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We give an elaborated treatment of discrete isothermic surfaces and theiranalogs in different geometries (projective, M\"obius, Laguerre, Lie). We findthe core of the theory to be a novel projective characterization of discreteisothermic nets as Moutard nets. The latter belong to projective geometry andare nets with planar faces defined through a five-point property: a vertex andits four diagonal neighbors span a three dimensional space. Analytically thisproperty is equivalent to the existence of representatives in the space ofhomogeneous coordinates satisfying the discrete Moutard equation. Restrictingthe projective theory to quadrics, we obtain Moutard nets in sphere geometries.In particular, Moutard nets in M\"obius geometry are shown to coincide withdiscrete isothermic nets. The five-point property in this particular case saysthat a vertex and its four diagonal neighbors lie on a common sphere, which isa novel characterization of discrete isothermic surfaces. Discrete Laguerreisothermic surfaces are defined through the corresponding five-plane propertywhich requires that a plane and its four diagonal neighbors share a commontouching sphere. Equivalently, Laguerre isothermic surfaces are characterizedby having an isothermic Gauss map. We conclude with Moutard nets in Liegeometry.Comment: 29 pages, 8 figure

Isothermic surfaces in sphere geometries as Moutard nets