By H. S. M. Coxeter
It is a reissue of Professor Coxeter's vintage textual content on non-Euclidean geometry. It starts off with a ancient introductory bankruptcy, after which devotes 3 chapters to surveying actual projective geometry, and 3 to elliptic geometry. After this the Euclidean and hyperbolic geometries are equipped up axiomatically as designated situations of a extra common 'descriptive geometry'. this is often crucial studying for anyone with an curiosity in geometry.
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Additional info for Non-Euclidean Geometry (6th Edition)
In view of 2. 1), we obtain the density function of Hz, which is proportional to p+1Fq [a 1, ... m; b1, ... ,bq;H~BHz(H~AHz)-1]IH~AHzl-m/2. 4), and the condition A = 1m gives the density function proportional to PHFq(a1, ... m; b1, ... , bq; H'zBH z ). 10) l=O >'I-! (B) are zonal polynomials (see Appendix A), and we can put d(O) = 1 without loss of generality. k; b1, . m; B) x pFq(a 1, ... , ap; b1, . k\ . B) etr(X' BX), 1 2 ' 2m, 1 (1 I ) F. 13) 1 (1 . I ) F (! k;X' BX). 3. 4, respectively. 2.
15)(P). 13)(P), having the density function 1 1 F (lk' 1 . 22) is a slight modification of the Downs (1972) distribution on the Stiefel manifold, and may be called the matrix Langevin distribution on Pk,m-k' which is denoted by L
X' BrX) = :E: >'[r];q, = 1, ... -C~[r] (X' Bl X, . [r] Il1i! 19) Here we use the notation in the theory of invariant polynomials j that is, C~[r] (B l , ... , Br) with the matrix arguments B l , ... , B r , and the d~[r are suitable coefficients. 2. 9) on V,. 21) 38 2. 11). 17)(P). 15)(P). 13)(P), having the density function 1 1 F (lk' 1 . 22) is a slight modification of the Downs (1972) distribution on the Stiefel manifold, and may be called the matrix Langevin distribution on Pk,m-k' which is denoted by L
Non-Euclidean Geometry (6th Edition) by H. S. M. Coxeter