Maclaurin Trisectrix


The Maclaurin trisectrix is 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 angle trisection, whence the name trisectrix. The Maclaurin trisectrix is an anallagmatic curve, and the origin is a crunode.

The Maclaurin trisectrix has Cartesian equation


or the parametric equations


The asymptote has equation x=a, and the center of the loop is at (-2a,0). If P is a point on the loop so that the line CP makes an angle of 3alpha with the negative y-axis, then the line OP will make an angle of alpha with the negative y-axis.

The Maclaurin trisectrix is given in polar coordinates as


Another form of the polar equation is the polar equation


which is a version shifted by two units along the x-axis so that the origin is inside the loop.

The tangents to the curve at the origin make angles of +/-60 degrees with the x-axis. The area and arc length of the loop are


(OEIS A138499), where E(x,k) is an elliptic integral of the second kind.

The negative x-intercept is (-3a,0) (MacTutor Archive).

The arc length, curvature, and tangential angle of the Maclaurin trisectrix (in the parametric representation given above) are


The Maclaurin trisectrix is the pedal curve of the parabola where the pedal point is taken as the reflection of the focus in the conic section directrix.

See also

Angle Trisection, Conchoid of de Sluze, Conchoid of Nicomedes, Right Strophoid, Tschirnhausen Cubic

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Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 103-106, 1972.Loy, J. "Trisection of an Angle." History of Mathematics Archive. "Trisectrix of Maclaurin.", N. J. A. Sequences A138499 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Maclaurin Trisectrix." From MathWorld--A Wolfram Web Resource.

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