GENERIC COORDINATE SYSTEMS IN THE COMPUTER GEOMETRY COURSE

Abstract

The article presents an approach to describe generic coordinate systems as a part of the course “Computer Geometry and Geometric Modeling”, which is taught to third-year students majoring in mathematics at the Lobachevsky State University of Nizhni Novgorod. Describing geometric mappings using coordinates and using transitions to other coordinate systems are the main tools in drawing images on the computer screen. The mathematical foundation of these operations come from analytic geometry and linear algebra courses, which mathematics majors take during their first year. However, many computer graphics textbooks do not make full use of theoretical concepts from these courses and do not provide proofs of correctness of coordinate transformations. Even when these proofs are present, they use linear algebra methods, which often consist of manipulating nested sums and numerous indices. This article demonstrates that the mains facts studied in linear algebra and used in constructing computer images can be generalized to arbitrary coordinate systems. The proofs of these facts use commutative diagrams. The helps abstract away from unnecessary details and clarify the main idea of the proof. We also advocate an approach that actively uses material covered by earlier mathematical courses and provides proofs that coordinate transformations used in drawing computer images are correct.

Keywords

• Сoordinate system
• commutative diagram
• affine transform
• computer graphics