Algebras and formal specifications in geometric modelling

We revisit the theoretical foundations of geometric modelling from the algebraic and functional points of view. We present algebras as a powerful tool to describe old and recent geometric models and their operations of construction, manipulation and retrieval. They can be systematically and algebraically specified by a first-order equational logic. This contributes to accurately defining adequateness, ambiguity, correctness, uniqueness and implementation classes. It gives a firm guide to programming geometric modelling systems. We show that generic, unevaluated, geometric objects and histories of constructions can be systematically built and handled in functional specifications by an extended lambda-calculus based on the algebraic specifications.