# Steiner conic

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  (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 product of perspective mappings.

Simple example: If one shifts in the first diagram point $U$  and its pencil of lines onto $V$  and rotates the shifted pencil around $V$  by a fixed angle $\varphi$  then the shift (translation) and the rotation generate a projective mapping $\pi$  of the pencil at point $U$  onto the pencil at $V$ . From the inscribed angle theorem one gets: The intersection points of corresponding lines form a circle.

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, 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).

## 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 $y=1/x$ .

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.

## 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.

### Examples

Dual Steiner conic defined by two perspectivities $\pi _{A},\pi _{B}$

(1) Projectivity given by two perspectivities:
Two lines $u,v$  with intersection point $W$  are given and a projectivity $\pi$  from $u$  onto $v$  by two perspectivities $\pi _{A},\pi _{B}$  with centers $A,B$ . $\pi _{A}$  maps line $u$  onto a third line $o$ , $\pi _{B}$  maps line $o$  onto line $v$  (see diagram). Point $W$  must not lie on the lines ${\overline {AB}},o$ . Projectivity $\pi$  is the composition of the two perspectivities: $\ \pi =\pi _{B}\pi _{A}$ . Hence a point $X$  is mapped onto $\pi (X)=\pi _{B}\pi _{A}(X)$  and the line $x={\overline {X\pi (X)}}$  is an element of the dual conic defined by $\pi$ .
(If $W$  would be a fixpoint, $\pi$  would be perspective .)

(2) Three points and their images are given:
The following example is the dual one given above for a Steiner conic.
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 $U,V$ .