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Collinearity-preserving functions between Desarguesian planes.

D S Carter¹, A Vogt

¹Department of Mathematics, Oregon State University, Corvallis, Oregon 97331.

Proceedings of the National Academy of Sciences of the United States of America

|July 1, 1980

Summary

We characterized collinearity-preserving functions between affine and projective Desarguesian planes. Our findings extend previous work to include affine planes and cases with fewer points, offering a comprehensive understanding of these geometric transformations.

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Area of Science:

Geometry
Algebraic Geometry
Valuation Theory

Background:

Collinearity-preserving functions are fundamental in geometry.
Previous research focused on projective planes with quadrangles.
Affine and projective Desarguesian planes have distinct properties.

Purpose of the Study:

To characterize all collinearity-preserving functions between affine or projective Desarguesian planes.
To extend existing results to include affine planes.
To analyze cases with minimal configurations like triangles.

Main Methods:

Application of concepts from valuation theory.
Development of a new characterization theorem.
Analysis of function extensions between affine and projective planes.

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Main Results:

A complete characterization of collinearity-preserving functions is obtained.
Results cover cases involving affine planes and configurations with triangles.
A key theorem demonstrates the extendability of functions from affine to projective planes, barring specific small cases.

Conclusions:

The study provides a comprehensive framework for understanding collinearity-preserving functions in Desarguesian planes.
The findings generalize and deepen existing knowledge in geometric transformation theory.
The extendability theorem offers a powerful tool for relating affine and projective geometries.