Parametrizing geometric objects using λ-calculus

Abstract

This paper focuses on the general parameterization in geometric modeling. We have adapted the -calculus formalism to an existing geometrical model which has an applicative representation: the generalized maps embedded in the plane. We investigated how this allows us to parameterize geometric objects by other objects, which can themselves be functional, rather than just by values such as size, shape or position. Thus, conditional, iterative, recursive and shared objects can be built and managed in an homogenous way. We have based the study on an interactive prototype implemented in Objective Caml. Its interface offers two consistent working views. The first interacts with geometric objects while the second interacts with the corresponding programs. We have examined in detail various higher-order operations where -calculus abstraction and application are used extensively to demonstrate how such objects can be built and manipulated. We will illustrate the force of our proposals using more complex constructions.