In geometry, a trisectrix is a curve which can be used to

trisect an arbitrary angle Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and ...

with ruler and compass and this curve as an additional tool. Such a method falls outside those allowed by

compass and straightedge constructions In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an ideali ...

, 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 In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose (mathematics), rose, ...

(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 In arithmetic and algebra, the cube of a number is its third power, that is, the result of multiplying three instances of together. The cube of a number or any other mathematical expression is denoted by a superscript 3, for example or . ...

* Hyperbola with eccentricity 2 * Rose curve specified by a

sinusoid A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...

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 The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...

* Quadratrix of Hippias * Sectrix of Maclaurin * Sectrix of Ceva * Sectrix of Delanges

References

Loy, Jim "Trisection of an Angle", Part VI
*
"Sectrix curve" at Encyclopédie des Formes Mathématiques Remarquables
(In French) * {{commonscat Curves

See also

References