Gauss map

Clearly the spherical image under the Gauss map of a highly curved surface patch will be larger than that of less curved patches of the same area, since the divergence in direction spanned by the normal vectors is wider for the highly curved patch. An extreme example is the plane, which is mapped onto a single point, whose location depends on the orientation of the plane. 

The Gauss map is closely related to the Gaussian curvature of the surface. In fact, the surface area of die Gauss-mapped region on the unit sphere is equal... 

Figure 1.6 The Gauss map of a surface. The normal vectors in the triangular ABC region of the saddle-shaped surface define a region on the unit sphere, A B C, given by the intersection of the unit sphere with the collection of normal vectors (each placed at the centre of a unit sphere) within the ABC region. Notice that for the example illustrated the bounding curve on the surface and on the uiut sphere are traversed in opposite senses. This is a necessary feature of saddle-shaped surfaces, with negative Gaussian curvature.

A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a mapping from the minimal surface to a Riemann surface. The mapping is a composite one first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except flat points), characterised by cartesian coordinates (x,y,z) is described by the complex number (o, which... 

Figure 1.16 Mappiitg of a minimal surface from real >ace to the complex plane. A point P on the surface, whose normal vector at P is n, is transformed to a point P by the Gauss map, given by the intersection of n (placed at the origin of the unit sphere O) with the sphere. P is mapped into a point P" on die complex plane (real and imaginary axes aand rresp.) by stereographic projection from the north pole of die sphere, N, onto the complex plane, which intersects the sphere in its equator.

The Gauss map of an IPMS (which is a function of the surface orientation only through the normal vectors) must be periodic, since a translationally periodic surface is necessarily orientationally periodic. (The converse, however, is not true.) Consequently, the Gauss map of IPMS must lead to periodic tilings of the sphere. This principle has been used to construct all the simpler IPMS, and has recently been generalised to allow explicit parametrisation of more complex "irregular" IPMS [13-24]. Some of these examples are illustrated in the Appendix to this chapter. 

Two surfaces x and y are parallel if they have an identical distribution of normal vectors i.e. their Gauss maps are indistinguishable. Thus, a family of parallel surfaces can be produced by translating a surface in the direction of its normal vectors by m equal amoimt everywhere on the surface. If x and y are parallel surfaces separated by a distance c, it can be shown that their Gaussian and mean curvatures are related 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... 

Figure 1 The Gauss map g between a surface E around a point P of position vector r and the unit sphere Si. Q ow Si corresponds to P on Z. The vector tangent plane n is the same. dQ transforms dL into dS = dG dT), changing its shape and area.

If C is hyperelliptic, other arguments are needed. Or one may use variants of the lemma where a is infinitesimal. The geometric meaning of 0f 0a, a infinitesimal, is the following let P be the projective space of (g — l)-dimensional subspaces of Tjac,o- Then we get the so-called Gauss map ... 

Aboites, V., Boltyanskii, V. G., Wilson, M. (2012). A model for co-discovery in science based on the synchronization of gauss maps. International Journal of Pure and Applied Mathematics, 79(2), 357-373. Available from http //ijpam. eu/contents/2012-79-2/15/15.pdf... 

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