Projectivity
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:
- 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).
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 .
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
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 .
Examples in the inhomogeneous model : The projective continuation
- the stretching at the zero point is a homology with a center and an axis .
- the stretching on the x-axis is a homology with the center and the x-axis as the axis.
- Translation is an elation with a center and an axis .
- the shear is an elation with the center and the x-axis as the axis.
- For a fixed point and a fixed straight line , the set of perspectives forms a group.
- If a fixed straight line is the set of all perspectives, then: is a group.
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 .
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.
(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 ).
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 .
- 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.
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.
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:
- Four projectively independent points of a Pappus projective plane can always be described in suitable homogeneous coordinates by:
- .
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 .
Web links
- Projective geometry. (PDF; 1.5 MB). Script (Uni Darmstadt), p. 7
- WP Barth: Geometry , Uni Marburg, p. 85
Individual evidence
- ^ Albrecht Beutelspacher, Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 96
- ^ Albrecht Beutelspacher, Ute Rosenbaum: Projective geometry. 2nd Edition. Vieweg, Wiesbaden 2004, ISBN 3-528-17241-X , p. 98
- ^ 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