In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
, 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:
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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, conchoid or epitro ...
(some sources refer to this curve as simply the trisectrix.)
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Trisectrix of Maclaurin
In algebraic geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a ...
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Equilateral trefoil (a.k.a. Longchamps' Trisectrix)
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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 von ...
(a.k.a. Catalan's trisectrix and L'Hôpital's cubic)
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Durer's folium
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Cubic parabola
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Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth 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, ca ...
with eccentricity 2
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Rose curve specified by a
sinusoid with
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of one-third.
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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 exact ...
A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. Examples include:
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Archimedean Spiral
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Quadratrix of Hippias
The quadratrix or trisectrix of Hippias (also quadratrix of Dinostratus) is a curve which is created by a uniform motion. It is one of the oldest examples for a kinematic curve (a curve created through motion). Its discovery is attributed to the ...
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Sectrix of Maclaurin
In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined a ...
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Sectrix of Ceva
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Sectrix of Delanges
See also
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Doubling the cube
Doubling the cube, also known as the Delian problem, is an ancient geometric problem. Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first. As with the related probl ...
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Neusis construction
In geometry, the neusis (; ; plural: grc, νεύσεις, neuseis, label=none) is a geometric construction method that was used in antiquity by Greek mathematicians.
Geometric construction
The neusis construction consists of fitting a line e ...
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Quadratrix
In geometry, a quadratrix () is a curve having ordinates which are a measure of the area (or quadrature) of another curve. The two most famous curves of this class are those of Dinostratus and E. W. Tschirnhaus, which are both related to the circ ...
References
Loy, Jim "Trisection of an Angle", Part VI*
"Sectrix curve" at Encyclopédie des Formes Mathématiques Remarquables(In French)
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{{commonscat
Curves