Cissoid of Diocles
A curve invented by Diocles in about 180 BC 
in connection with his attempt to duplicate the cube by geometricalmethods. The name ``cissoid'' first appears in the work of Geminus about 100 years later. Fermat 
and Robervalconstructed the tangent in 1634. Huygens and Wallis found, in 1658, that the Area between the curve and itsasymptote was 
(MacTutor Archive). From a given point there are either one or three Tangents tothe cissoid.
Given an origin 
and a point 
on the curve, let 
be the point where the extension of the line 
intersects theline 
and 
be the intersection of the Circle of Radius 
and center 
with the extension of. Then the cissoid of Diocles is the curve which satisfies 
.
The cissoid of Diocles is the Roulette of the Vertex of a Parabola rolling on anequal Newton gave a method of drawing the cissoid of Diocles using two line segments of equallength at Right Angles. If they are moved so that one line always passes through a fixed point and theend of the other line segment slides along a straight line, then the Midpoint of the sliding line segment traces out acissoid of Diocles.
The cissoid of Diocles is given by the parametric equations
 |  |  | (1) |
 | |  | (2) |
Converting these to Polar Coordinates gives
 | |  | |
|  | (3) |
so
 | (4) |
 | |  | |
|  | (5) |
An equivalent form is
 | (6) |
Using the alternative parametric form
 | |  | (7) |
 | |  | (8) |
(Gray 1993), gives the Curvature as
 | |  | (9) |
References
Gray, A. ``The Cissoid of Diocles.'' §3.4 in Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, pp. 43-46, 1993.
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 98-100, 1972.
Lee, X. ``Cissoid of Diocles.''http://www.best.com/~xah/SpecialPlaneCurves_dir/CissoidOfDiocles_dir/cissoidOfDiocles.html.
Lockwood, E. H. A Book of Curves. Cambridge, England: Cambridge University Press, pp. 130-133, 1967.
MacTutor History of Mathematics Archive. ``Cissoid of Diocles.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cissoid.html.
Yates, R. C. ``Cissoid.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 26-30, 1952.
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