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 known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them.

is rational, otherwise the curve is not algebraic and is dense in the plane.

then, by the law of sines, so is the equation in polar coordinates.

is an integer greater than 2 gives arachnida or araneidan curves The case

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

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.

are integers and the fraction is in lowest terms.

This can also be written from which it is relatively simple to derive the Cartesian equation given m and n. The function

is analytic so the orthogonal trajectories of the family

Then converting the polar equation above to parametric equations produces Applying the angle addition rule for sine produces So if the origin is shifted to the right by a/2 then the parametric equations are These are the equations for Plateau curves when

, or The inverse with respect to the circle with radius a and center at the origin of is This is another curve in the family.

are integers in lowest terms and assume

be a given angle and suppose that the sectrix of Maclaurin has been drawn with poles

be the point of intersection of the ray and the sectrix and draw

Thus, the curve is an m-sectrix, meaning that with the aid of the curve an arbitrary angle can be divided by any integer.

This is a generalization of the concept of a trisectrix and examples of these will be found below.

be the point of intersection of this ray with the curve.

so any point on the circle forms an angle of

be the point intersection of this circle and the curve.

so Applying a Euclidean algorithm a third time gives an angle of

It has polar equation It is the inverse with respect to the origin of the q = 0 case.

These form the Apollonian circles with poles

From the polar equation it is evident that the curves has asymptotes at

So the conics are, in fact, rectangular hyperbolas.

which is the family of Cassini ovals with foci

The construction above gives a method that this curve may be used as a 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.