Trisectricix

Maclaurin's Trisektrix for a = 1: ${\displaystyle x(x^{2}+y^{2})=(3x^{2}-y^{2})}$

A trisffektix (derived from the Latin tri for three and sectus for divided ) is a curve that enables the (exact) third of any angle with a compass and ruler . It is not possible to divide any angle into thirds with a compass and ruler alone, but if you allow a TrisTRRIX as the (only) additional aid, it is possible to divide any angle into three. If such a curve enables not only the third of an angle, but more generally the division into n parts of equal size, it is also referred to as a spectrix .

The oldest examples of a trisectrix have been known since antiquity, to them belong the trisectrix of Hippias and the spiral of Archimedes , both of which are also sectrices. Above all, the Maclaurin Trisektrix is ​​known , which is often given in the literature as a standard example of a Trisektrix. It can be described by the equation and goes back to the mathematician Colin Maclaurin (1698–1746). ${\displaystyle x(x^{2}+y^{2})=a(3x^{2}-y^{2})}$

Further examples:

Trisectricix

• Tschirnhausen-Kubik / Catalan Trisektrix ( ) ${\displaystyle y^{2}=x^{3}+3x^{2}}$
• Limaçon Trisektrix ( ) ${\displaystyle (x^{2}+y^{2}-2bx)^{2}-b^{2}(x^{2}+y^{2})\,=\,0\,,b\in \mathbb {R} ^{+}}$
• Trisffektix from Longchamps
• Parabola (as trisectrix)
• Hyperbola (as trisectrix)
• Cycloids from Ceva

Sektix

• Quadratrix from Tschirnhaus
• Sektrix from Maclaurin
• Sektrix from Ceva
• Sektrix from Delanges

See also

• Quadratrix

literature

• Steven Schwartzmann: The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English . MAA, 1994, ISBN 0-88385-511-9 ( excerpt (Google) )

Web links

Commons : Trisectrix  - collection of images, videos and audio files

• Jim Loy: Trisection of an Angle. Part VI - Cheating (using curves other than circles) ( Memento from November 4, 2013 in the Internet Archive )
• Trisffektix on an archived website of the University of Lüneburg
• Regina Bruischütz: Angle division into three parts - construction with additional aids
• Eric W. Weisstein : Trisectrix . In: MathWorld (English).
• Trisection using Special Curves