Chapter 1 introduced several constructions of barycentric coordinates as well as their properties. In this chapter, we explore the deep connection between barycentric coordinates and higher order parametric representations of curves, surfaces, and volumes in arbitrary dimension. In the case of curves, these curves are known as Bézier curves, which are used in applications from font representations to controlling animations. The extension to convex surface patches, called S-patches , is more recent. As originally proposed, S-patches were parametric, multi-sided surface patches restricted to convex domains. However, these restrictions were more of a function of the limited set of generalized barycentric coordinates, namely Wachspress coordinates, available at that time. Today we have generalized barycentric coordinate functions that do not require convexity and extend to arbitrary dimension. Hence, we will investigate S-patches within their full generality, afforded by modern barycentric coordinates with generalized domains and in arbitrary dimension.