Imagine two lines revolving at a constant rate round two different points (poles).
The intersection of the two lines make a sectrix of Maclaurin.
The name comes from the trisectrix of Maclaurin (see below), together with the sectrix quality (able to divide an angle).
Parameters a and 1/a give the same curves, so we can confine ourselves to the case a>=1.
For a=1, the curve is a line.
For b=0, two curve groups can be distinguished:
 parameter a is an integer: the curve is called an arachnida^{1)} or
araneidan.
For a=2, the curve is a circle.
For a=3: we see the cubic trisectrix of Maclaurin^{2)}.
Colin Maclaurin 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 ^{3}√3, and the distance from the origin to the point where the curve cuts the xaxis is equal to 3.
Some relationships with other curves are the following:
The curve is an anallagmatic curve,
and also an epi spiral (a=1/3).
Freeth (18191904) described in a paper published by the London
Mathematical Society (1879) the strophoid
of the trisectrix.
For larger values for the parameter a we see better the spiderly character of the curve:
A negative parameter a gives a variant on the curve, consisting of a number of (a) hyperbolic branches:
 parameter a is a rational number: the curve is an Plateau curve.
The curve is named after the Belgian physicist Joseph Antoine Ferdinand Plateau (18011883).
Some examples:
For a=3/2 the curve is the (limaçon) trisectrix.
When parameter a is positive, the curve consists of a number of elliptic forms.
The number of elliptics is equal to the denominator of the parameter.
When parameter a is negative, the curve consists of a number of hyperbolic forms.
The number of hyperbolics is equal to the sum of the nominator and denominator minus 1:
notes
1) From the Greek word αραχνη that means spider.
2) Cartesian equation: x^{3} + xy^{2}  3x^{2}  y^{2} = 0,
or: 2x^{3} + 2xy^{2}  3x^{2} + y^{2} = 0.
