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Journal for Geometry and Graphics 11 (2007), No. 2, 187--198
Copyright Heldermann Verlag 2007

A Geometrical Approach to the Numerical Stability Analysis of Some Projective Collinear Mapping Methods

Branislav Popkonstantinovic
Dept. of Machine Theory and Mechanisms, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Belgrade, Serbia

Dragan Petrovic
Faculty of Mechanical Engineering, University of Kragujevac, Dositejeva 19, 36000 Kraljevo, Serbia

This paper exposes the geometrical approach to the numerical stability analysis of the projective collinear mapping method based on Laguerre's points of the involution mapped. The ill-conditioned and corresponding unstable zones for this mapping method are defined and some alternative procedures for their corrections are proposed. As a result of the analysis, the stable and well conditioned collinear mapping technique, which can be used in the design of some 3D imaging software algorithms, are created and explained. The exposed analysis is a contribution to the theory of projective and computational geometry. Moreover, it proposes a numerical stability criterion for the practical mapping procedures in software applications which are based on the principles of the projective collineation.

Keywords: Involution, Laguerre's points, numerical stability, projective collineation.

MSC: 51M04; 51N15

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