Maclaurin Trisectrix
A curve first studied by Colin Maclaurin 
in 1742. It was studied to provide a solution to one of theGeometric Problems of Antiquity, in particular Trisection of an Angle, whence the name trisectrix.The Maclaurin trisectrix is an Anallagmatic Curve, and the origin is a Crunode.
The Maclaurin trisectrix has Cartesian equation
 | (1) |
 |  |  | (2) |
 | |  | (3) |
The Asymptote has equation
, and the center of the loop is at 
. If 
is a point on the loop so thatthe line 
makes an Angle of 
with the negative y-Axis, then the line 
will make anAngle of 
with the negative y-Axis.The Maclaurin trisectrix is sometimes defined instead as
 | (4) |
 | (5) |
 | (6) |
 | (7) |
The tangents to the curve at the origin make angles of 
with the x-Axis. The Area of the loop is
 | (8) |
-intercept is 
(MacTutor Archive).The Maclaurin trisectrix is the Pedal Curve of the Parabola where the Pedal Point is taken as thereflection of the Focus in the Directrix.
See also Right Strophoid, Tschirnhausen Cubic
References
Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 103-106, 1972. Lee, X. ``Trisectrix.''http://www.best.com/~xah/SpecialPlaneCurves_dir/Trisectrix_dir/trisectrix.html. Lee, X. ``Trisectrix of Maclaurin.''http://www.best.com/~xah/SpecialPlaneCurves_dir/TriOfMaclaurin_dir/triOfMaclaurin.html MacTutor History of Mathematics Archive. ``Trisectrix of Maclaurin.''http://www-groups.dcs.st-and.ac.uk/~history/Curves/Trisectrix.html.
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