Trisectrix of Maclaurin 

The Trisectrix of Maclaurin showing the angle trisection property

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

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).

In Cartesian coordinates the equation is

2x(x2 + y2) = a(3x2 − y2)

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

r = \frac{a}{2 cos{\theta \over 3}}

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

Contents

History

Colin Maclaurin investigated the curve in 1742.

The trisection property

Given an angle φ, draw a ray from (a,0) whose angle with the x-axis is φ. 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 φ / 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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