Quadratrix of Hippias
QuadratrixofHippias
The quadratrix was discovered by Hippias of Elias in 430 BC, and later studied by Dinostratus in 350 BC (MacTutor Archive). It can be used for angle trisection or, more generally, division of an angle into any integral number of equal parts, and circle squaring.
It has polar equation
with corresponding parametric equation
x = [画像:(2atcott)/pi]
(2)
y = [画像:(2at)/pi,]
(3)
| [画像: x=ycot((piy)/(2a)). ] |
(4)
|
Using the parametric representation, the curvature and tangential angle are given by
kappa = [画像:(pi(sint-tcost))/(a(1-2tcott+t^2csc^2t)^(3/2))]
(5)
![画像:1/2[t+cot^(-1)(cott-tcsc^2t)-tan^(-1)(t/(tcott-1))]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/QuadratrixofHippias/Inline12.svg){kind=link}
for 0<t<pi.
See also
Angle trisection, CochleoidExplore with Wolfram|Alpha
WolframAlpha
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References
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 223, 1987.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 195 and 198, 1972.Loomis, E. S. "The Quadratrix." §2.1 in The Pythagorean Proposition: Its Demonstrations Analyzed and Classified and Bibliography of Sources for Data of the Four Kinds of "Proofs," 2nd ed. Reston, VA: National Council of Teachers of Mathematics, pp. 19-20, 1968.Loy, J. "Trisection of an Angle." https://web.archive.org/web/20030402133520/http://www.jimloy.com/geometry/trisect.htm#curves.MacTutor History of Mathematics Archive. "Quadratrix of Hippias." https://mathshistory.st-andrews.ac.uk/Curves/Quadratrix/.Referenced on Wolfram|Alpha
Quadratrix of HippiasCite this as:
Weisstein, Eric W. "Quadratrix of Hippias." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/QuadratrixofHippias.html