Trisectrix of Maclaurin

In geometry, the trisectrix of Maclaurin is a cubic plane curve defined by the equation in polar coordinates

:$$r= a frac{sin 3 heta}{sin 2 heta} = {a over 2} frac{4 cos^2 heta - 1} {cos heta} = {a over 2} (4 cos heta - sec heta).

In Cartesian coordinates the equation is

:$$2x(x^2+y^2)=a(3x^2-y^2)

If the origin is moved to (a, 0) then the equation of the curve in polar coordinated becomes

:$$r = frac{a}{2 cos{ heta over 3

It is a trisectrix, meaning it can used to trisect an angle.

History

Colin Maclaurin investigated the curve in 1742.

The trisection property

Given an angle $$phi, draw a ray from $$(a, 0) whose angle with the $$x-axis is $$phi. Draw a ray from the origin to the point where the first ray intersects the curve. Then the angle between the second ray and the $$x-axis is $$phi/ 3

Notable points and features

The curve has an x-intercept at $$3a over 2 and a double point at the origin. The vertical line $$x={-{a over 2 is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at $$(a,{pm {1 over sqrt{3 a}). As a nodal cubic, it is of genus zero.

Relationship to other curves

The inverse with respect to the origin is a hyperbola with eccentricity 2. The inverse with respect to the point $$(a, 0) is the Limaçon trisectrix. The trisectrix of Maclaurin is a member of the Conchoid of de Sluze family of curves. The trisectrix of Maclaurin is related to the Folium of Descartes by affine transformation.

References

*

External links

*
* [http://www.jimloy.com/geometry/trisect.htm#curves Loy, Jim "Trisection of an Angle", Part VI]
* [http://www.2dcurves.com/cubic/cubictr.html "Trisectrix of MacLaurin" on 2dcurves.com]
* [http://www.mathcurve.com/courbes2d/courbes2d.shtml "Trisectrix of Maclaurin" at Encyclopédie des Formes Mathématiques Remarquables]