sectrix of Maclaurin

trigonometric

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 arachnida1) or araneidan.

    For a=2, the curve is a circle.

    For a=3: we see the cubic trisectrix of Maclaurin2).
    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 x-axis 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 (1819-1904) 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 (1801-1883).

    Some examples:










    For a=3/2 the curve is the (limašon) 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: x3 + xy2 - 3x2 - y2 = 0,
or: 2x3 + 2xy2 - 3x2 + y2 = 0.