trisectrix of MacLaurin

 

where sec stands for the secant.

Colin MacLaurin who was the first to study the curve (in 1742), while looking at the ancient Greek problem of the trisection of an angle: the angle formed by points ABP is three times the angle formed by AOP for points P of the trisectrix.

The area of the loop is equal to , and the distance from the origin to the point where the curve cuts the x-axis is equal to 3.
The cubic curve ¹⁾ has a second well-known polar form:


Some relationships with other curves are the following:

• the trisectrix is a pedal of the parabola
• it is the polar inverse of Tschirnhausen's cubic (with the focus as center of inversion)
• it is the polar inverse of the hyperbola (with the nod as center of inversion)

The curve is an anallagmatic curve, and also an epi spiral (a=1/3).

Freeth (1819-1904) described in a paper published by the London Mathematical Society (1879) the strophoid of the trisectrix.

notes

1) With Cartesian coordinates:

y² = x²(3+x)/(1-x).