Sectrix of Maclaurin

In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases are also known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them.
Equations in polar coordinates.
We are given two lines rotating about two poles formula_1 and formula_2. By translation and rotation we may assume formula_3 and formula_4. At time formula_5, the line rotating about formula_1 has angle formula_7 and the line rotating about formula_2 has angle formula_9, where formula_10, formula_11, formula_12 and formula_13 are constants. Eliminate formula_5 to get 
formula_15 where formula_16 and formula_17. We assume formula_18 is rational, otherwise the curve is not algebraic and is dense in the plane. Let formula_19 be the point of intersection of the two lines and let formula_20 be the angle at formula_19, so formula_22. If formula_23 is the distance from formula_1 to formula_19 then, by the law of sines, 
so
is the equation in polar coordinates.
The case formula_28 and formula_29 where formula_30 is an integer greater than 2 gives arachnida or araneidan curves
The case formula_28 and formula_33 where formula_30 is an integer greater than 1 gives alternate forms of arachnida or araneidan curves
A similar derivation to that above gives
as the polar equation (in formula_37 and formula_38) if the origin is shifted to the right by formula_39. Note that this is the earlier equation with a change of parameters; this to be expected from the fact that two poles are interchangeable in the construction of the curve.
Equations in the complex plane, rectangular coordinates and orthogonal trajectories.
Let formula_40 where formula_41 and formula_30 are integers and the fraction is in lowest terms. In the notation of the previous section, we have 
formula_15 or 
formula_44.
If formula_45 then formula_46, so the equation becomes
formula_47 or 
formula_48. This can also be written
from which it is relatively simple to derive the Cartesian equation given m and n. The function 
formula_50 is analytic so the orthogonal trajectories of the family formula_51 are the curves formula_52, or
formula_53 These form the Apollonian circles with poles formula_54 and formula_55.
"q" = -1.
These curves have polar equation 
complex equation formula_57 In rectangular coordinates this becomes
formula_58 which is a conic. From the polar equation it is evident that the curves has asymptotes at formula_59 and formula_60 which are at right angles. So the conics are, in fact, rectangular hyperbolas. The center of the hyperbola is always formula_61. The orthogonal trajectories of this family are given by 
formula_62 which is the family of Cassini ovals with foci 
formula_54 and formula_55.
Trisectrix of Maclaurin.
In the case where formula_65 (or formula_66 by switching the poles) and formula_28, the equation is
This is the Trisectrix of Maclaurin which is specific case whose generalization is the sectrix of Maclaurin. The construction above gives a method that this curve may be used as a trisectrix.
Limaçon trisectrix and rose.
In the case where formula_69 (or formula_70 by switching the poles) and formula_28, the equation is
This is the Limaçon trisectrix.
The equation with the origin take to be the other pole is the rose curve that has the same shape
The 3 in the numerator of "q" and the construction above give a method that the curve may be used as a trisectrix.