Trisectrix of Maclaurin.html

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

2 x (x 2 + y 2) = a (3 x 2 - y 2)

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

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

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

1 History
2 The trisection property
3 Notable points and features
4 Relationship to other curves
5 References
6 External links

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

J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36,95,104–106. ISBN 0-486-60288-5.

External links

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