Limaçon trisectrix

If the first line is rotating about the origin, forming angle ${\displaystyle \theta }$ with the ${\displaystyle x}$-axis, and the second line is rotating about the point ${\displaystyle (a,0)}$ with angle ${\displaystyle {3\theta \over 2}}$, then the angle between them is ${\displaystyle {\theta \over 2}}$, and the Law of Sines can be used to determine the distance from the point of intersection to the origin as

${\displaystyle r=a{\frac {\sin {\tfrac {3}{2}}\theta }{\sin {\tfrac {1}{2}}\theta }}=a(3\cos ^{2}{\tfrac {1}{2}}\theta -\sin ^{2}{\tfrac {1}{2}}\theta )=a(1+2\cos \theta )}$.

This is the equation with polar coordinates, showing that the curve is a Limaçon. The curve crosses itself at the origin, the rightmost point of the outer loop is at ${\displaystyle (3a,0)}$ and the tip of the inner loop is at ${\displaystyle (a,0)}$.

If the curve is shifted so that the origin is at the tip of the inner loop then the equation becomes

${\displaystyle r=2a\cos {\theta \over 3}}$

so it is also in the rose family of curves.

second method: ${\displaystyle \beta ={\tfrac {1}{3}}\alpha }$

There are several ways to use the curve to trisect an angle. Let ${\displaystyle \varphi }$ be the angle to be trisected. First, draw a ray from the tip of the small loop at ${\displaystyle (a,0)}$ with angle ${\displaystyle \varphi }$ with the ${\displaystyle x}$-axis. Let ${\displaystyle P}$ be the point where the ray intersects the curve, assumed to be on the outer loop if ${\displaystyle \varphi }$ is small. Draw another ray from the origin to ${\displaystyle P}$. Then the angle between the two rays at ${\displaystyle P}$ trisects ${\displaystyle \varphi }$. This follows easily from the construction of the curve given above.

For the second method, draw a circle of radius ${\displaystyle a}$ and center at the origin. Draw a ray from the origin with angle ${\displaystyle \varphi }$ with the ${\displaystyle x}$-axis. Let ${\displaystyle C}$ be the point where this ray intersects the circle and draw the line from ${\displaystyle C}$ to ${\displaystyle (a,0)}$. Let ${\displaystyle P'}$ be the point where this line intersects the curve, assumed to be on the inner loop if ${\displaystyle \varphi }$ is small. The line from the origin to ${\displaystyle P'}$ has angle ${\displaystyle {\varphi \over 3}}$ with the ${\displaystyle x}$-axis.

By rotating the curve, the second form of the equation becomes

${\displaystyle r=a\sin {\theta \over 3}}$.

So if a right triangle is constructed with side ${\displaystyle r}$ and hypotenuse ${\displaystyle a}$ then the angle between them will be ${\displaystyle {\theta \over 3}}$. It is straightforward to generate a third method from this.

Wikisource has the text of the 1911 Encyclopædia Britannica article Trisectrix.

• "Limaçon" at 2dcurves.com
• "Trisectrix" at A Visual Dictionary of Special Plane Curves
• "Limaçon Trisecteur" at Encyclopédie des Formes Mathématiques Remarquables
• Jim "Trisection of an Angle", Part VI (archived version) Gives 5 different ways to trisect an angle using this curve.