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., $$Char\ne2$$).

Definition of a Steiner conic
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).
 * 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 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 $$\R$$, the rational numbers $$\Q$$ or the complex numbers $$\C$$. 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.

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 homogeneous 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 $$\operatorname{Char} =2$$ 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 $$\operatorname{Char}\ne2$$ is the dual of a non-degenerate point conic a non-degenerate line conic.

Examples
(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 $$\pi_B,\pi_A$$: 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 $$U,V$$.
 * 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 $$A$$.

Intrinsic conics in a linear incidence geometry
The Steiner construction defines the conics in a planar linear incidence geometry (two points determine at most one line and two lines intersect in at most one point) intrinsically, that is, using only the collineation group. Specifically, $$E(T,P)$$ is the conic at point $$P$$ afforded by the collineation $$T$$, consisting of the intersections of $$L$$ and $$T(L)$$ for all lines $$L$$ through $$P$$. If $$T(P)=P$$ or $$T(L)=L$$ for some $$L$$ then the conic is degenerate. For example, in the real coordinate plane, the affine type (ellipse, parabola, hyperbola) of $$E(T,P)$$ is determined by the trace and determinant of the matrix component of $$T$$, independent of $$P$$.

By contrast, the collineation group of the real hyperbolic plane $$\mathbb{H}^2$$consists of isometries. Consequently, the intrinsic conics comprise a small but varied subset of the general conics, curves obtained from the intersections of projective conics with a hyperbolic domain. Further, unlike the Euclidean plane, there is no overlap between the direct $$E(T,P);$$ $$T$$ preserves orientation – and the opposite $$E(T,P);$$ $$T$$ reverses orientation. The direct case includes central (two perpendicular lines of symmetry) and non-central conics, whereas every opposite conic is central. Even though direct and opposite central conics cannot be congruent, they are related by a quasi-symmetry defined in terms of complementary angles of parallelism. Thus, in any inversive model of $$\mathbb{H}^2$$, each direct central conic is birationally equivalent to an opposite central conic. In fact, the central conics represent all genus 1 curves with real shape invariant $$j\geq1$$. A minimal set of representatives is obtained from the central direct conics with common center and axis of symmetry, whereby the shape invariant is a function of the eccentricity, defined in terms of the distance between $$P$$ and $$T(P)$$. The orthogonal trajectories of these curves represent all genus 1 curves with $$j\leq1$$, which manifest as either irreducible cubics or bi-circular quartics. Using the elliptic curve addition law on each trajectory, every general central conic in $$\mathbb{H}^2$$decomposes uniquely as the sum of two intrinsic conics by adding pairs of points where the conics intersect each trajectory.