Projectivity

Central collineation: For each point are collinear ${\displaystyle P}$ ${\displaystyle Z,P,\pi (P)}$

A projectivity or projective collineation is a special collineation of a projective plane or a projective space in geometry . In the simplest case, a projectivity is a central collineation or perspectivity , i. That is, there is a fixed point (the center), and all straight lines through are fixed lines . One defines: ${\displaystyle Z}$ ${\displaystyle Z}$

• A projectivity (a projective plane or a projective space) is a collineation that can be represented by a product (execution) from a finite number of perspectives. In the general case there are other collineations in addition to the projectivities. In a real projective space, however, every collineation is already a projectivity. A useful feature of the projectivities is:
• The projectivities of a projective space over a body are precisely the collineations that can be described in the homogeneous model by linear images (matrices). ${\displaystyle {\mathcal {P}}^{n}(K)}$ ${\displaystyle K}$

Thus, the tools of linear algebra can be used to investigate projectivities.

A collineation that is not projectivity exists e.g. B. in the projective plane above the complex numbers : The projective continuation of the collineation of the complex affine plane is not projectivity. In the homogeneous model, it can only be represented by a semi-linear mapping . ${\displaystyle {\mathcal {P}}^{2}(\mathbb {C} )}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle (x,y)\to ({\overline {x}},{\overline {y}})}$

Projective collineation should not be confused with projective mapping . The latter maps one projective space onto another .

Properties of central collineations of a projective plane

→ Main article : central collineation

Central collineation: homology

Central collineation: elation

The properties of projectivities of a projective plane listed here can be transferred relatively easily to higher dimensions. Therefore it is assumed from here: The projective space is a projective plane .

So that the set of projectivities forms a group , one defines:

• Identity is a projectivity.

When projectivity is mentioned below, we usually tacitly assume that it is not a question of identity.

• A central collineation with the center always has an axis , i. That is, there is a straight line that remains fixed at certain points. They say: is a perspective. If on is called Elation , in the other case ( ) homology . ${\displaystyle \pi }$ ${\displaystyle Z}$ ${\displaystyle a}$ ${\displaystyle \pi }$ ${\displaystyle (Z,a)}$ ${\displaystyle Z}$ ${\displaystyle a}$ ${\displaystyle \pi }$ ${\displaystyle Z\notin a}$

Examples in the inhomogeneous model : ${\displaystyle {\mathcal {P}}^{2}(K)}$ The projective continuation

1. the stretching at the zero point is a homology with a center and an axis . ${\displaystyle (x,y)\to (sx,sy)}$ ${\displaystyle Z=(0,0)}$ ${\displaystyle g_{\infty }}$
2. the stretching on the x-axis is a homology with the center and the x-axis as the axis. ${\displaystyle (x,y)\to (x,sy)}$ ${\displaystyle Z=(\infty )}$
3. Translation is an elation with a center and an axis . ${\displaystyle (x,y)\to (x+c,y)}$ ${\displaystyle Z=(0)}$ ${\displaystyle g_{\infty })}$
4. the shear is an elation with the center and the x-axis as the axis. ${\displaystyle (x,y)\to (x+cy,y)}$ ${\displaystyle Z=(0)}$

• For a fixed point and a fixed straight line , the set of perspectives forms a group. ${\displaystyle Z}$ ${\displaystyle a}$ ${\displaystyle \Pi (Z,a)}$ ${\displaystyle (Z,a)}$
• If a fixed straight line is the set of all perspectives, then: is a group. ${\displaystyle a}$ ${\displaystyle \Pi (a)}$ ${\displaystyle (Z,a)}$ ${\displaystyle \Pi (a)}$

The last statement means: The sequential execution of a -Perspectivity and a -Perspectivity is again a perspectivity with an axis and a center on the straight line . ${\displaystyle (Z,a)}$ ${\displaystyle (Z',a)}$ ${\displaystyle a}$ ${\displaystyle Z''}$ ${\displaystyle ZZ'}$

From the duality principle for projective levels it follows:

• Every axial collineation (there is a straight line that only contains fixed points) is a central collineation. Axial collineations do not have to be treated separately.

The following sentence makes a statement about the existence and uniqueness of central collineations:

Theorem ( Baer ):

• Applies in a projective plane theorem of Desargues , then: Is a straight line and are three points collinear with , so there is exactly one -Perspektivität (central collineation with the center , axis ), which on maps. ${\displaystyle a}$ ${\displaystyle Z,P,P'}$ ${\displaystyle P,P'\notin a}$ ${\displaystyle (Z,a)}$ ${\displaystyle Z}$ ${\displaystyle a}$ ${\displaystyle P}$ ${\displaystyle P'}$

(A Desargue's plane can be coordinated with an oblique body .)

Properties of projectivities in a projective plane over a body

In this case:

• The set of projectivities of a projective plane over a body (Pappus plane) is the set of collineations that are induced by regular matrices in the homogeneous model. They are called ( projective linear group ). ${\displaystyle {\mathcal {P}}^{2}(K)}$ ${\displaystyle K}$ ${\displaystyle (3\times 3)}$ ${\displaystyle PGL(3,K)}$

Since multiples of the identity matrix only induce identity, the matrix of a projectivity can be multiplied by such a multiple of the identity matrix without changing the effect of the projectivity.

• A matrix induces a central collineation if and only if it has an eigenspace of the dimension . ${\displaystyle A\in PGL(3,K)}$ ${\displaystyle A}$ ${\displaystyle 2}$

It is a homology if it has another eigenvalue and the matrix can therefore be diagonalized. It is an elation if it cannot be diagonalized. ${\displaystyle A}$ ${\displaystyle A}$

A characteristic invariant of the projectivities is the double ratio :

• The projectivities of a Pappus projective plane are those collineations that leave the double ratio invariant.

Four projective independent points in the projective plane

The role of base points of a projective plane is assumed by quadruples of projectively independent points (points in general position ). Four points are projectively independent if no three are on a straight line. The following applies: ${\displaystyle A,B,C,D}$

• Four projectively independent points of a Pappus projective plane can always be described in suitable homogeneous coordinates by: ${\displaystyle A,B,C,D}$

${\displaystyle A=(1\colon 0\colon 0),\ B=(0\colon 1\colon 0),\ C=(0\colon 0\colon 1),D=(1\colon 1\colon 1)}$.

An important statement for the investigation of projectivities is the fundamental theorem :

• If a projectivity of a Pappus projective plane allows four projectively independent points to be established, then it is identity.

One consequence of the main theorem is:

• If and are each four projectively independent points of a Pappus projective plane, then there is exactly one projectivity with . ${\displaystyle A,B,C,D}$ ${\displaystyle A',B',C',D'}$ ${\displaystyle \pi }$ ${\displaystyle \pi (A)=A',\pi (B)=B',\pi (C)=C',\pi (D)=D'}$

Web links

• Projective geometry. (PDF; 1.5 MB). Script (Uni Darmstadt), p. 7
• WP Barth: Geometry , Uni Marburg, p. 85

Individual evidence

1. ^ Albrecht Beutelspacher, Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 96
2. ^ Albrecht Beutelspacher, Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 98
3. ^ Albrecht Beutelspacher, Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 125

literature

• Albrecht Beutelspacher , Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , pp. 93, 124.
• Wilhelm Blaschke : Projective Geometry. Wolfenbüttel Publishing House, Wolfenbüttel-Hanover, 1947.
• Wendelin Degen and Lothar Profke: Fundamentals of affine and Euclidean geometry , Teubner, Stuttgart, 1976, ISBN 3-519-02751-8
• P. Dembowski, Finite Geometries , Springer-Verlag (1968) ISBN 3-540-61786-8
• 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 .
• Rolf Lingenberg : Fundamentals of Geometry I , Bibliograph. Institute for University Pocket Books 158 / 158a, 1969, p. 83
• Günter Pickert : Projective levels . 2nd Edition. Springer, Berlin a. a. 1975, ISBN 3-540-07280-2 .
• Hermann Schaal: Linear Algebra and Analytical Geometry, Volume II , Vieweg 1980, ISBN 3-528-13057-1
• Olaf Tamaschke: Projective Geometry I , Bibliogr. Institute, Hochschulskripten 829 / 829a, 1968, p. 84