Spiral curve

Spiral curves as plane sections of a torus

In geometry, a spiritual curve , also called the spiritual curve of Perseus or spiritual line , is a plane curve of the fourth order with an equation of form

${\displaystyle (x^{2}+y^{2})^{2}=dx^{2}+ey^{2}+f.}$

An equivalent definition is:

A spiral curve is the plane intersection of a torus with a plane that is parallel to the axis of rotation.

The last definition gives one a good idea of ​​the possible shape of a spiritual curve. The spiral curves are not all possible intersection curves of a torus with a plane. But they contain the Cassinian curves as a subset and thus also the lemniscates of Bernoulli . The lemniscate of Booth are special spirische curves.

Spiral curves were first identified by the Greek geometer Perseus around 150 BC. Studied as plane sections of a torus. The name spirisch comes from the Greek name spira for the torus.

Equations

${\displaystyle r=1,R=2,y_{0}=0;\,0{,}8;\,1}$

If you cut the torus with the equation

${\displaystyle \left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y^{2}\right)}$

with the plane , we get first ${\displaystyle y=y_{0}}$

${\displaystyle \left(x^{2}+y_{0}^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4R^{2}\left(x^{2}+y_{0}^{2}\right)}$.

If you partially open the left bracket, you get

${\displaystyle (x^{2}+z^{2})^{2}=dx^{2}+ez^{2}+f}$

With:

${\displaystyle d=2(r^{2}+R^{2}-y_{0}^{2})}$,

${\displaystyle e=2(r^{2}-R^{2}-y_{0}^{2})}$

${\displaystyle f=-(r+R+y_{0})\;(r+R-y_{0})\;(r-R+y_{0})\;(r-R-y_{0})}$

This is the equation of a spiral curve in the coordinates. ${\displaystyle x-z}$

In polar coordinates results ${\displaystyle \rho ,\;\varphi }$

${\displaystyle (\rho ^{2}-r^{2}+R^{2}+y_{0}^{2})^{2}=4r^{2}(\rho ^{2}\cos ^{2}\varphi +y_{0}^{2})}$

and from this the implicit representation

${\displaystyle \rho ^{4}-\rho ^{2}(d\cos ^{2}\varphi +e\sin ^{2}\varphi )-f=0}$.

Spiral curves on a spindle torus

Spiral curves on a spindle torus

Spiral curves of a spindle torus, the planes of which also intersect the spindle (the inner part), consist of an outer and an inner curve (see picture).

Spiral curves as isoptic curves

Isoptic curves of ellipses and hyperbolas are spiritual curves. (See also: Weblink The Mathematics Enthusiast. )

See also

• Villarceau circles

literature

• Kuno Fladt , Arnold Baur: Analytical geometry of special surfaces and space curves (=  Collection Vieweg . Volume 136 ). Vieweg Verlag, Braunschweig 1975, ISBN 978-3-528-08278-9 , p. 94 ( MR0430974 ).
• Gino Loria : Special algebraic and transcendent plane curves. Theory and history (=  BG Teubner's collection of textbooks in the field of mathematical sciences, including their applications . V, 1). 2nd Edition. Volume 1: The Algebraic Curves . BG Teubner Verlag, Leipzig / Berlin 1910, p. 124-127 .

Web links

• R. Böttcher: Introduction to the theory of algebraic curves. (PDF) p. 41
• About Perseus and his curves
• "Circle" for drawing a spiritual curve
• TU Vienna: torus cuts
• Eric W. Weisstein : Spiric Section . In: MathWorld (English).
• MacTutor history
• spiric section . 2Dcurves.com
• Spirique de Persée at Encyclopédie des Formes Mathématiques Remarquables
• MacTutor biography of Perseus
• The Mathematics Enthusiast. Number 9, article 4

Individual evidence

1. ^ John Stillwell: Mathematics and Its History. Springer-Verlag, 2010, ISBN 978-1-4419-6053-5 , p. 33.
2. ^ Wilbur Richard Knorr: The Ancient Tradition of Geometric Problems. Dover Publ., New York 1993, ISBN 0-486-67532-7 , p. 268.