Steiner's theorem (geometry)

1. Definition of the Steiner generation of a conic section
2. Perspective mapping between lines
3. Example of a Steiner generation: generation of a point

Steiner's theorem concerns a property of non-degenerate conic sections in a projective plane defined over a field (a pappian projective plane). It is named after the Swiss mathematician Jakob Steiner and provides a non-metric way to define and construct conic sections in these planes.[1]

Statement of theorem

Steiner's Theorem: In a pappian projective plane, If U and V are any two distinct points of a conic, and P is a variable point of the conic, the lines PU and PV are projectively, but not perspectively, related.[2][3]

A perspective mapping (perspectivity) ${\displaystyle \pi }$ of a pencil ${\displaystyle B(U)}$ onto a pencil ${\displaystyle B(V)}$ is a bijection (1-1 correspondence) such that corresponding lines intersect on a fixed line ${\displaystyle a}$, which is called the axis of the perspectivity ${\displaystyle \pi }$ (figure 2). When a perspectivity exists between two pencils of lines, the lines are said to be perspectively related.

A projective mapping (projectivity) is a finite sequence of perspective mappings. In general, the composition of two perspectivities is not a perspectivity and the type of mapping obtained in this manner is called a projectivity. When a projectivity exists between two pencils of lines, the lines are said to be projectively related. A perspectivity is a projectivity.

Projective generation of conics

Steiner's theorem suggests an alternate way to define conic sections in pappian projective planes that does not involve measurement (is non-metric). This is sometimes referred to as Steiner's definition of conics or the projective generation of conics.

Given two pencils ${\displaystyle B(U),B(V)}$ of lines at two points ${\displaystyle U,V}$ (all lines containing ${\displaystyle U}$ and ${\displaystyle V}$ resp.) and a projective but not perspective mapping ${\displaystyle \pi }$ of ${\displaystyle B(U)}$ onto ${\displaystyle B(V)}$. Then the intersection points of corresponding lines form a non-degenerate projective conic section.[4][5] (figure 1)

Fields such as the real numbers ${\displaystyle \mathbb {R} }$, the rational numbers ${\displaystyle \mathbb {Q} }$, the complex numbers ${\displaystyle \mathbb {C} }$ or finite fields are commonly used in the construction of pappian projective planes.

It is well known that five points determine a conic in the metric setting. Actually, it is five pieces of data about the conic that will determine it, such as four points and a tangent line, or three points and two tangent lines.[6] This remains true for the Steiner definition of conics since the fundamental theorem for projective planes [7] states, that a projectivity in a pappian projective plane is uniquely determined by prescribing the images of three lines. This means, for the projective generation of a conic section, besides the two points ${\displaystyle U,V}$ only the images of 3 lines have to be given and from these 5 items (2 points, 3 lines) the conic section is uniquely determined.

Example

For the following example the images of the lines ${\displaystyle a,u,w}$ (see figure 3) are given: ${\displaystyle \pi (a)=b,\pi (u)=w,\pi (w)=v}$. The projective mapping ${\displaystyle \pi }$ is the product of the perspective mappings ${\displaystyle \pi _{b},\pi _{a}}$ where ${\displaystyle \pi _{b}}$ is the perspective mapping of the pencil at point ${\displaystyle U}$ onto the pencil at point ${\displaystyle O}$ with axis ${\displaystyle b}$ and ${\displaystyle \pi _{a}}$ is the perspective mapping of the pencil at point ${\displaystyle O}$ onto the pencil at point ${\displaystyle V}$ with axis ${\displaystyle a}$. First one should check that ${\displaystyle \pi =\pi _{a}\pi _{b}}$ has the properties: ${\displaystyle \pi (a)=b,\pi (u)=w,\pi (w)=v}$. Hence, for any line ${\displaystyle g}$, the image ${\displaystyle \pi (g)=\pi _{a}\pi _{b}(g)}$ can be constructed and therefore the images of an arbitrary set of points. The lines ${\displaystyle u}$ and ${\displaystyle v}$ contain only the conic points ${\displaystyle U}$ and ${\displaystyle V}$ resp.. Hence ${\displaystyle u}$ and ${\displaystyle v}$ are tangent lines of the generated conic section.

The proof that this method generates a conic section follows from switching to the affine restriction with line ${\displaystyle w}$ as line at infinity, point ${\displaystyle O}$ as the origin of a coordinate system with points ${\displaystyle U,V}$ as points at infinity of the x- and y-axis resp. and point ${\displaystyle E=(1,1)}$. The affine part of the generated curve appears to be the hyperbola ${\displaystyle y=1/x}$.[5]

Remark:

1. The Steiner generation of conic sections provides simple methods for the construction of ellipses, parabolas and hyperbolas which are commonly called the parallelogram methods.
2. The figure, which appears while constructing a point on a conic (figure 3), is the 4-point-degeneration of Pascal's theorem.[8]

Notes

1. Jacob Steiner’s Vorlesungen über synthetische Geometrie, B. G. Teubner, Leipzig 1867 (bei Google Books: [1])
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References

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