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Some questions concerning the problem of determining of projective collineations

Some questions concerning the problem of determining of projective collineations DEMONSTRATIO MATHEMATICAVol. XXVIINo 21994Matgorzata Buba-BrzozowaSOME QUESTIONS CONCERNING THE PROBLEMOF D E T E R M I N I N G OF P R O J E C T I V E COLLINEATIONSIntroductionIt is a well-known fact that in a pappian plane any projectivity is uniquelydetermined by four points, and their images, no three of which are collinear.R. Sturm [1] considered, among others, the problem of unique determinationof any projective collineation and correlation by another set of elements. Inparticular he studied this problem for a collineation and for the set of eightpoints and eight lines containing their images; for a correlation and the setof eight unhomogenous elements (points, lines, polars). He also posed thefollowing question: Can another set of elements-unnecessarily of the samekind-be given to determine a projectivity in a pappian plane (for example:to determine a collineation-six points, two of their images and four linescontaining the images of the remaining four points)? This paper is devotedto the problems of the same kind.Let Pn denote the n-dimensional pappian projective space. Projectivecollineations in Pn will be written in the matrix form:n+l(1)A[x] = A[y] o - ^ a i j X j = Xyi,i = 1 , 2 , . . . ,ra http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

Some questions concerning the problem of determining of projective collineations

Buba-Brzozowa, Matgorzata
Demonstratio Mathematica , Volume 27 (2): 6 – Apr 1, 1994
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Publisher
de Gruyter
Copyright
© by Matgorzata Buba-Brzozowa
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1994-0206
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXVIINo 21994Matgorzata Buba-BrzozowaSOME QUESTIONS CONCERNING THE PROBLEMOF D E T E R M I N I N G OF P R O J E C T I V E COLLINEATIONSIntroductionIt is a well-known fact that in a pappian plane any projectivity is uniquelydetermined by four points, and their images, no three of which are collinear.R. Sturm [1] considered, among others, the problem of unique determinationof any projective collineation and correlation by another set of elements. Inparticular he studied this problem for a collineation and for the set of eightpoints and eight lines containing their images; for a correlation and the setof eight unhomogenous elements (points, lines, polars). He also posed thefollowing question: Can another set of elements-unnecessarily of the samekind-be given to determine a projectivity in a pappian plane (for example:to determine a collineation-six points, two of their images and four linescontaining the images of the remaining four points)? This paper is devotedto the problems of the same kind.Let Pn denote the n-dimensional pappian projective space. Projectivecollineations in Pn will be written in the matrix form:n+l(1)A[x] = A[y] o - ^ a i j X j = Xyi,i = 1 , 2 , . . . ,ra

Journal

Demonstratio Mathematica – de Gruyter

Published: Apr 1, 1994

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