Curve which could be used to trisect an angle with compass and straightedge
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 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.)
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...
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...
{\displaystyle 3a^{2}/2} . The folium of Descartes is related to the trisectrix of Maclaurin by affine transformation. To see this, start with the equation...
Trisection using the Archimedean spiral Trisection using the Maclaurin trisectrix There are certain curves called trisectrices which, if drawn on the plane...
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...
lines at any singular point. For example, the equation of the limaçon trisectrix shown to the right is ( x 2 + y 2 − 2 a x ) 2 = a 2 ( x 2 + y 2 ) . {\displaystyle...
squaring the circle. To solve this problem, Dinostratus made use of the trisectrix of Hippias, for which he proved a special property (Dinostratus' theorem)...
equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member...
the family) a = −1, cissoid of Diocles a = −2, right strophoid a = −4, trisectrix of Maclaurin Smith, David Eugene (1958), History of Mathematics, Volume...
Archibald, though it is sometimes known as de L'Hôpital's cubic or the trisectrix of Catalan. Put t = tan ( θ / 3 ) {\displaystyle t=\tan(\theta /3)}...
Hyperboloid of one sheet Hyperboloid of two sheets A parabola can be used as a trisectrix, that is it allows the exact trisection of an arbitrary angle with straightedge...
Constructible polygon Pierpont prime Quadratrix Steel square Tomahawk (geometry) Trisectrix Benjamin, Elliot; Snyder, C (May 2014). "On the construction of the regular...
cubic curves containing several well-known examples. Specifically: The Trisectrix of Maclaurin given by 2 x ( x 2 + y 2 ) = a ( 3 x 2 − y 2 ) {\displaystyle...
family includes, in addition to the cissoid of Diocles listed above, the trisectrix of Maclaurin (d = −c/3) and the right strophoid (d = −c). Inverting the...
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