Trisectrix

In geometry, a trisectrix is a curve which can be used to trisect an arbitrary angle with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by compass and straightedge constructions, so they do not contradict the well known theorem which states that an arbitrary angle cannot be trisected with that type of construction. There is a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:

• Limaçon trisectrix (some sources refer to this curve as simply the trisectrix.) ^{[1] [2]}
• Trisectrix of Maclaurin ^[3]
• Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic) ^[4]
• Cubic parabola (the graph of the cube function) ^[2]
• Hyperbola with eccentricity 2 ^{[5] [2]}
• Parabola ^[2]
• Cycloid of Ceva ^[2]

A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. ^[6] Examples include:

• Archimedean spiral ^[7]
• Quadratrix of Hippias ^{[8] [2]}

See also

• Sectrix of Maclaurin, a family of curves different members of which can divide angles into different numbers of parts
• Neusis construction, the use of a marked ruler in constructions such as angle trisection
• Quadratrix, a curve used for squaring the circle

References

1.   Chisholm, Hugh, ed. (1911), "Trisectrix", Encyclopædia Britannica , vol. 27 (11th ed.), Cambridge University Press
2. Yates, Robert C. (January 1941), "The trisection problem, chapter II: Solutions by means of curves", National Mathematics Magazine, 15 (4): 191–202, JSTOR   3028133
3. Dudley, Underwood (1994), The Trisectors, Cambridge University Press, p. 12, ISBN   0883855143 ; excerpt , p. 12, at Google Books
4. Farouki, Rida T. (2008), Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Geometry and Computing, vol. 1, Springer, pp. 398–399, doi:10.1007/978-3-540-73398-0, ISBN   978-3-540-73397-3, MR   2365013
5. Wright, J. M. F. (1836), "257. To trisect any angle by the hyperbola", An Algebraic System of Conic Sections, and Other Curves, London: Black and Armstrong, p. 206
6. Ferréol, Robert (2017), "Sectrix curve", Encyclopédie des formes mathématiques remarquables, retrieved 2025-10-20
7. Merzbach, Uta B.; Boyer, Carl B. (2011), A History of Mathematics (Third ed.), John Wiley & Sons, pp. 113–114, ISBN   978-0470525487
8. Harper, Suzanne; Driskell, Shannon (July 2006), "An investigation of historical geometric constructions", Convergence, Mathematical Association of America

External links

• Weisstein, Eric W., "Trisectrix", MathWorld