Limaçon trisectrix

In geometry, a limaçon trisectrix, or simply trisectrix is a member of the Limaçon family of curves which is named from its trisectrix or angle trisection property. Up to rotation and translation, the equation in polar coordinates is

: $r=b(1+2cos\; heta),,$

where the origin at the point where the curve crosses itself. If the origin is taken to be the tip of the inner loop then the equation becomes

: $r\; =\; 2bcos\{\; heta\; over\; 3\}.$

To demonstrate the trisection property, draw a ray at an angle of "θ" to the "x"-axis, beginning at point "C" (1,0) (the center of the small loop) which intersects the large loop of the trisectrix at point "P". Draw another ray from the origin O (0,0) to point "P". These two rays intersect at an angle of (1/3)"θ".

To prove the trisection property, label point D (3,0) (the center of the large loop of the trisectrix), and draw a unit circle centered at C (1,0), whose polar equation is $r=2cos\; heta$. Label point Q where OP intersects the circle. Observe that $riangle\; OCQ$ and $riangle\; CQP$ are isosceles. (OQ and OP differ in length by 1, as seen from the two equations.) By the properties of isosceles triangles, $angle\; OQC$ is twice $angle\; OPC$, and $angle\; QCD$ is twice $angle\; OQC$. Thus $angle\; PCD$ is three times $angle\; OPC$.