Modeling Topological Relations Based on Euler Characteristics

The criteria on the description and determination of topological relations are topological invariants, which may be dimension, separations, Euler characteristics etc. This paper is concentrated on some basic issues of modeling topological relations using Euler characteristics. Geometricaly, a detailed investigation on the relations among simplex, complex and spatial object is made. With the introduction of the concept of pure complex, a generic representation of spatial object in the GIS field is given. Topological components of simplex, i.e. boundary and interior, are then defined through the algebraic topology. A suitable and sound pre-processing approach is put forward prior to calculation of Euler characteristics. Compared with the classic 4/9-intersection model, the model based on Euler characteristics can distinguish more details in topological aspect.