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 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 . The inner and outer loops of the curve intersect at the pole.

Specification and loop structure

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The limaçon trisectrix specified as a polar equation is

 .[4]

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

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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

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See the article Sectrix of Maclaurin on the limaçon as an instance of the sectrix.

Trisection properties

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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

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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  .

The construction of the outer loop of   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  .

  • 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

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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.

The inner loop of the limaçon trisectrix has the desirable property that the trisection of an angle is internal to the angle being trisected.[6] 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  .

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  , where  . Thus, the polar coordinate   of   is determined by

 
 
 
 
 .

The last equation has two solutions, the first being:  , which results in  , the polar axis, a line that intersects both curves but not at   on the unit circle.

The second solution is based on the identity   which is expressed as

 , which implies  ,

and shows that   demonstrating the larger angle has been trisected.

The upper half of the inner loop can trisect any central angle of   because   implies   which is in the domain of the re-defined loop.

Line segment trisection property

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The limaçon trisectrix   trisects the line segment on the polar axis that serves as its axis of symmetry. Since the outer loop extends to the point   and the inner loop to the point  , the limaçon trisects the segment with endpoints at the pole (where the two loops intersect) and the point  , where the total length of   is three times the length running from the pole to the other end of the inner loop along the segment.

Relationship with the trisectrix hyperbola

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Given the limaçon trisectrix  , the inverse   is the polar equation of a hyperbola with eccentricity equal to 2, a curve that is a trisectrix. (See Hyperbola - angle trisection.)

References

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  1. ^ Xah Lee. "Trisectrix". Retrieved 2021-02-20.
  2. ^ Oliver Knill. "Chonchoid of Nicomedes". Harvard College Research Program project 2008. Retrieved 2021-02-20.
  3. ^ Weisstein, Eric W. "Cycloid of Ceva". MathWorld.
  4. ^ Xah Lee. "Trisectrix". Retrieved 2021-02-20.
  5. ^ Yates, Robert C. (1942). The Trisection Problem (PDF) (The National Council of Teachers of Mathematics ed.). Baton Rouge, Louisiana: Franklin Press. pp. 23–25.
  6. ^ "Trisectrix" . Encyclopædia Britannica. Vol. 27 (11th ed.). 1911. p. 292.
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  • "The Trisection Problem" by Robert C. Yates published in 1942 and reprinted by the National Council of Teachers of Mathematics available at the U.S. Dept. of Education ERIC site.