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Discrete parametric surfaces

Discrete parametric surfaces are discrete analogues of smooth parametric surfaces. They are, however, not simply discrete approximations of their smooth counterparts, but are the subject of a separate discrete theory. As it turns out, a systematic theory of parametric surfaces can be based on integrable systems, and the discrete case can be interpreted as a “master” case which contains smooth surfaces as a limit. This section on discrete parametric surfaces is organized as follows: We first introduce notation. Two particular kinds of discrete surfaces are discussed next: circular nets in §2, and K-nets in §3 are examples of a 3-system and a 2-system, respectively. In the case of K-nets, we also discuss the relation to the sine-Gordon equation. We then show applications within mathematics in §4, cf. [BHS06], and the connection with freeform architecture in §5, cf. [PW16]. The main source for this chapter is the monograph [BS09].