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.

1. Definition of the Steiner generation of a conic section

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., ${\displaystyle Char\neq 2}$).

Definition of a Steiner conic

• 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^[1][2][3][4] (figure 1)

 

2. Perspective mapping between lines

A perspective mapping ${\displaystyle \pi }$  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).

A projective mapping is a finite product of perspective mappings.

Simple example: If one shifts in the first diagram point ${\displaystyle U}$  and its pencil of lines onto ${\displaystyle V}$  and rotates the shifted pencil around ${\displaystyle V}$  by a fixed angle ${\displaystyle \varphi }$  then the shift (translation) and the rotation generate a projective mapping ${\displaystyle \pi }$  of the pencil at point ${\displaystyle U}$  onto the pencil at ${\displaystyle V}$ . 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 ${\displaystyle \mathbb {R} }$ , the rational numbers ${\displaystyle \mathbb {Q} }$  or the complex numbers ${\displaystyle \mathbb {C} }$ . 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 ${\displaystyle U,V}$  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 ${\displaystyle a}$  from a center ${\displaystyle Z}$  onto a line ${\displaystyle b}$  is called a perspectivity (see below).^[5]

 

3. Example of a Steiner generation: generation of a point

Example

For the following example the images of the lines ${\displaystyle a,u,w}$  (see picture) are given: ${\displaystyle \pi (a)=b,\pi (u)=w,\pi (w)=v}$ . The projective mapping ${\displaystyle \pi }$  is the product of the following perspective mappings ${\displaystyle \pi _{b},\pi _{a}}$ : 1) ${\displaystyle \pi _{b}}$  is the perspective mapping of the pencil at point ${\displaystyle U}$  onto the pencil at point ${\displaystyle O}$  with axis ${\displaystyle b}$ . 2) ${\displaystyle \pi _{a}}$  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.

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

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 ${\displaystyle u,v}$  and a projective but not perspective mapping ${\displaystyle \pi }$  of ${\displaystyle u}$  onto ${\displaystyle v}$ . Then the lines connecting corresponding points form a dual non-degenerate projective conic section.

A perspective mapping ${\displaystyle \pi }$  of the point set of a line ${\displaystyle u}$  onto the point set of a line ${\displaystyle v}$  is a bijection (1-1 correspondence) such that the connecting lines of corresponding points intersect at a fixed point ${\displaystyle Z}$ , which is called the centre of the perspectivity ${\displaystyle \pi }$  (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 ${\displaystyle \operatorname {Char} =2}$  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 ${\displaystyle \operatorname {Char} \neq 2}$  is the dual of a non-degenerate point conic a non-degenerate line conic.

Examples

 

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

 

example of a Steiner generation of a dual conic

(1) Projectivity given by two perspectivities:
Two lines ${\displaystyle u,v}$  with intersection point ${\displaystyle W}$  are given and a projectivity ${\displaystyle \pi }$  from ${\displaystyle u}$  onto ${\displaystyle v}$  by two perspectivities ${\displaystyle \pi _{A},\pi _{B}}$  with centers ${\displaystyle A,B}$ . ${\displaystyle \pi _{A}}$  maps line ${\displaystyle u}$  onto a third line ${\displaystyle o}$ , ${\displaystyle \pi _{B}}$  maps line ${\displaystyle o}$  onto line ${\displaystyle v}$  (see diagram). Point ${\displaystyle W}$  must not lie on the lines ${\displaystyle {\overline {AB}},o}$ . Projectivity ${\displaystyle \pi }$  is the composition of the two perspectivities: ${\displaystyle \ \pi =\pi _{B}\pi _{A}}$ . Hence a point ${\displaystyle X}$  is mapped onto ${\displaystyle \pi (X)=\pi _{B}\pi _{A}(X)}$  and the line ${\displaystyle x={\overline {X\pi (X)}}}$  is an element of the dual conic defined by ${\displaystyle \pi }$ .
(If ${\displaystyle W}$  would be a fixpoint, ${\displaystyle \pi }$  would be perspective.^[7])

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

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

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

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, ${\displaystyle E(T,P)}$  is the conic at point ${\displaystyle P}$  afforded by the collineation ${\displaystyle T}$ , consisting of the intersections of ${\displaystyle L}$  and ${\displaystyle T(L)}$  for all lines ${\displaystyle L}$  through ${\displaystyle P}$ . If ${\displaystyle T(P)=P}$  or ${\displaystyle T(L)=L}$  for some ${\displaystyle L}$  then the conic is degenerate. For example, in the real coordinate plane, the affine type (ellipse, parabola, hyperbola) of ${\displaystyle E(T,P)}$  is determined by the trace and determinant of the matrix component of ${\displaystyle T}$ , independent of ${\displaystyle P}$ .

By contrast, the collineation group of the real hyperbolic plane ${\displaystyle \mathbb {H} ^{2}}$ 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 ${\displaystyle E(T,P);}$  ${\displaystyle T}$  preserves orientation – and the opposite ${\displaystyle E(T,P);}$  ${\displaystyle T}$  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 ${\displaystyle \mathbb {H} ^{2}}$ , each direct central conic is birationally equivalent to an opposite central conic.^[8] In fact, the central conics represent all genus 1 curves with real shape invariant ${\displaystyle j\geq 1}$ . 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 ${\displaystyle P}$  and ${\displaystyle T(P)}$ . The orthogonal trajectories of these curves represent all genus 1 curves with ${\displaystyle j\leq 1}$ , which manifest as either irreducible cubics or bi-circular quartics. Using the elliptic curve addition law on each trajectory, every general central conic in ${\displaystyle \mathbb {H} ^{2}}$ decomposes uniquely as the sum of two intrinsic conics by adding pairs of points where the conics intersect each trajectory.^[9]

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
7. H. Lenz: Vorlesungen über projektive Geometrie, BI, Mannheim, 1965, S. 49.
8. Sarli, John (April 2012). "Conics in the hyperbolic plane intrinsic to the collineation group". Journal of Geometry. 103 (1): 131–148. doi:10.1007/s00022-012-0115-5. ISSN 0047-2468. S2CID 119588289.
9. Sarli, John (2021-10-22). "The Elliptic Curve Decomposition of Central Conics in the Real Hyperbolic Plane". doi:10.21203/rs.3.rs-936116/v1. {{cite journal}}: Cite journal requires |journal= (help)

References

 

Wikimedia Commons has media related to Steiner conic.

• 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