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.
Specification and loop structure
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
, .
Relationship with rose curves
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.
Relationship with the sectrix of Maclaurin
See the article Sectrix of Maclaurin on the limaçon as an instance of the sectrix.
Trisection properties
The outer and inner loops of the limaçon trisectrix have angle trisection properties. Theoretically, an angle may be trisected using a method with either property, though practical considerations may limit use.
Outer loop trisectrix property
The construction of the outer loop of reveals its angle trisection properties. The outer loop exists on the interval . Here, we examine the trisectrix property of the portion of the outer loop above the polar axis, i.e., defined on the interval .
First, note that polar equation is a circle with radius , center on the polar axis, and has a diameter that is tangent to the line at the pole . Denote the diameter containing the pole as , where is at .
Second, consider any chord of the circle with the polar angle . Since is a right triangle, . The corresponding point on the outer loop has coordinates , where .
Given this construction, it is shown that and two other angles trisect as follows:
, as it is the central angle for on the circle .
The base angles of isosceles triangle measure – specifically, .
The apex angle of isosceles triangle is supplementary with , and so, . Consequently the base angles, and measure .
. Thus is trisected, since .
Note that also , and .
The upper half of the outer loop can trisect any central angle of because implies which is in the domain of the outer loop.
Inner loop trisectrix property
The inner loop of the limaçon trisectrix has the desirable property that the trisection of an angle is internal to the angle being trisected. Here, we examine the inner loop of that lies above the polar axis, which is defined on the polar angle interval . The trisection property is that given a central angle that includes a point lying on the unit circle with center at the pole, , has a measure three times the measure of the polar angle of the point at the intersection of chord and the inner loop, where is at .
In Cartesian coordinates the equation of is , where , which is the polar equation
, where and . (Note: atan2(y,x) gives the polar angle of the Cartesian coordinate point (x,y).)
Since the normal line to is , it bisects the apex of isosceles triangle , so and the polar coordinates of is .