Steiner conic

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 conic or more precisely Steiner's 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.

The usual definition of a conic uses a quadratic form (see Quadric (projective geometry)). Another alternative definition of a conic uses a hyperbolic polarity. It is due to K. G. C. von Staudt and sometimes called a von Staudt conic. The disadvantage of von Staudt's definition is that it only works when the underlying field has odd characteristic (i.e., $Char\neq 2$).

Definition of a Steiner conic

• 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][2][3] [4]} (figure 1)

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 $\pi$ (figure 2).

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 $\mathbb {C}$. The construction also works over finite fields, providing examples in finite projective planes.

Remark: The fundamental theorem for projective planes states,^[5] 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 that, for the Steiner generation of a conic section, besides two points $U,V$ only the images of 3 lines have to be given. These 5 items (2 points, 3 lines) uniquely determine the conic section.

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 a perspectivity (see below).^[5]

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.

A proof that this method generates a conic section follows from switching to the affine restriction with line $w$ as the 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 $y=1/x$.^[2]

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 figure that appears while constructing a point (figure 3) is the 4-point-degeneration of Pascal's theorem.^[6]

Steiner generation of a dual conic

dual ellipse

Steiner generation of a dual conic

definition of a perspective mapping

example of a Steiner generation of a dual conic

Definitions and the dual generation

Dualizing (see duality (projective geometry)) a projective plane means exchanging the points with the lines and the operations intersection and connecting. The dual structure of a projective plane is also a projective plane. The dual plane of a pappian plane is pappian and can also be coordinatized by homogenous coordinates. A nondegenerate dual conic section is analogously defined by a quadratic form.

A dual conic can be generated by Steiner's dual method:

• Given the point sets of two lines $u,v$ and a projective but not perspective mapping $\pi$ of $u$ onto $v$. Then the lines connecting corresponding points form a dual non-degenerate projective conic section.

A perspective mapping $\pi$ of the point set of a line $u$ onto the point set of a line $v$ is a bijection (1-1 correspondence) such that the connecting lines of corresponding points intersect at a fixed point $Z$, which is called the centre of the perspectivity $\pi$ (see figure).

A projective mapping is a finite sequence of perspective mappings.

It is usual, when dealing with dual and common conic sections, to call the common conic section a point conic and the dual conic a line conic.

In the case that the underlying field has $Char=2$ all the tangents of a point conic intersect in a point, called the knot (or nucleus) of the conic. Thus, the dual of a non-degenerate point conic is a subset of points of a dual line and not an oval curve (in the dual plane). So, only in the case that $Char\neq 2$ is the dual of a non-degenerate point conic a non-degenerate line conic.

Example

For the following example the images of the points $A,U,W$ are given: $\pi (A)=B,\,\pi (U)=W,\,\pi (W)=V$. The projective mapping $\pi$ can be represented by the product of the following perspectivities $\pi _{B},\pi _{A}$:

1) $\pi _{B}$ is the perspectivity of the point set of line $u$ onto the point set of line $o$ with centre $B$.

2) $\pi _{A}$ is the perspectivity of the point set of line $o$ onto the point set of line $v$ with centre $A$.

One easily checks that the projective mapping $\pi =\pi _{A}\pi _{B}$ fulfills $\pi (A)=B,\,\pi (U)=W,\,\pi (W)=V$. Hence for any arbitrary point $G$ the image $\pi (G)=\pi _{A}\pi _{B}(G)$ can be constructed and line $\overline {G\pi (G)}$ is an element of a non degenerate dual conic section. Because the points $U$ and $V$ are contained in the lines $u$, $v$ resp.,the points $U$ and $V$ are points of the conic and the lines $u,v$ are tangents at $U,V$.

Notes

1. Coxeter 1993, p. 80
2. Hartmann, p. 38
3. Merserve 1983, p. 65
4. Jacob Steiner’s Vorlesungen über synthetische Geometrie, B. G. Teubner, Leipzig 1867 (from Google Books: (German) Part II follows Part I) Part II, pg. 96
5. Hartmann, p. 19
6. Hartmann, p. 32

References

• Coxeter, H. S. M. (1993), The Real Projective Plane, Springer Science & Business Media
• Hartmann, Erich, Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes (PDF), retrieved 20 September 2014 (PDF; 891 kB).
• Merserve, Bruce E. (1983) [1959], Fundamental Concepts of Geometry, Dover, ISBN 0-486-63415-9