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 c ...
, 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:
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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 Ehrenf ...
(a.k.a. Catalan's trisectrix and L'Hôpital's cubic)
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Durer's folium
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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 .
...
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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, cal ...
with eccentricity 2
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Rose curve
A rose is either a woody perennial flowering plant of the genus ''Rosa'' (), in the family Rosaceae (), or the flower it bears. There are over three hundred species and tens of thousands of cultivars. They form a group of plants that can be ...
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
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 mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One descript ...
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
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 ...
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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
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 ...
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Sectrix of Delanges
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 ...
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 pro ...
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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 Ehrenfried Walther von Tschirnhaus, E. W. Tschirnhaus, ...
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