Trisectrix of Maclaurin
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In geometry, the trisectrix of Maclaurin is a cubic plane curve defined by the equation in polar coordinates
.
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
It is a trisectrix, meaning it can be used to trisect an angle.
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[edit] History
Colin Maclaurin investigated the curve in 1742.
[edit] 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
[edit] Notable points and features
The curve has an x-intercept at 
and a double point at the origin. The vertical line 
is an asymptote. The curve intersects the line x = a, or the point corresponding to the trisection of a right angle, at 
. As a nodal cubic, it is of genus zero.
[edit] 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.
[edit] References
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36,95,104–106. ISBN 0-486-60288-5.
