Geodesic surfaces

It is useful to test our understanding of determinate structures by considering the simple, yet powerful class of 3-d objects called geodesic structures. These are structures that can be mapped onto the surface of a sphere. There is a beautiful structural theorem about geodesic structures that we will now prove. 

The second approach is based on the idea of synthesizing bowl-shaped hydrocarbons in which curved networks of trigonal C-atoms map out the same patterns of five- and six-membered rings as those found on the surfaces of Cjq and/or the higher fullerenes [145-152]. An example for such an open geodesic polyarene is circum-trindene (5), generated by flash-vacuum pyrolysis (FVP) of trichlorodecacyclene 4 (Scheme 1.5) [152], Circumtrindene represents 60% of the framework of Cjq. 

Global chemicals giant Mitsubishi Chemical created a company named Frontier Carbon Corporation to manufacture tiny, high-strength carbon structures known as fiillerenes or bucky balls. Technically speaking, a fiillerene is a microscopic, round structure that is geodesic in nature. It has 60 joints across the rounded surface,... 

Another entity that we shall need belongs to the realm of intrinsic geometry geodesic curvature. Consider a surface x, a point P on x and a curve on x passing through P. The curvature vector of at P joins P to the centre of curvature of This curvature vector may be decomposed into mutually orthogonal components. These components are given by projection of the... 

Figure 1.7 Decomposition of a curve in a surface (left) into orthogonal geodesic and normal curvatures (right).

The torsion of a curve describes its pitch a helix exhibits both constant curvature and torsion. Its curvature is measured by its projection in the tangent plane to the curve - which is a circle for a helix - while its torsion describes the degree of non-planarity of the curve. Thus a curve on a surface (even a geodesic), generally displays both curvature and torsion. 

For a surface characterised by ki=-K2, the Gaussian curvature is simply related to the normal curvature and geodesic torsion ... 

In this case, the magnitude of the geodesic torsion at a point on a straight line lying in the surface is equal to the magnitude of the principal curvatures of the surface at that point. 

Figure 1.8 Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (flj) and the geodesic curvature along the arcs AB, BC, CD and DA.

Choose a triangle traced on a surface, whose three edges are geodesics. From the theorem, we have... 

If the (curved) edges lie along the principal directions on the surface the geodesic torsion of the network vanishes (xg=0). In this case, the density is ... 

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