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
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or the parametric equations
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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 given in polar coordinates as
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Another form of the polar equation is the polar equation
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which is a version shifted by two units along the -axis so that the origin is inside the loop.
The tangents to the curve at the origin make angles of with the x-axis. The area and arc length of the loop are
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(OEIS A138499), where is an elliptic integral of the second kind.
The negative -intercept is (MacTutor Archive).
The arc length, curvature, and tangential angle of the Maclaurin trisectrix (in the parametric representation given above) are
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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.