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By Walter Benz

This booklet relies on genuine internal product areas X of arbitrary (finite or endless) size more than or equivalent to two. Designed as a time period graduate path, the e-book is helping scholars to appreciate nice rules of classical geometries in a contemporary and normal context. a true profit is the dimension-free method of vital geometrical theories. the one necessities are easy linear algebra and simple 2- and three-dimensional genuine geometry.

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2 ( y j) = 2 ϕ2 x k So we get ψ 2 (αj) = ψ 2 (αj ) for all real α ≥ 0 and all j, j ∈ H with j 2 = 1 = j 2 . 32) for all h, h ∈ H with h2 = (h )2 . 27) does not depend on the chosen j. F. a) There exists a constant δ ≥ 0 such that for all h ∈ H, ψ (h) = 1 + δh2 . b) For all x, y ∈ X with x = 0, ϕ2 d (x, y) k = λ + ϕ2 (η − ξ) · (1 + δλ) holds true, where x =: ϕ (ξ), λx2 := x2 y 2 − (xy)2 and √ xy =: ϕ (ξ) ϕ (η) 1 + δλ. Proof. 27) does not depend on the chosen j ∈ H satisfying j 2 = 1, and thus η does not depend on this j.

13) holds true. Remark. 14), (h, α) · (0, β) = (h, β) · (0, α) is equivalent with Tα (h) · Tβ (0) = Tβ (h) · Tα (0), by noticing h ∈ e⊥ . 13) can be replaced by this latter equation which, of course, also holds true in the case αβ = 0. 13) is equivalent with Tα (h1 ) · Tβ (h2 ) = Tα (h2 ) · Tβ (h1 ) 16 Chapter 1. Translation Groups (for all α, β ∈ R and all h1 , h2 ∈ H) as well, and also with Tα (h) · T1 (0) = Tα (0) · T1 (h) for all α ∈ R and h ∈ H. 13), for instance Tα (h1 ) − h1 Tα (h2 ) − h2 = Tβ (h1 ) − h1 Tβ (h2 ) − h2 for all real α, β with β = 0 and all h1 , h2 ∈ H.

7) holds true, but not for λ ∈] − 1, 0[ or λ < −1. Hence [a, b] = {a + µ (b − a) | 0 ≤ µ ≤ 1}, and l (a, b) = {a + µ (b − a) | µ ∈ R}. In the case (X, eucl) the Menger lines are thus exactly the previous lines. The same holds true for (X, hyp) as will be proved in Theorem 6. 8) pq = 0, q 2 = 1, is the hyperbolic line through a, b, then a = p cosh α + q sinh α, b = p cosh β + q sinh β with uniquely determined reals α, β. 8) by ξ = −ξ and q by q = −q. So without loss of generality we may assume α < β.

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