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.,
C
h
a
r
≠
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
π
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
π
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
π
(figure 2).
A projective mapping is a finite sequence of perspective mappings.
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).
For the following example the images of the lines
a
,
u
,
w
(see picture) are given:
π
(
a
)
=
b
,
π
(
u
)
=
w
,
π
(
w
)
=
v
. The projective mapping
π
is the product of the following perspective mappings
π
b
,
π
a
: 1)
π
b
is the perspective mapping of the pencil at point
U
onto the pencil at point
O
with axis
b
. 2)
π
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
π
=
π
a
π
b
has the properties:
π
(
a
)
=
b
,
π
(
u
)
=
w
,
π
(
w
)
=
v
. Hence for any line
g
the image
π
(
g
)
=
π
a
π
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:
- 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.
- 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 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
π
of
u
onto
v
. Then the lines connecting corresponding points form a dual non-degenerate projective conic section.
A perspective mapping
π
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
π
(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
C
h
a
r
=
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
C
h
a
r
≠
2
is the dual of a non-degenerate point conic a non-degenerate line conic.
For the following example the images of the points
A
,
U
,
W
are given:
π
(
A
)
=
B
,
π
(
U
)
=
W
,
π
(
W
)
=
V
. The projective mapping
π
can be represented by the product of the following perspectivities
π
B
,
π
A
:
1)
π
B
is the perspectivity of the point set of line
u
onto the point set of line
o
with centre
B
.
2)
π
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
π
=
π
A
π
B
fulfills
π
(
A
)
=
B
,
π
(
U
)
=
W
,
π
(
W
)
=
V
. Hence for any arbitrary point
G
the image
π
(
G
)
=
π
A
π
B
(
G
)
can be constructed and line
G
π
(
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
.
