The Sphere by Itself: Can We Distribute Points on It Evenly?

Chapter III The sphere by itself: can we distribute points on it evenly? III.1. The metric of the sphere and spherical trigonometry As we shall see throughout this chapter, the geometry of the “ordinary sphere” S2 two dimensional in a space of three dimensions harbors many pitfalls. It’s much more subtle than we might think, given the nice roundness and all the sym- metries of the object. Its geometry is indeed not made easier at least for certain questions by its being round, compact, and bounded, in contrast to the Euclidean plane. Sect. III.3 will be the most representative in this regard; but, much simpler and more fundamental, we encounter the “impossible” problem of maps of Earth, which we will scarcely mention, except in Sect. III.3; see also 18.1 of [B]. One of the reasons for the difficulties the sphere poses is that its group of isometries is not at all commutative, whereas the Euclidean plane admits a commutative group of trans- lations. But this group of rotations about the origin of E3 the orthogonal group O.3/ is crucial for our lives. Here is the place to quote (Gromov, 1988b): “O.3/ pervades all the essential properties of the physical world. But we remain intellectually blind to this symmetry, even if we encounter it frequently and use it in everyday life, for instance when we experience or engender mechanical movements, such as walking. This is due in part to the non commutativity of O.3/,whichis difficult to grasp.” Since spherical geometry is not much treated in the various curricula, we permit ourselves at the outset some very elementary recollections, which are treated in detail in Chap. 18 of [B]. Typically we will deal with the sphere S2 of radius 1 centered at the origin of the Euclidean space E3, but all spherical geometries are the same within a change of scale. The interesting distance between two points p and q of S2 isn’t their distance in E3 but that on S2, i.e. the length of a shortest path on S2 that joins p to q. We are dealing here with the metric that is called intrinsic or internal, as opposed to the induced metric that is the distance between these points in E3 and evidently without much interest for Earth’s inhabitants: digging tunnels is expensive. It’s classical that this path is unique and that is the arc of a great circle that joins p to q, which can be proved rigorously by a symmetry argument, or here with the fundamental formula (III.1.1) below. Note that in saying the arc of a great circle we necessarily exclude antipodes, for which all the great circles starting from the one arrive at the other after a distance .Weletd.p;q/ denote this distance, which varies from 0 to . It is given in terms of the scalar product by the formula: M. Berger, Geometry Revealed, DOI 10.1007/978-3-540-70997-8_3, 141 c Springer-Verlag Berlin Heidelberg 2010 142 CHAPTER III. THE SPHERE BY ITSELF cos.d.p; q// D p q. But it may also be regarded as the angle through which we see the points p and q from the origin O. We can say, in an entirely equivalent manner, that we are dealing here with seeing the sphere as a set of half-lines emanating from the origin, which is crucial in descriptive astronomy: the sphere is the celestial vault and there is an essential need of the formulas that will follow. As astronomy is very old, these formulas date from several centuries ago, but we will read them on S2, not as would astronomers. Fig. III.1.1. We remain on S2. It is clear that when we have three points p, q, r of S2 we may speak in analogy to Euclidean geometry of the spherical triangle fp; q; rg. Now if we know the distances of d.p;q/ and d.p;r/, as well as the angle between the sides that emanate from p, we sense in advance that the distance of d.q;r/ is determined, thus theoretically calculable. The formula that provides this distance is called the first fundamental formula of spherical trigonometry. Let us call a, b, c the lengths (distances) of the sides of fp; q; rg, and ˛, ˇ, its angles (between 0 and ). Then we always have: .III:1:1/ cos a D cos b cos c C sin b sin c cos ˛: The proof is a direct application of the definitions and of the scalar product, whose neglect in school mathematics explains why this formula isn’t better known to stu- dents. It was prohibited in the French lycées until about 1950, which explains the incredible contortions in treating the inequality of days and nights, which was part of the required program. From it we can deduce cautioning that the value of the sine does not determine the angle unambiguously all the formulary given in detail in [B]. The practical importance is considerable; this or analogous formulas provide the solution to the following problem relating to three directions (or three half-lines) in space. Such a configuration is associated, as is the corresponding spherical triangle, with six numbers: the three angles between each pair of lines, and also the angles between the planes determined by pairs of these lines. This play with formulas allows us to calculate all these elements as a function of only three of them, but with some precaution. Think about the “dubious” case of equality of triangles. Readers III.1. THE METRIC OF THE SPHERE AND SPHERICAL TRIGONOMETRY 143 will be able to investigate some practical applications: to astronomy, computation of the length of days in the course of the year, positions of artificial satellites, geodesy, petroleum research, GPS, etc. Fig. III.1.2. Construction of the spherical triangle dual to fp; q; rg. In the order: construction of p0, construction of q0 and r0 in the plane orthogonal to the ray Op (p , q , r represent the projections onto the plane of points p, q, r), the triangle fp; q; rg and the dual triangle. II [B] Géométrie. Nathan (1977, 1990) réimp. Cassini (2009) c Nathan Édition It is important to mention the duality that exists for spherical triangles, that simplifies the derivation of the formulas. This duality associates with the triangle fp; q; rg, by taking appropriate perpendiculars, a (the) dual triangle fp0;q0;r0g whose elements (denoted using apostrophes) are simply the angles complementary to , but exchanging the roles of sides and angles: a C ˛0 D b C ˇ0 D c C 0 D a0 C ˛ D b0 C ˇ D c0 C D : The triangle dual to fp0;q0;r0g is the original triangle fp; q; rg. This duality is the spherical version of that given in Sect. I.7 for the real projective plane P . In the lan- guage of half-lines, i.e. of trihedra, it is expressed thus: with the trihedron fD; E; Fg we associate the trihedron fD0; E0; F0g, called supplementary, defined by the conditions: F0 is perpendicular to D and E and is located on the same side of the plane defined by D and E as F, and likewise for the two other half-lines. An historical note on “school” mathematics, interesting but doubly depressing: in French instruction the duality of supplementary trihedra was included in the program for the terminal classes at the lycées. Until 1950 the proof of duality (involutivity) was done geomet- rically, since use of the scalar product was strictly forbidden. Now with the notion of “same side” involutivity was rather obscure to prove, even in the very good work (Hadamard, 1911). It took several pages, with complicated reasoning about the poles associated with a great circle, etc. Such reasoning is always subtle; we will see this clearly in Sect. III.4 for the problem of “the thirteenth sphere”. Now this involutivity is trivial with the use of the scalar product: if fx;y;zg and fx0;y0;z0g are nonzero vectors representing our trihedra of our spherical triangles, the six conditions defin- ing fx0;y0;z0g are: 144 CHAPTER III. THE SPHERE BY ITSELF x0 y D x0 z D 0; y0 x D y0 z D 0; z0 x D z0 y D 0 x0 x>0; y0 y>0; z0 z>0: The set in its totality is clearly symmetric. The author learned this algebraic formulation from his classmate André Aragnol when he was preparing for his “agré- gation” (French competition for specially privileged teaching positions at the sec- ondary level). We ask ourselves why so many schoolchildren and their teachers need suffer because of a perverse sort of love for pure geometry. We have seen this phenomenon above regarding the fundamental formula, we will encounter it again regarding the duality with respect to a circle in Sect. IV.2. Avoiding use of coordi- nates and still worse the scalar product, is an obsession that has long haunted “pure” geometers. Now the scalar product figured in Grassmann’s work from 1840, but in writing that was far too prophetic; it wasn’t really popularized until Gibbs in 1860. But resistance to Gibbs raged from the disciples of Hamilton, who wanted to keep the scalar product exclusively for quaternions (thus in dimension 4) and pre- pared the way for a long and bitter correspondence, which is today hilarious: see Gibbs (1961). The second point, sadder perhaps, is that this discussion is hardly relevant today, since geometry, especially that of space, has all but completely disappeared from the mathematics curriculum of schools and universities this while space visualiza- tion is becoming more and more necessary for practical applications such as three dimensional animation and robotics.

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