This chapter comprises two relatively independent topics: As an application of the Clifford algebra formulation of the classical model that we developed in Chapter 6, we want to discuss smooth and discrete (triply) orthogonal systems and their Ribaucour transformations; and as an application of the Vahlen matrix enhancement of the Clifford algebra model that we elaborated on in Chapter 7, we will discuss (smooth) isothermic surfaces of arbitrary codimension. However, (Darboux pairs of) isothermic surfaces can be considered as a special case of (Ribaucour pairs of) orthogonal systems, which may justify treating both topics in one chapter.

Technically, an orthogonal system can be considered as an orthogonal coordinate system on ℝn (or the conformal n-sphere Sn); geometrically, it is a system of n 1-parameter families of hypersurfaces so that any two hypersurfaces from different families intersect orthogonally. Triply orthogonal systems in Euclidean space have been studied intensively in classical time (see for example [17], [94], or [136]; an excellent overview is given in [248]). However, the notion of an orthogonal system is obviously conformally invariant, and important properties (as, for example, Dupin's theorem; cf. §2.4.8) can be formulated and proven in a conformal setting — remember that we already briefly touched on triply orthogonal systems in the context Guichard nets (see §2.4.4).