Limaçon trisectrix

Limaçon Trisectrix

In geometry, a limaçon trisectrix (called simply a trisectrix by some authors) is a member of the limaçon family of curves which has the trisectrix, or angle trisection, property. 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 2:3 and the lines initially coincide with the line between the two points. Thus, it is an example of a sectrix of Maclaurin.

Equations

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

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

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 $(3a,0)$ and the tip of the inner loop is at $(a,0)$.

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

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

so it is also in the rose family of curves.

The trisection property

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

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

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

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

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

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

References

• "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.