Trisectrix
Encyclopedia
In geometry
, a trisectrix is a curve which can be used to trisect
an arbitrary angle. 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 are a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:
A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. Examples include:
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a trisectrix is a curve which can be used to trisect
Angle trisection
Angle trisection is a classic problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one-third of a given arbitrary angle, using only two tools: an un-marked straightedge, and a compass....
an arbitrary angle. 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 are a variety of such curves and the methods used to construct an angle trisector differ according to the curve. Examples include:
- Limaçon trisectrixLimaçon trisectrixIn geometry, a limaçon trisectrix is a member of the Limaçon family of curves which has the trisectrix, or angle trisection, property...
(some sources refer to this curve as simply the trisectrix.) - Trisectrix of MaclaurinTrisectrix of MaclaurinIn 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 points of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the...
- Equilateral trefoil (aka Longchamps' Trisectrix)
- Tschirnhausen cubicTschirnhausen cubicIn geometry, Tschirnhausen cubic, is a plane curve defined by the polar equationr = a\sec^3.-History:The curve was studied by von Tschirnhaus, de L'Hôpital and Catalan. It was given the name Tschirnhausen cubic in a 1900 paper by R C Archibald, though it is sometimes known as de L'Hôpital's cubic...
(aka Catalan's trisectrix and L'hospital's cubic) - Durer's folium
- Cubic parabola
- HyperbolaHyperbolaIn mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
with eccentricity 2 - Rose with 3 petalsRose (mathematics)In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, thesecurves can all be expressed by a polar equation of the form\!\,r=\cos.If k is an integer, the curve will be rose shaped with...
- ParabolaParabolaIn mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
A related concept is a sectrix, which is a curve which can be used to divide an arbitrary angle by any integer. Examples include:
- Quadratrix of Hippias
- Sectrix of MaclaurinSectrix of MaclaurinIn 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 as a curve whose equation in biangular coordinates is...
- Sectrix of Ceva
- Sectrix of Delanges