Redshift conjecture

1. Definition of the Steiner generation of a conic section

2. Perspective mapping between lines

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 projective conic section in a projective plane over a field:

• Given two pencils $B(U),B(V)$ of lines at two points $U,V$ (all lines containing $U$ and $V$ resp.) and a projective but not perspective mapping $\pi$ of $B(U)$ onto $B(V)$. Then the intersection points of corresponding lines form a non-degenerate projective conic section (1. picture)

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

A projective mapping is a finite sequence of perspective mappings.

Examples of commonly used fields are the real numbers $\mathbb {R}$, the rational numbers $\mathbb {Q}$ or the complex numbers $\mathbb {C}$. Even finite fields are allowed.

Remark: The fundamental theorem for projective planes 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 $U,V$ 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 $a$ from a center $Z$ onto a line $b$ is called perspective.

Example

For the following example the images of the lines $a,u,w$ (see picture) are given: $\pi (a)=b,\pi (u)=w,\pi (w)=v$. The projective mapping $\pi$ is the product of the following perspective mappings $\pi _{b},\pi _{a}$: 1) $\pi _{b}$ is the perspective mapping of the pencil at point $U$ onto the pencil at point $O$ with axis $b$. 2) $\pi _{a}$ is the perspective mapping of the pencil at point $O$ onto the pencil at point $V$ with axis $a$. First one should check that $\pi =\pi _{a}\pi _{b}$ has the properties: $\pi (a)=b,\pi (u)=w,\pi (w)=v$. Hence for any line $g$ the image $\pi (g)=\pi _{a}\pi _{b}(g)$ can be constructed and therefore the images of an arbitrary set of points. The lines $u$ and $v$ contain only the conic points $U$ and $V$ resp.. Hence $u$ and $v$ 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 $w$ as line at infinity, point $O$ as the origin of a coordinate system with points $U,V$ as points at infinity of the x- and y-axis resp. and point $E=(1,1)$. The affine part of the generated curve appears to be the hyperbola $y=1/x$.

Remark:

1. 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.
2. The figur, which appears while constructing a point (3. picture) is the 4-point-degeneration of Pascal's theorem.

Notes

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