Appendix...A catalogue of some minimal surfaces

Let us finish this chapter with a catalogue of some minimal surfaces that can be generated using the Weierstrass parametrisation (eqs. (1.18)). 

This representation enables a catalogue of minimal surfaces to be built. 

The least number of flat points a minimal surface may possess is one. An example of such a surface is Enneper s surface, which is asymptotically flat (Fig 1.23). In this case the flat point is not an isolated point on the surface. However, only a single surface orientation is displayed by the asymptotically flat boundary. 

The surface is unique in that the Bonnet transformation applied to this surface does not produce distinct surfaces. All the members of the isometric family related to Enneper s surface are equivalent - they differ only in their relative orientation in space. The Weierstrass parametrisation for this surface is given by 

The next possibility is a Gauss map containing two singularities due to flat points. Examples of this case are the helicoid and the catenoid (Figs. 1.13 and 1.14). The normals of the flat points on these surfaces (at the asymptotic ends of the surfaces) are antiparallel, and hence the Weierstrass parametrisation is given by 

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