Limaçon trisectrix

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.

[1] The curve is one among a number of plane curve trisectrixes that includes the Conchoid of Nicomedes,[2] the Cycloid of Ceva,[3] 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

The limaçon trisectrix is composed of two loops.

As a rose, the curve has the structure of a single petal with two loops that is inscribed in the circle

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.

reveals its angle trisection properties.

[5] 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

as follows: The upper half of the outer loop can trisect any central angle of

The inner loop of the limaçon trisectrix has the desirable property that the trisection of an angle is internal to the angle being trisected.

The trisection property is that given a central angle that includes a point

lying on the unit circle with center at the pole,

, which is the polar equation (Note: atan2(y,x) gives the polar angle of the Cartesian coordinate point (x,y).)

, it bisects the apex of isosceles triangle

With respect to the limaçon, the range of polar angles

that defines the inner loop is problematic because the range of polar angles subject to trisection falls in the range

Furthermore, on its native domain, the radial coordinates of the inner loop are non-positive.

The inner loop then is equivalently re-defined within the polar angle range of interest and with non-negative radial coordinates as

, the polar axis, a line that intersects both curves but not at

demonstrating the larger angle has been trisected.

The upper half of the inner loop can trisect any central angle of

trisects the line segment on the polar axis that serves as its axis of symmetry.

Since the outer loop extends to the point

, the limaçon trisects the segment with endpoints at the pole (where the two loops intersect) and the point

is three times the length running from the pole to the other end of the inner loop along the segment.

is the polar equation of a hyperbola with eccentricity equal to 2, a curve that is a trisectrix.

The limaçon trisectrix specified as the polar equation where a > 0 . When a < 0 , the resulting curve is the reflection of this curve with respect to the line As a function, r has a period of 2π . The inner and outer loops of the curve intersect at the pole.
Angle trisection property of the (green) outer loop of the limaçon trisectrix . The (blue) generating circle is required to prove the trisection of . The (red) construction results in two angles, and , that have one-third the measure of ; and one angle, , that has two-thirds the measure of .
Angle trisection property of the (green) inner loop of the limaçon trisectrix . Given a point on the (blue) unit circle centered at the pole with at , where (in red) intersects the inner loop at , trisects . The (black) normal line to is , so is at . The inner loop is re-defined on the interval as because its native range is greater than where its radial coordinates are non-positive.