ABSTRACT
Requirements for shape representation by surfaces arise from the same types of circumstances as with curves. Sometimes the designer has the shape of a complex object in mind and must quantify that shape with a mathematical representation. Other times, shapes are required to meet certain constraints, such as to pass through specific points in space or to pass through certain curves already created. Sometimes the points or curves being used as a starting point for surface creation have some regularity or neighborhood associativity, while at other times they are more randomly distributed. In this chapter we introduce representation forms that can be used to solve such problems for subsets of general problems. However, design involves more than just representation of the surface. We must be able to create models and query them for rendering, analysis, and manufacture. Further, for representations to be useful, the designers must find them easy to manipulate and modify. Hence, even though we introduce a variety of representations, we will stress and show how to both interpolate and approximate surfaces using mainly parametric tensor product surface representations.
