دانشنامه ریاضی و کامپیوتر

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

(1)

or the parametric equation s

			(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 given in polar coordinates as

			(4)
			(5)

Another form of the polar equation is the polar equation

(6)

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

			(7)
			(8)
			(9)

(Sloane's 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

			(10)
			(11)
			(12)

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

REFERENCES:

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 103-106, 1972.

Loy, J. "Trisection of an Angle." http://www.jimloy.com/geometry/trisect.htm#curves.

MacTutor History of Mathematics Archive. "Trisectrix of Maclaurin." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Trisectrix.html.

Sloane, N. J. A. Sequences A138499 in "The On-Line Encyclopedia of Integer Sequences."