Trisectrix of Maclaurin

The trisection property

Given an angle ${\displaystyle \varphi }$, draw a ray from ${\displaystyle (a,0)}$ whose angle with the ${\displaystyle x}$-axis is ${\displaystyle \varphi }$. 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 ${\displaystyle \varphi /3}$.

Notable points and features

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

Relationship to other curves

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 ${\displaystyle x=\frac{a}{2}}$ relative to the origin.

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

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

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.