Redshift conjecture
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 projective conic section in a projective plane over a field:
- Given two pencils of lines at two points (all lines containing and resp.) and a projective but not perspective mapping of onto . Then the intersection points of corresponding lines form a non-degenerate projective conic section [1][2] (1. picture)
A perspective mapping of a pencil onto a pencil is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line , which is called the axis of the perspectivity (2. picture).
A projective mapping is a finite sequence of perspective mappings.
Examples of commonly used fields are the real numbers , the rational numbers or the complex numbers . Even finite fields are allowed.
Remark: The fundamental theorem for projective planes [3] 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 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 from a center onto a line is called perspective.[4]
Example
For the following example the images of the lines (see picture) are given: . The projective mapping is the product of the following perspective mappings : 1) is the perspective mapping of the pencil at point onto the pencil at point with axis . 2) is the perspective mapping of the pencil at point onto the pencil at point with axis . First one should check that has the properties: . Hence for any line the image can be constructed and therefore the images of an arbitrary set of points. The lines and contain only the conic points and resp.. Hence and 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 as line at infinity, point as the origin of a coordinate system with points as points at infinity of the x- and y-axis resp. and point . The affine part of the generated curve appears to be the hyperbola .[5]
Remark:
- The Steiner generation of a conic section provides simple methods for the construction of ellipses, parabolas and hyperbolas 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.[6]
Notes
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- ↑ Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.(PDF; 891 kB), p. 38.
- ↑ Jacob Steiner’s Vorlesungen über synthetische Geometrie, B. G. Teubner, Leipzig 1867 (bei Google Books: [1]), 2. Teil, p. 96
- ↑ Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.(PDF; 891 kB), p. 19.
- ↑ Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.(PDF; 891 kB), p. 19.
- ↑ Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.(PDF; 891 kB), p. 38.
- ↑ Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.(PDF; 891 kB), p. 32.