On Projections in Geometric Design

We discuss using projected manifolds in geometric design and processing. Curves and surfaces are defined in n-space as algebraic sets of the appropriate dimensionality, and their projection into a three-dimensional subspace is considered. In this approach, the additional variables serve to express simply geometric constraints, and so complex surfaces such as offsets, equi-distance surfaces, rounds and fillets caD be represented exactly. We describe the definitional method and discuss some of the algorithmic infrastructure available to manipulate and interrogate curves and surfaces 80 represented. This representation generalizes both implicit and parametric representations, and can deal with surfaces that otherwise could be obtained exactly only through elimination computations of forbidding complexity.