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.)
• Trisectrix of Maclaurin
• Equilateral trefoil (a.k.a. Longchamps' Trisectrix)
• Tschirnhausen cubic (a.k.a. Catalan's trisectrix and L'Hôpital's cubic)
• Durer's folium
• Cubic parabola
• Hyperbola with eccentricity 2
• Rose curve specified by a sinusoid with angular frequency of one-third.
• Parabola

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

• Archimedean Spiral
• Quadratrix of Hippias
• Sectrix of Maclaurin
• Sectrix of Ceva
• Sectrix of Delanges

See also

• Doubling the cube
• Neusis construction
• Quadratrix

References

• Loy, Jim "Trisection of an Angle", Part VI
• Weisstein, Eric W. "Trisectrix". MathWorld.
• "Sectrix curve" at Encyclopédie des Formes Mathématiques Remarquables (In French)
• This article incorporates text from a publication now in the public domain: Chisholm, Hugh, ed. (1911). "Trisectrix". Encyclopædia Britannica. Vol. 27 (11th ed.). Cambridge University Press.