Maclaurin Trisectrix

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

$$y^2={x^2(x+3a)\over a-x},$$ (1)

or the parametric equations

$$x = a{t^2-3\over t^2+1}$$ (2)

$$y = a{t(t^2-3)\over t^2+1}.$$ (3)

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 $3\alpha$ with the negative y-Axis, then the line $OP$ will make an Angle of $\alpha$ with the negative y-Axis.

The Maclaurin trisectrix is sometimes defined instead as

$$x(x^2+y^2)=a(y^2-3x^2)$$ (4)

$$y^2={x^2(3a+x)\over a-x}$$ (5)

$$r={2a\sin(3\theta)\over \sin(2\theta)}.$$ (6)

Another form of the equation is the Polar Equation

$$r = a \sec({\textstyle{1\over 3}}\theta),$$ (7)

where the origin is inside the loop and the crossing point is on the Negative x-Axis.

The tangents to the curve at the origin make angles of $\pm 60^\circ$ with the x-Axis. The Area of the loop is

$$A_{\rm loop}=3\sqrt{3}\,a^2,$$ (8)

and the Negative $x$-intercept is $(-3a,0)$ (MacTutor Archive).

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.

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.