Trisectrix
Template:Short description Template:CS1 config 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:
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
Page Template:Reflist/styles.css has no content.
- ^ Script error: No such module "template wrapper".
- ^ a b c d e f Page Module:Citation/CS1/styles.css has no content.Yates, Robert C. (January 1941), "The trisection problem, chapter II: Solutions by means of curves", National Mathematics Magazine, 15 (4): 191–202, doi:10.2307/3028133, JSTOR 3028133
- ^ Page Module:Citation/CS1/styles.css has no content.Dudley, Underwood (1994), The Trisectors, Cambridge University Press, p. 12, ISBN 0883855143; excerpt, p. 12, at Google Books
- ^ Page Module:Citation/CS1/styles.css has no content.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
- ^ Page Module:Citation/CS1/styles.css has no content.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
- ^ Page Module:Citation/CS1/styles.css has no content.Ferréol, Robert (2017), "Sectrix curve", Encyclopédie des formes mathématiques remarquables, retrieved 2025-10-20
- ^ Page Module:Citation/CS1/styles.css has no content.Merzbach, Uta B.; Boyer, Carl B. (2011), A History of Mathematics (Third ed.), John Wiley & Sons, pp. 113–114, ISBN 978-0470525487
- ^ Page Module:Citation/CS1/styles.css has no content.Harper, Suzanne; Driskell, Shannon (July 2006), "An investigation of historical geometric constructions", Convergence, Mathematical Association of America
- ^ Page Module:Citation/CS1/styles.css has no content.Sheng, Hung Tao (1969), "A method of trisection of an angle and X-section of an angle", Mathematics Magazine, 42 (2): 73–80, doi:10.1080/0025570X.1969.11975925, JSTOR 2689193, MR 0240707
External links
- Script error: No such module "Template wrapper".
Page Module:Side box/styles.css has no content.Page Template:Sister project/styles.css has no content.
