Abstract
This chapter will review some work done at the Computer Graphics Laboratory of the University of Waterloo to build software abstractions with which ideas for new techniques in spline surface design can be supported. Our goal has been to facilitate the construction of prototype surface editors through the use of these abstractions. These editors are concerned with the definition of surfaces for industrial and artistic purposes.The work built upon the most fundamental of mathematical abstractions: affine geometry; involving one of the commonest definitions of manifolds: parametric splines. Graph and tree abstractions that are linked to geometry are employed to organize the parts of the models that are edited.