Trisectrix of Maclaurin

Let two lines rotate about the points ${\displaystyle P=(0,0)}$ and ${\displaystyle P_{1}=(a,0)}$ so that when the line rotating about ${\displaystyle P}$ has angle ${\displaystyle \theta }$ with the x axis, the rotating about ${\displaystyle P_{1}}$ has angle ${\displaystyle 3\theta }$. Let ${\displaystyle Q}$ be the point of intersection, then the angle formed by the lines at ${\displaystyle Q}$ is ${\displaystyle 2\theta }$. By the law of sines,

${\displaystyle {r \over \sin 3\theta }={a \over \sin 2\theta }\!}$

so the equation in polar coordinates is (up to translation and rotation)

${\displaystyle r=a{\frac {\sin 3\theta }{\sin 2\theta }}={a \over 2}{\frac {4\cos ^{2}\theta -1}{\cos \theta }}={a \over 2}(4\cos \theta -\sec \theta )\!}$.

The curve is therefore a member of the Conchoid of de Sluze family.

In Cartesian coordinates the equation of this curve is

${\displaystyle 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})\!}$.

If the origin is moved to (a, 0) then a derivation similar to that given above shows that the equation of the curve in polar coordinates becomes

${\displaystyle r=2a\cos {\theta \over 3}\!}$

making it an example of a limacon with a loop.

The Trisectrix of Maclaurin showing the angle trisection property

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

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

The trisectrix of Maclaurin can be defined from conic sections in three ways. Specifically:

• It is the inverse with respect to the unit circle of the hyperbola

${\displaystyle 2x=a(3x^{2}-y^{2})}$.

• It is cissoid of the circle

${\displaystyle (x+a)^{2}+y^{2}=a^{2}}$

and the line ${\displaystyle x={a \over 2}}$ relative to the origin.

• It is the pedal with respect to the origin of the parabola

${\displaystyle y^{2}=2a(x-{\tfrac {3}{2}}a)}$.

In addition:

• The inverse with respect to the point ${\displaystyle (a,0)}$ is the Limaçon trisectrix.
• The trisectrix of Maclaurin is related to the Folium of Descartes by affine transformation.

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 95, 104–106. ISBN 0-486-60288-5.
• Weisstein, Eric W. "Maclaurin Trisectrix". MathWorld.
• "Trisectrix of Maclaurin" at MacTutor's Famous Curves Index
• Maclaurin Trisectrix at mathcurve.com
• "Trisectrix of Maclaurin" at Visual Dictionary Of Special Plane Curves

Wikimedia Commons has media related to Maclaurin's Trisectrix.

• Loy, Jim "Trisection of an Angle", Part VI