A representation for modeling functional knowledge in geometric structures

Abstract

Most geometric models are quantitative (half-spaces and transformations), making it difficult to perform the kind of abstraction needed to model the underlying functional knowledge. Typically, users have used ad hoc subjective notions to perform this abstraction, which we call the "get-beneath-the-geometry" syndrome.

In this work we describe a systematic representation scheme that builds spatial maps based on local relations between objects. It derives relations that are more "functionally relevant" - i.e. those that involve accidental alignments, or can be described based on such alignments. The principal advantages of this representation in building functional descriptions is that

• a) it is free of subjective bias,

• b) it is complete in the qualitative sense of distinguishing all overlap/tangency/nocontact geometries.

In addition, the model is capable of handling uncertainty in the initial system (e.g. "the fuse box is somewhere behind the compressor") by constructing bounded inferences from disjunctive input data. Two kinds of uncertainty can be handled — those arising from deliberate imprecision in the interest of compactness ("down the road from"), or those caused by an inadequacy of data (sensors, spatial descriptions, or maps).

The representation is an extension to two and higher dimensions of the one-dimensional interval logic [Allen 83]. In orthogonal domains, this extension is straightforward, but a new non-commutative algebra has been developed for handling angular relations.