Visual space geometry derived from occlusion axioms

Abstract

In our previous work [1] occlusion phenomena were considered as a foundation for development of geometrical structures in the visual field. The laws of occlusion were adapted to form the basis of an axiomatic system. Occlusion is formalized as a ternary relation and its properties include symmetry and transistivity. This leads to the definition of an abstract visual space (AVS). The geometry of an AVS is the subject of this research. Typical examples of an AVS are the convex open subsets of an n-dimensional real affine space for n≧2 or more generally, the convex open subsets of any n-dimensional affine space over some totally ordered field F, commutative or noncommutative. In an AVS we define such geometrical objects as lines, planes, convex bodies, closed and open subsets, etc. For any n-dimensional (n≧3) AVS X we construct an n-dimensional projective space P _F ⁿ over a totally ordered commutative or noncommutative field F and produce an embedding i: X x↪P _F ⁿ such that the image i(X) is open in P _F ⁿ and every line l in X is of the form l=i ^-1 (L∩i(X)) for some line L in P _F ⁿ. The mapping i and the field F depend on X and are unique up to isomorphism. We obtain a characterization of X via this embedding which is complete in the case of archimedean fields F. If F=R, the field of real numbers, i induces an embedding j: X ¶ A _R ⁿ and j(X) is a convex open subset of an affine space A _R ⁿ. If F≠R there can exist other types of AVS.