Trisffektix from Maclaurin

Definition based on a rotating straight line

Definition based on an outer circle

Definition based on two rotating straight lines

The trisectrix of Maclaurin , named after Colin Maclaurin (1698-1746) is a plane curve which the trisection of angles can be used (hence trisectrix ).

Geometric definitions

In the literature, the Maclaurin trisectrix is usually defined as a locus , unless it is only given as a parametric form or equation . However, there is no standard construction for generating the locus curve, but different geometrical constructions can be found in the literature, which of course all generate the same curve.

Definition based on a rotating straight line

On a straight line you first choose a point and on the same side of two further points and , which have the distance or . In one sets a vertical line and in one lets a straight line rotate. The rotating straight line cuts through the vertical at a point , at this point a perpendicular to the rotating straight line is established. This perpendicular intersects the parallel to the rotating straight line at a point . The Trisffektix is now the locus curve of the point , which is created by the rotation of the straight line through . ${\ displaystyle O}$ ${\ displaystyle O}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle O}$ ${\ displaystyle 3a}$ ${\ displaystyle 4a}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle A}$ ${\ displaystyle E}$ ${\ displaystyle O}$ ${\ displaystyle F}$ ${\ displaystyle F}$ ${\ displaystyle B}$

Definition based on an outer circle

A point can be rotated on a circle with a radius , center point and diameter . The locus of the intersection of the mid-perpendicular of the line and the straight line is the Maclaurin TrisTRRIX. ${\ displaystyle 2a}$ ${\ displaystyle M}$ ${\ displaystyle AB}$ ${\ displaystyle E}$ ${\ displaystyle EM}$ ${\ displaystyle EA}$

Definition based on two rotating straight lines

In contrast to the two previous definitions, the Trisektrix does not arise from a uniform movement or rotation, but from two with different speeds. On a straight line you first select a point and then a point that has the distance . Now you let straight lines rotate at a uniform speed in both points, with the straight line in rotating at three times the speed of the straight line in . The locus of the intersection of the two rotating straight lines is the Trisektrix of Maclaurin. ${\ displaystyle O}$ ${\ displaystyle M}$ ${\ displaystyle O}$ ${\ displaystyle 2a}$ ${\ displaystyle M}$ ${\ displaystyle O}$

Equation and parameter curve

Taking the axis of symmetry of trisectrix on the x-axis of a coordinate system and placed while the colon from the above definitions, the origin and the points , and in the corresponding intervals on the positive x-axis, we get the following figures as an equation or Parameter curve. ${\ displaystyle O}$ ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle M}$ ${\ displaystyle O}$

Equation in polar coordinates

{\ displaystyle r = 2a {\ frac {\ sin (3 \ varphi)} {\ sin (2 \ varphi)}} = a \ left (4 \ cos (\ varphi) - {\ frac {1} {\ cos (\ varphi)}} \ right)}

.

Equation in Cartesian coordinates

{\ displaystyle x (x ^ {2} + y ^ {2}) = a (3x ^ {2} -y ^ {2})}

.

Parameter curves

The following representation is obtained as a parameter curve with trigonometric functions based on the equation in polar coordinates: ${\ displaystyle \ gamma (t) = {\ begin {pmatrix} x (t) \\ y (t) \ end {pmatrix}}}$

{\ displaystyle \ gamma: \ left (0, {\ frac {\ pi} {2}} \ right) \ cup \ left ({\ frac {\ pi} {2}}, \ pi \ right) \ rightarrow \ mathbb {R} ^ {2}}

with and

{\ displaystyle x (t) = a (\ cos (\ varphi) ^ {2} -1)}

{\ displaystyle y (t) = a (2 \ sin (\ varphi) - \ tan (\ varphi))}

There is also a representation based on rational functions:

{\ displaystyle \ gamma: (- \ infty, \ infty) \ rightarrow \ mathbb {R} ^ {2}}

with and:

{\ displaystyle x (t) = a {\ frac {3-t ^ {2}} {1 + t ^ {2}}}}

{\ displaystyle y (t) = at {\ frac {3-t ^ {2}} {1 + t ^ {2}}}}

Angular trisection

To divide an angle into three, one leg is placed on the axis of symmetry of the Trisektrix, so that the tip of the angle is in , the center point is from the above definition, i.e. it lies within the loop of the Trisektrix and is at the distance from its colon . The point of intersection of the other leg with the trisectrix is connected with the colon . The angle that this connecting line forms with the axis of symmetry is exactly one third of the starting angle . ${\ displaystyle M}$ ${\ displaystyle M}$ ${\ displaystyle O}$ ${\ displaystyle 2a}$ ${\ displaystyle F}$ ${\ displaystyle O}$ ${\ displaystyle \ angle MOF}$ ${\ displaystyle O}$

Other properties

Trisffektix (red) as the base curve of a parabola (green) as well as a hyperbola (blue) as the inverse of the Trisektrix and the asymptote of the Trisektrix (orange)

As an asymptote, the Trisektrix has a straight line that is perpendicular to the axis of symmetry of the Trisektrix and has the distance from the colon . For the above representation of the trisectrix in the coordinate system we get for the asymptote: ${\ displaystyle O}$ ${\ displaystyle a}$

{\ displaystyle x = -a}

The inverse of the trisectrix ( reflection on the unit circle ) is a hyperbola with the following equation:

{\ displaystyle x = a (3x ^ {2} -y ^ {2})}

The Trisffektix can also be generated as a base point curve of a parabola. This is how the above representations of the Trisektrix arise in the coordinate system as a base point curve of the following parabola with a pole at the origin:

{\ displaystyle y ^ {2} = 4a (x-3a)}

The area enclosed by the loop of the Trisektrix is units of area, the loop having a length of approximately units of length. ${\ displaystyle 3 {\ sqrt {3}} a ^ {2}}$ ${\ displaystyle 8 {,} 2446532a}$

literature

Dörte Haftendorn: Exploring and understanding curves: With GeoGebra and other tools . Springer, 2016, ISBN 9783658147495 , pp. 62–64
Gino Loria: Special Algebraic and Transcendent Plane Curves: Theory and History . Teubner, 1902, pp. 81-83
Underwood Dudley: The Trisectors . MAA, 1994, ISBN 9780883855140 , pp. 12-14
Daniele Ritelli, Aldo Scimone: A New Way for Old Loci . International Journal of Geometry, Volume 6 (2017), No. 2, pp. 86-92
Jack Eidswick: Two Trisectrices for the Price of One Rolling Coin. The College Mathematics Journal, Vol. 24, No. 5, 1993, pp. 422-430 ( JSTOR )

Web links

Commons : Trisektrix by Maclaurin - collection of images, videos and audio files

Eric W. Weisstein : Maclaurin Trisectrix . In: MathWorld (English).
Trisectrix of MacLaurin in the MacTutor History of Mathematics archive
Maclaurin Trisectrix on mathcurve.com
Justin Seago: The Maclaurin Trisectrix

Individual evidence

↑ Gino Loria: Special algebraic and transcendent level curves: theory and history . Teubner, 1902, pp. 81-83
↑ Dörte Haftendorn: Exploring and understanding curves: With GeoGebra and other tools . Springer, 2016, ISBN 9783658147495 , pp. 62–64
^ Anthony Lo Bello: Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots . JHU Press, 2013, ISBN 9781421410999 , p. 265

[Loria-1] Gino Loria: Special algebraic and transcendent level curves: theory and history . Teubner, 1902, pp. 81-83

[Haftendorn-2] Dörte Haftendorn: Exploring and understanding curves: With GeoGebra and other tools . Springer, 2016, ISBN 9783658147495 , pp. 62–64

[Bello-3] Anthony Lo Bello: Origins of Mathematical Words: A Comprehensive Dictionary of Latin, Greek, and Arabic Roots . JHU Press, 2013, ISBN 9781421410999 , p. 265