We have introduced a general mathematical framework for transforming functionally defined shapes. The proposed model of extended space mappings considers transformations of a hypersurface in coordinate-function space with its projection onto geometric space. This model covers coordinate space mappings, metamorphosis, and algebraic operations on defining functions, and introduces several new types of transformations, such as function-dependent space mappings and combined mappings. The approach is illustrated by new local deformations created by means of function mappings, feature-based space mapping, offsetting along the normal, generation of a thin shell of a solid (figure above), 2D shape blending, and collision-free metamorphosis.

See the formulation of the extended space mapping and examples of applications in the paper

Savchenko V.V., Pasko A.A. “ Transformation of functionally defined shapes by extended space mappings ”, The Visual Computer, vol. 14, No. 5/6, 1998, pp. 257-270. Electronic version: PDF (581 K)