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

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External links

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* [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]