A curve first studied by Colin Maclaurin in 1742. It was studied to provide a solution to one of the Geometric 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 that the line makes an Angle of with the negative y-Axis, then the line will make an Angle 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) |

The Maclaurin trisectrix is the Pedal Curve of the Parabola where the Pedal Point is taken as the reflection of the Focus in the Directrix.

**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.

© 1996-9

1999-05-26