Projective conic section

Projective plane with conic sections and straight line

A non-degenerate (na) projective conic section is a curve in a Pappusian projective plane, which can be described as a hyperbola (see image: c2) or parabola (image: c1) with a suitable choice of a distance line . The equation does not always describe a na conic section. ${\displaystyle g_{\infty }}$ ${\displaystyle y={\tfrac {1}{x}}}$ ${\displaystyle y=x^{2}}$ ${\displaystyle x^{2}+y^{2}=1}$

• A conic section can be described in homogeneous coordinates (see below) by an equation of the form and is therefore also a projective quadric . ${\displaystyle x_{1}x_{2}-x_{3}^{2}=0}$

Geometrically, one can imagine a projective conic section resembling a circle with the essential properties: 1) a straight line meets at 0, 1 or 2 points, 2) at every point of there is exactly one tangent , ie . However, these two properties do not yet determine a na conic section. In addition to the geometric properties 1), 2), a na conic section has many symmetries (see below). ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle P}$ ${\displaystyle k}$ ${\displaystyle t_{P}}$ ${\displaystyle |t_{P}\cap k|=1}$

The advantage of a projective na conic section is the fact that all na projective conic sections are projectively equivalent to the curve with the equation . Affin the affine are conic sections ellipse, parabola and hyperbole are not equivalent: A parable can not be a affine transformation into an ellipse or hyperbola convict. ${\displaystyle x_{1}x_{2}=x_{3}^{2}}$

Projective plane over a body K

The projective extension of the affine plane over a body K provides the descriptive inhomogeneous model of the projective plane over K. Each straight line or a point that also belongs to all straight lines parallel to it is added. The new points are called far points and the set of new points is called the far line . In the projective extension there is no longer a parallel relation between straight lines. The geometry has become “simpler”: 1) There is exactly one connecting line for every two points . 2) Two straight lines intersect at exactly one point. The initially inhomogeneous description (i.e. the distant line seems to play a special role) is eliminated by the homogeneous model: A point is a straight line through the origin, a straight line is a plane of origin in . The advantage of the homogeneous model is: The most important collineations are induced by linear maps. ${\displaystyle y=mx+d}$ ${\displaystyle x=c}$ ${\displaystyle K^{3}}$

Projective plane: inhomogeneous model

Definition: Let K be a body and

${\displaystyle P_{1}:=K^{2}\cup K\cup \{\infty \},\ \infty \notin K,}$the amount of points
${\displaystyle G_{1}:=\ \{\{(x,y)\in K^{2}\ |\ {\color {red}y=mx+d}\}\cup \{(m)\}\ |\ m,d\in K\}}$

${\displaystyle \cup \ \{\{(x,y)\in K^{2}\ |\ {\color {red}x=c}\}\cup \{(\infty )\}\ |\ c\in K\}\ }$

${\displaystyle \cup \ \{\{(m)\ |\ m\in K\}\cup \{(\infty )\}\}}$the set of straight lines ,

${\displaystyle {\color {red}g_{\infty }}:=\{(m)\ |\ m\in K\}\cup \{(\infty )\}}$the distant line , its points are the distant points .

${\displaystyle {\mathfrak {P}}_{1}(K):=(P_{1},G_{1},\in )}$is called the inhomogeneous model of the projective plane over the body K.

Definition: It is a body of the vector space and , , ${\displaystyle K}$ ${\displaystyle V}$ ${\displaystyle K^{3}}$ ${\displaystyle {\vec {0}}:=(0,0,0)^{\top }}$ ${\displaystyle P_{2}:=\{{\text{1-dim. Unterräume von V}}\}=\{<{\vec {x}}>\ |\ {\vec {0}}\neq {\vec {x}}\in V\}}$

where is the subspace spanned by. ${\displaystyle <{\vec {x}}>}$ ${\displaystyle {\vec {x}}}$

${\displaystyle G_{2}:=\{{\text{2-dim. Unterräume von V}}\}}$

${\displaystyle =\{\{<(x_{1},x_{2},x_{3})^{\top }>\ \in P_{2}\ |\ {\color {blue}ax_{1}+bx_{2}+cx_{3}=0}\}\ |\ {\vec {0}}\neq (a,b,c)^{\top }\in K^{3}\}}$.

${\displaystyle {\mathfrak {P}}_{2}(K):=(P_{2},G_{2},\in )}$is called homogeneous model of the projective plane above . ${\displaystyle K}$

Theorem: and are isomorphic projective planes. ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$

The following figure maps to . The projective straight line with the equation is mapped to: ${\displaystyle \phi }$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$ ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle {\color {blue}x_{3}=0}}$ ${\displaystyle {\color {red}g_{\infty }}}$

${\displaystyle <(x_{1},x_{2},x_{3})^{\top }>\ \rightarrow ({\frac {x_{1}}{x_{3}}},{\frac {x_{2}}{x_{3}}})=(x,y)}$if ${\displaystyle x_{3}\neq 0\ ,}$

${\displaystyle <(x_{1},x_{2},0)^{\top }>\ \rightarrow ({\frac {x_{2}}{x_{1}}})=(m)}$if , if . ${\displaystyle x_{1}\neq 0,\ \ <(0,x_{2},0)^{\top }>\ \rightarrow (\infty )}$ ${\displaystyle x_{2}\neq 0}$

The reverse is:

${\displaystyle (x,y)\rightarrow \ <(x,y,1)^{\top }>,\ \ (m)\rightarrow \ <(1,m,0)^{\top }>,\ \ (\infty )\rightarrow \ <(0,1,0)^{\top }>\ .}$

Definition:

1. Permutations of the set of points that map straight lines on straight lines are called collineations . ${\displaystyle P_{i}}$
2. Collineations of induced by linear mappings are called projective . ${\displaystyle {\mathfrak {P}}_{2}(K)}$

Note: Pappos's theorem applies to the projective planes and . That's why they are called pappussch . ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$

Definition of a non-degenerate projective conic section

Projective conic section in homogeneous coordinates including inhomogeneous designations ${\displaystyle k_{1}}$

Projective conic section in inhomogeneous coordinates: hyperbola and far points ${\displaystyle k_{1}}$

Projective conic section in homogeneous coordinates including inhomogeneous designations ${\displaystyle k_{2}}$

Projective conic section in inhomogeneous coordinates: parabola and far point ${\displaystyle k_{2}}$

First the curves are defined as quadrics in (homogeneous coordinates). The association between the homogeneous model and the inhomogeneous model explained in the previous section finally provides clearer inhomogeneous descriptions of . ${\displaystyle k_{1},k_{2}}$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$ ${\displaystyle \phi }$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$ ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle k_{1},k_{2}}$

Definition: It is a body. In be ${\displaystyle K}$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$

${\displaystyle k_{1}:=\{<(x_{1},x_{2},x_{3})^{\top }>\ |\ {\color {blue}x_{1}x_{2}=x_{3}^{2}}\}}$.

In is : . ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle k_{1}}$ ${\displaystyle \{(x,y)\ |\ {\color {red}y={\frac {1}{x}}},x\neq 0\}\cup \{(0),(\infty )\}}$

Every image from under a collineation of is called a non-degenerate projective conic section . ( Degenerate conic sections are: the empty set, 1 point, 1 straight line or 2 straight lines.) ${\displaystyle k_{1}}$ ${\displaystyle {\mathfrak {P}}_{i}(K)}$

Definition: . ${\displaystyle k_{2}:=\{<(x_{1},x_{2},x_{3})^{\top }>\ |\ {\color {blue}x_{2}x_{3}=x_{1}^{2}}\}}$

In is : . ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle k_{2}}$ ${\displaystyle \{(x,y)\ |\ {\color {red}y=x^{2}}\}\cup \{(\infty )\}}$

Note: The equations describe in a cone with points at the zero point (see pictures). contains the and axes, contains the and axes. ${\displaystyle x_{1}x_{2}=x_{3}^{2},\ x_{2}x_{3}=x_{1}^{2}}$ ${\displaystyle K^{3}}$ ${\displaystyle k_{1}}$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle k_{2}}$ ${\displaystyle x_{2}}$ ${\displaystyle x_{3}}$

Lemma: The na conic sections in are projectively equivalent to (or ). (In other words, they can be converted into one another through projective collineation.) ${\displaystyle {\mathfrak {P}}_{i}(K)}$ ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$

Note: The linear map induces projective collineation that on maps. In the inhomogeneous model, this collineation is described by. ${\displaystyle (x_{1},x_{2},x_{3})^{\top }\rightarrow (x_{2},x_{3},x_{1})^{\top }}$ ${\displaystyle k_{1}}$ ${\displaystyle k_{2}}$ ${\displaystyle (x,y)\rightarrow \left({\frac {1}{y}},{\frac {x}{y}}\right)}$

Comment:

1. The “unit circle” is in the case (i.e. 1 + 1 = 0) not a na conic section, since in this case the equation describes a straight line . ${\displaystyle \color {red}x^{2}+y^{2}=1}$ ${\displaystyle CharK=2}$ ${\displaystyle x^{2}+y^{2}=(x+y)^{2}=1}$
2. In this case , the equation can be converted into the equation by means of a suitable coordinate transformation , i.e. H. the unit circle is a na conic section only in the case . ${\displaystyle CharK\neq 2}$ ${\displaystyle x_{1}^{2}+x_{2}^{2}=x_{3}^{2}}$ ${\displaystyle x_{1}x_{2}=x_{3}^{2}}$ ${\displaystyle CharK\neq 2}$
3. In the case , the unit circle intersects the distance line in two points and can be compared with a hyperbola in the affine part. ${\displaystyle K=\mathbb {C} }$

Properties of a na projective conic section

Sentence:

1. A na conic section ${\displaystyle k}$
   • is intersected by a straight line in a maximum of 2 points. In the case means passer , in the case tangent and in the case secant . ${\displaystyle g}$ ${\displaystyle |g\cap k|=0}$ ${\displaystyle g}$ ${\displaystyle |g\cap k|=1}$ ${\displaystyle |g\cap k|=2}$
   • has exactly one tangent at each point.
2. A na conic is symmetrical about any point a secant passes through, i.e. H. there is an involutive central collineation with a center that leaves invariant. ${\displaystyle k}$ ${\displaystyle P\notin k}$ ${\displaystyle \sigma _{P}}$ ${\displaystyle P}$ ${\displaystyle k}$
3. If is, a na conic has points. ${\displaystyle |K|=n<\infty }$ ${\displaystyle n+1}$
4. Pascal's theorems apply .

Examples of symmetries in the case : ${\displaystyle CharK\neq 2}$

1. ${\displaystyle (x,y)\leftrightarrow (-x+b,y-2bx+b^{2}),\quad (m)\leftrightarrow (2b-m),\quad (\infty )\leftrightarrow (\infty )}$is for each an oblique reflection on the straight line that fixes as a whole. are fixed points . In this case , the oblique reflection is the normal reflection on the y-axis. ${\displaystyle b\in K}$ ${\displaystyle x={\tfrac {b}{2}}}$ ${\displaystyle k_{2}}$ ${\displaystyle ({\tfrac {b}{2}},{\tfrac {b^{2}}{4}}),(\infty )}$ ${\displaystyle k_{2}}$ ${\displaystyle b=0}$
2. The involution is the "mirroring" (involutive central collineation) with the axis and center . She leaves stuck as a whole. are fixed points . ${\displaystyle (x,y)\leftrightarrow (\textstyle {\frac {x}{y}},{\tfrac {1}{y}}),y\neq 0,\quad (x,0),\ x\neq 0\leftrightarrow ({\tfrac {1}{x}}),\quad (0,0)\leftrightarrow (\infty )}$ ${\displaystyle y=1}$ ${\displaystyle (0,-1)}$ ${\displaystyle k_{2}}$ ${\displaystyle (1,1),(-1,1)}$ ${\displaystyle k_{2}}$

Examples of symmetries in the case : ${\displaystyle CharK=2}$

1. ${\displaystyle (x,y)\leftrightarrow (x+b,y+b^{2}),\quad (m)\leftrightarrow (m),\quad (\infty )\leftrightarrow (\infty )}$is for each an involutive central collineation with a center on the axis that fixes as a whole. is the only fixed point on . ( This mapping acts as a translation in direction .) ${\displaystyle b\in K}$ ${\displaystyle (b)}$ ${\displaystyle g_{\infty }}$ ${\displaystyle k_{2}}$ ${\displaystyle (\infty )}$ ${\displaystyle k_{2}}$ ${\displaystyle K^{2}}$ ${\displaystyle (b)}$
2. Involution is the involutive central collineation with the center on the axis . She leaves stuck as a whole. is the only fixed point on . ${\displaystyle (x,y)\leftrightarrow (\textstyle {\frac {x}{y}},{\tfrac {1}{y}}),y\neq 0,\quad (x,0),\ x\neq 0\leftrightarrow ({\tfrac {1}{x}}),\quad (0,0)\leftrightarrow (\infty )}$ ${\displaystyle (0,1)}$ ${\displaystyle y=1}$ ${\displaystyle k_{2}}$ ${\displaystyle (1,1)}$ ${\displaystyle k_{2}}$

Comment:

1. The tangent at the point of the conic section has the equation . In the case , the equation simplifies to , i. H. all tangents go through the point . is called the knot of . ${\displaystyle <(a,b,c)>}$ ${\displaystyle k_{1}:x_{1}x_{2}=x_{3}^{2}}$ ${\displaystyle ax_{2}+bx_{1}-2cx_{3}=0}$ ${\displaystyle CharK=2}$ ${\displaystyle ax_{2}+bx_{1}=0}$ ${\displaystyle N:<(0,0,1)>}$ ${\displaystyle N}$ ${\displaystyle k_{1}}$
2. In the inhomogeneous model, the point has the tangent . The tangents in the far points are the coordinate axes. In the case , the equation simplifies to , i. H. all tangents go through the point . ${\displaystyle k_{1}:y={\tfrac {1}{x}}}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle y=-{\tfrac {1}{x_{0}^{2}}}x+{\tfrac {2}{x_{0}}}}$ ${\displaystyle (0),(\infty )}$ ${\displaystyle CharK=2}$ ${\displaystyle y={\tfrac {1}{x_{0}^{2}}}x}$ ${\displaystyle (0,0)}$
3. In the inhomogeneous model, the point has the tangent . The tangent at the far point is the distance line. In the case , the equation simplifies to , i. H. all tangents go through the point (far point of the x-axis). ${\displaystyle k_{2}:y=x^{2}}$ ${\displaystyle (x_{0},y_{0})}$ ${\displaystyle y=2x_{0}x-x_{0}^{2}}$ ${\displaystyle (\infty )}$ ${\displaystyle CharK=2}$ ${\displaystyle y=x_{0}^{2}}$ ${\displaystyle (0)}$

Note: A set of points with the properties ${\displaystyle {\mathfrak {o}}}$

• ${\displaystyle {\mathfrak {o}}}$ is cut by a garade in a maximum of 2 points.
• ${\displaystyle {\mathfrak {o}}}$has exactly one tangent in each point (straight line that has only one point in common). ${\displaystyle {\mathfrak {o}}}$

is called oval . Every na conic section is an oval, but not the other way around. In the real case there are many ovals that are not conic sections: e.g. B. the curve or in the case of a conic section you replace the parabola with the curve or you put two halves of different ellipses together smoothly. It takes a lot of symmetries to turn an oval into a conic section. ${\displaystyle x^{4}+y^{4}=1}$ ${\displaystyle k_{2}}$ ${\displaystyle y=x^{4}}$

Steiner generation of the conic sections k ₁ , k ₂

Steiner generation of the conic section : specifications ${\displaystyle k_{2}}$

Steiner's generation of the conic section ${\displaystyle k_{2}}$

Steiner's generation of the conic section ${\displaystyle k_{1}}$

A na projective conic section can also be generated according to Steiner as follows (see Steiner's theorem ):

• If one has a projective but not perspective mapping of one tuft to the other for two straight tufts in two points (all straight lines through the point or ) , the intersection points of assigned straight lines form a non-degenerate projective conic section. ${\displaystyle U,V}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle \pi }$

Generation of : ${\displaystyle k_{2}}$

In order to generate the projective conic section (parabola), we specify the 3 points in the inhomogeneous model of the projective plane , the x-axis as a tangent at the point and the straight line as a tangent at the point (see figure). We use the tufts in and as straight tufts . With the help of the two straight lines and as axes for perspectives (see Steiner's theorem ), we first map the straight tuft in with the tuft in the far point (parallels to the straight line ) and then with the tuft in (parallels to the y-axis). The straight line is first intersected with the straight line . The intersection is . The parallel to through this point is . The intersection with is . From this it follows . If you run through all the numbers you get all the points of the parabola . ${\displaystyle k_{2}}$ ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle (0,0),(x_{0},x_{0}^{2}),(\infty )}$ ${\displaystyle (0,0)}$ ${\displaystyle g_{\infty }}$ ${\displaystyle (\infty )}$ ${\displaystyle (0,0)}$ ${\displaystyle (\infty )}$ ${\displaystyle a:y=x_{0}^{2}}$ ${\displaystyle b:x=x_{0}}$ ${\displaystyle \pi _{a},\ \pi _{b}}$ ${\displaystyle (0,0)}$ ${\displaystyle \pi _{b}}$ ${\displaystyle (-x_{0})}$ ${\displaystyle AB}$ ${\displaystyle \pi _{a}}$ ${\displaystyle (\infty )}$ ${\displaystyle g:y=mx,m\in K,\ }$ ${\displaystyle b:x=x_{0}}$ ${\displaystyle (x_{0},mx_{0})}$ ${\displaystyle AB}$ ${\displaystyle y=-x_{0}(x-x_{0})+mx_{0}=-x_{0}x+x_{0}^{2}+mx_{0}}$ ${\displaystyle a:y=x_{0}^{2}}$ ${\displaystyle (m,x_{0}^{2})}$ ${\displaystyle G=\pi _{a}\pi _{b}(g)\cap g=(m,m^{2})}$ ${\displaystyle m}$ ${\displaystyle K}$ ${\displaystyle y=x^{2}}$

Note: In projective imaging , the x-axis is mapped onto the y-axis and the y-axis onto the distance line. ${\displaystyle \pi _{a}\pi _{b}}$

Note: The Steiner generation of provides a simple method to generate many points of a parabola. See: parabola . ${\displaystyle k_{2}}$

Generation of : ${\displaystyle k_{1}}$

In order to generate the projective conic section (hyperbola), we specify the 3 points in the inhomogeneous model of the projective plane , the x-axis as a tangent in the point and the y-axis as a tangent in the point . We use the tufts in and as straight tufts . first maps the tuft to the auxiliary tuft at the point . Due to the symmetry, this case is easier to compute. It is easy to calculate that the straight line is mapped onto the straight line by the projective mapping (see picture). ${\displaystyle k_{1}}$ ${\displaystyle {\mathfrak {P}}_{1}(K)}$ ${\displaystyle (0),(\infty ),(x_{0},{\tfrac {1}{x_{0}}})}$ ${\displaystyle (0)}$ ${\displaystyle (\infty )}$ ${\displaystyle (0)}$ ${\displaystyle (\infty )}$ ${\displaystyle \pi _{b}}$ ${\displaystyle (\infty )}$ ${\displaystyle (0,0)}$ ${\displaystyle g:x=c,c\neq 0}$ ${\displaystyle \pi _{a}\pi _{b}}$ ${\displaystyle y={\tfrac {1}{c}}}$

Comment:

1. In projective imaging, the y-axis is mapped onto and onto the x-axis. ${\displaystyle \pi _{a}\pi _{b}}$ ${\displaystyle g_{\infty }}$ ${\displaystyle g_{\infty }}$
2. The figure also shows the connection between the Steiner generation and an affine version of the 4-point degeneration of Pascal's theorem .

Note: A generation of the hyperbola can be found here . ${\displaystyle x^{2}-y^{2}=1}$

Polarity and v. Staudt conic section

na conic section: polarity

According to Karl von Staudt, a na projective conic section can also be understood as the set of self-polar points of a hyperbolic projective polarity . ${\displaystyle Char\neq 2}$

For a vector space over a body let be a map of in with the following properties ${\displaystyle V(K)}$ ${\displaystyle K}$ ${\displaystyle \rho }$ ${\displaystyle V(K)}$ ${\displaystyle K}$

(Q1) for each and . ${\displaystyle \rho (x{\vec {x}})=x^{2}\rho ({\vec {x}})}$ ${\displaystyle x\in K}$ ${\displaystyle {\vec {x}}\in V(K)}$

(Q2) is a bilinear form . ${\displaystyle f({\vec {x}},{\vec {y}}):=\rho ({\vec {x}}+{\vec {y}})-\rho ({\vec {x}})-\rho ({\vec {y}})}$

${\displaystyle \rho }$is called a square shape . (The bilinear shape is even symmetrical, i.e.. ) ${\displaystyle f}$ ${\displaystyle f({\vec {x}},{\vec {y}})=f({\vec {y}},{\vec {x}})}$

In the case applies , d. H. and determine each other in a clear way. In the case is . ${\displaystyle charK\neq 2}$ ${\displaystyle f({\vec {x}},{\vec {x}})=2\rho ({\vec {x}})}$ ${\displaystyle f}$ ${\displaystyle \rho }$
${\displaystyle charK=2}$ ${\displaystyle f({\vec {x}},{\vec {x}})=0}$

The following is . Then is . ${\displaystyle V(K)=K^{3},\ \rho ({\vec {x}})=x_{1}x_{2}-x_{3}^{2},}$ ${\displaystyle \ f({\vec {x}},{\vec {y}})=x_{1}y_{2}+x_{2}y_{1}-2x_{3}y_{3}}$

For one point is ${\displaystyle P=<{\vec {p}}>}$

${\displaystyle P^{\perp }:=\{<{\vec {x}}>\in {\mathcal {P}}\ |\ f({\vec {p}},{\vec {x}})=0\}}$a straight line and is called the polar of . is called the pole of ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P^{\perp }}$

The assignment is a projective hyperbolic polarity . Hyperbolic means that there are points that lie on their polar. Such points are called self-polar . (If a polarity has no self-polar points, the polarity is called elliptical .) ${\displaystyle P\leftrightarrow P^{\perp }}$

Properties of polarity:

1. The polar of a conic intersection point is the tangent at this point.
2. ${\displaystyle A\in P^{\perp }\rightarrow P\in A^{\perp }}$ (see picture),
3. ${\displaystyle P^{\perp \perp }=P}$.

If one now starts vice versa with a projective hyperbolic polarity in the projective plane , so this is a regular symmetric bilinear form to be described. In the case there is then a square shape that describes a non-degenerate conic section . A conic section defined in this way is called v. Staudt conic section . ${\displaystyle \pi }$ ${\displaystyle {\mathfrak {P}}_{2}(K)}$ ${\displaystyle f}$ ${\displaystyle K^{3}}$ ${\displaystyle CharK\neq 2}$ ${\displaystyle \rho ({\vec {x}})=f({\vec {x}},{\vec {x}})}$ ${\displaystyle k}$

Projective conic section: symmetry ${\displaystyle \sigma _{P}}$

Note: The linear mapping induces the involutive central collineation with axis and center , which leaves invariant (see section “Properties of a na conic section”). ${\displaystyle {\vec {x}}\rightarrow {\vec {x}}-2{\frac {f({\vec {p}},{\vec {x}})}{f({\vec {p}},{\vec {p}})}}{\vec {p}}}$ ${\displaystyle \sigma _{P}}$ ${\displaystyle P^{\perp }}$ ${\displaystyle P=<{\vec {p}}>}$ ${\displaystyle k}$

Note: There are also polarities for the affine conic sections ellipse, parabola and hyperbola .

See also

• Pascal's theorem
• Qvist's theorem
• Segre's theorem

Individual evidence

1. ↑ CDKG: Computer-assisted Descriptive and Constructive Geometry (TU Darmstadt) (PDF; 3.4 MB), p. 249.
2. ↑ CDKG: Computer-Aided Descriptive and Constructive Geometry (TU Darmstadt) (PDF; 3.4 MB), p. 250.
3. ↑ Projective geometry. (PDF; 180 kB). Short script, Uni Darmstadt, p. 6.
4. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 24.
5. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 28.
6. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), pp. 29–34.
7. ^ Planar Circle Geometries, an Introduction to Moebius, Laguerre and Minkowski Planes. (PDF; 891 kB), p. 23.
8. ↑ Projective geometry. (PDF; 180 kB). Short script, Uni Darmstadt, p. 12.
9. ↑ Hanfried Lenz : Lectures on projective geometry , Akad. Verl. Leipzig, 1965, p. 67.

literature

• Albrecht Beutelspacher, Ute Rosenbaum: Projective geometry. 2., through and exp. Edition. Vieweg-Verlag, Wiesbaden 2004, ISBN 3-528-17241-X .
• Daniel R. Hughes, Fred C. Piper: Projective Planes . Springer, Berlin a. a. 1973, ISBN 3-540-90044-6 .
• Hanfried Lenz : Lectures on projective geometry. Akademie Verlag, Leipzig 1965, DNB 452996449 .
• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin a. a. 1975, ISBN 3-540-07280-2 .