniscate, Limaçon, Mandelbrot Set
References
Gray, A. ``Cardioids.'' §3.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 41-42, 1993.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 118-121, 1972. Lee, X. ``Cardioid.''http://www.best.com/~xah/SpecialPlaneCurves_dir/Cardioid_dir/cardioid.html. Lee, X. ``Cardioid.''http://www.best.com/~xah/SpecialPlaneCurves_dir/Cardioid_dir/cardioidGG.html. Lockwood, E. H. ``The Cardioid.'' Ch. 4 in A Book of Curves. Cambridge, England: Cambridge University Press, pp. 34-43, 1967. MacTutor History of Mathematics Archive. ``Cardioid.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cardioid.html. Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. xxvi-xxvii, 1995. Yates, R. C. ``The Cardioid.'' Math. Teacher 52, 10-14, 1959. Yates, R. C. ``Cardioid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 4-7, 1952.
|
