Trisectrix of Maclaurin

In geometry, the trisectrix of Maclaurin is a cubic plane curve notable for its trisectrix property, meaning it can be used to trisect an angle. It can be defined as locus of the point of intersection of two lines, each rotating at a uniform rate about separate points, so that the ratio of the rates of rotation is 1:3 and the lines initially coincide with the line between the two points. A generalization of this construction is called a sectrix of Maclaurin. The curve is named after Colin Maclaurin who investigated the curve in 1742.

Let two lines rotate about the points ${\displaystyle P=(0,0)}$ and ${\displaystyle P_{1}=(a,0)}$ so that when the line rotating about ${\displaystyle P}$ has angle ${\displaystyle \theta }$ with the x axis, the rotating about ${\displaystyle P_{1}}$ has angle ${\displaystyle 3\theta }$. Let ${\displaystyle Q}$ be the point of intersection, then the angle formed by the lines at ${\displaystyle Q}$ is ${\displaystyle 2\theta }$. By the law of sines,

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