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[[File:Steiner-erz-def.png|250px|thumb|1. Definition of the Steiner generation of a conic section]]
[[File:Persp-geradenb.png|250px|thumb|2. Perspective mapping between lines]]
[[File:Steiner-erz-beisp.png|250px|thumb|Example of a Steiner generation: generation of a point]]
The '''Steiner theorem''', or '''Steiner generation of a conic ''', named after the Swiss mathematician [[Jakob Steiner]], is an alternative method to define a non-degenerate [[Quadric (projective geometry)|projective conic section]] in a [[projective plane]] over a [[Field (mathematics)|field]]:

*Given two pencils <math>B(U),B(V)</math> of lines at two points <math>U,V</math> (all lines containing <math>U</math> and <math>V</math> resp.) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>. Then the intersection points of corresponding lines form a non-degenerate projective conic section <ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891 kB), p. 38.</ref><ref>''Jacob Steiner’s Vorlesungen über synthetische Geometrie'', B. G. Teubner, Leipzig 1867 (bei Google Books: [http://books.google.de/books?id=jCgPAAAAQAAJ]), 2. Teil, p. 96</ref> (1. picture)

A ''perspective'' mapping <math>\pi</math> of a pencil <math>B(U)</math> onto a pencil <math>B(V)</math> is a [[bijection]] (1-1 correspondence) such that corresponding lines intersect on a fixed line <math>a</math>, which is called the ''axis'' of the perspectivity <math>\pi</math> (2. picture).

A ''projective '' mapping is a finite sequence of perspective mappings.

Examples of commonly used fields are the real numbers <math>\R</math>, the rational numbers <math>\Q</math> or the complex numbers <math>\C</math>. Even finite fields are allowed.

''Remark:''
The fundamental theorem for projective planes <ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891 kB), p. 19.</ref> states, that a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines. That means for the Steiner generation of a conic section: besides two points <math>U,V</math> only the images of 3 lines have to be given. From these 5 items (2 points, 3 lines) the conic section is uniquely determined.

''Remark:''
The notation "perspective" is due to the dual statement: The projection of the points on a line <math> a</math> from a center <math>Z</math> onto a line <math>b</math> is called perspective.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891 kB), p. 19.</ref>

== Example ==
For the following example the images of the lines <math> a,u,w</math> (see picture) are given: <math>\pi(a)=b, \pi(u)=w, \pi(w)=v</math>. The projective mapping <math>\pi</math> is the product of the following perspective mappings <math>\pi_b,\pi_a</math>: 1) <math>\pi_b</math> is the perspective mapping of the pencil at point <math>U</math> onto the pencil at point <math>O</math> with axis <math>b</math>. 2) <math>\pi_a</math> is the perspective mapping of the pencil at point <math>O</math> onto the pencil at point <math>V</math> with axis <math>a</math>.
First one should check that <math>\pi=\pi_a\pi_b</math> has the properties: <math>\pi(a)=b, \pi(u)=w, \pi(w)=v</math>. Hence for any line <math>g</math> the image <math>\pi(g)=\pi_a\pi_b(g)</math> can be constructed and therefore the images of an arbitrary set of points. The lines <math>u</math> and <math>v</math> contain only the conic points <math>U</math> and <math>V</math> resp.. Hence <math>u</math> and <math>v</math> are tangent lines of the generated conic section.

The '''proof''', that this method generates a conic section can be done by switching to the affine restriction with line <math>w</math> as line at infinity, point <math>O</math> as the origin of a coordinate system with points <math>U,V</math> as points at infinity of the x- and y-axis resp. and point <math>E=(1,1)</math>. The affine part of the generated curve appears to be the hyperbola <math>y=1/x</math>.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891 kB), p. 38.</ref>

''Remark:''
#The Steiner generation of a conic section provides simple methods for the construction of [[ellipse]]s, [[parabola]]s and [[hyperbola]]s which are commonly called the ''parallelogram methods''.
#The figur, which appears while constructing a point (3. picture) is the 4-point-degeneration of [[Pascal's theorem]].<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891 kB), p. 32.</ref>

==Notes==

{{reflist}}

[[Category:Conic sections]]
[[Category:Theorems in projective geometry]]

Redshift conjecture - Revision history

en>K9re11: added Category:Conjectures using HotCat

en>Alvin Seville: removing and categorizing