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.

We are given two lines rotating about two poles ${\displaystyle P}$ and ${\displaystyle P_{1}}$. By translation and rotation we may assume ${\displaystyle P=(0,0)}$ and ${\displaystyle P_{1}=(a,0)}$. At time ${\displaystyle t}$, the line rotating about ${\displaystyle P}$ has angle ${\displaystyle \theta =\kappa t+\alpha }$ and the line rotating about ${\displaystyle P_{1}}$ has angle ${\displaystyle \theta _{1}=\kappa _{1}t+\alpha _{1}}$, where ${\displaystyle \kappa }$, ${\displaystyle \alpha }$, ${\displaystyle \kappa _{1}}$ and ${\displaystyle \alpha _{1}}$ are constants. Eliminate ${\displaystyle t}$ to get ${\displaystyle \theta _{1}=q\theta +\theta _{0}}$ where ${\displaystyle q=\kappa _{1}/\kappa }$ and ${\displaystyle \theta _{0}=\alpha _{1}-q\alpha }$. We assume ${\displaystyle q}$ is rational, otherwise the curve is not algebraic and is dense in the plane. Let ${\displaystyle Q}$ be the point of intersection of the two lines and let ${\displaystyle \psi }$ be the angle at ${\displaystyle Q}$, so ${\displaystyle \psi =\theta _{1}-\theta }$. If ${\displaystyle r}$ is the distance from ${\displaystyle P}$ to ${\displaystyle Q}$ then, by the law of sines,

...
Wikipedia