geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, 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 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 (some sources refer to this curve as simply the trisectrix.) * Trisectrix of Maclaurin * Equilateral trefoil (a.k.a. Longchamps' Trisectrix) *

Tschirnhausen cubic In algebraic geometry, the Tschirnhausen cubic, or Tschirnhaus' cubic is a plane curve defined, in its left-opening form, by the polar equation :r = a\sec^3 \left(\frac\right) where is the secant function. History The curve was studied by Ehrenf ...

(a.k.a. Catalan's trisectrix and L'Hôpital's cubic) * Durer's folium * Cubic parabola *

Hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

with eccentricity 2 * Rose curve specified by a

sinusoid A sine wave, sinusoidal wave, or sinusoid (symbol: ∿) is a periodic wave whose waveform (shape) is the trigonometric sine function. In mechanics, as a linear motion over time, this is '' simple harmonic motion''; as rotation, it correspond ...

with

angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...

of one-third. *

Parabola In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

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 Archimedes' spiral, the arithmetic spiral) is a spiral named after the 3rd-century BC Ancient Greece, Greek mathematician Archimedes. The term ''Archimedean spiral'' is sometimes used to refer to the more gene ...

Quadratrix of Hippias The quadratrix or trisectrix of Hippias (also called the quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is traced out by the crossing point of two Line (geometry), lines, one moving by translation (geometry), tran ...

* 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