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 equations
 |  |  |
(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)
|
 |  | ![-[1+2cos(2theta)]sectheta.](/images/equations/MaclaurinTrisectrix/Inline19.svg) |
(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)
|
(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
 |  |  |
(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.
