Limaçon trisectrix

Limaçon In geometry, a limaçon trisectrix is the name for the quartic plane curve that is a trisectrix that is specified as a limaçon. The shape of the limaçon trisectrix can be specified by other curves particularly as a rose, conchoid or epitrochoid. The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes, the Cycloid of Ceva, Quadratrix of Hippias, Trisectrix of Maclaurin, and Tschirnhausen cubic. The limaçon trisectrix a special case of a sectrix of Maclaurin. The limaçon trisectrix specified as a polar equation is The constant may be positive or negative. The two curves with constants and are reflections of each other across the line . The period of is given the period of the sinusoid . The limaçon trisectrix is composed of two loops. The outer loop is defined when on the polar angle interval , and is symmetric about the polar axis. The point furthest from the pole on the outer loop has the coordinates . The inner loop is defined when on the polar angle interval , and is symmetric about the polar axis. The point furthest from the pole on the inner loop has the coordinates , and on the polar axis, is one-third of the distance from the pole compared to the furthest point of the outer loop. The outer and inner loops intersect at the pole. The curve can be specified in Cartesian coordinates as and parametric equations In polar coordinates, the shape of is the same as that of the rose . Corresponding points of the rose are a distance to the left of the limaçon's points when , and to the right when . As a rose, the curve has the structure of a single petal with two loops that is inscribed in the circle and is symmetric about the polar axis. The inverse of this rose is a trisectrix since the inverse has the same shape as the trisectrix of Maclaurin. See the article Sectrix of Maclaurin on the limaçon as an instance of the sectrix. The outer and inner loops of the limaçon trisectrix have angle trisection properties.