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By Jean A. Dieudonne

ISBN-10: 0534037232

ISBN-13: 9780534037239

Publication via Dieudonne, Jean A.

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24) of the Lobachevsky 3 Straight lines can also be represented by diameters. 2. Model interpretations of Lobachevsky’s planimetry 39 plane (in the same variables), allows one to apply in their study the tools and methods of Riemannian geometry, of the theory of curves and surfaces, and so on. Based on these, in Chapter 2 we will obtain important geometric characteristics of various geometric elements of Lobachevsky planimetry. 5) conformally onto the upper half-plane Π = {w = u + iv ∈ W, Im w > 0}.

To this aim, as before, we consider in the Lobachevsky plane some straight line m and a point M not on it, as well as two straight lines b and c that pass through M parallel to m. 3). 3). In other words, drop from the point M the perpendicular M H to m and denote it by h ≡ M H. Consider the angles that arise in this way. 1). Note also that in the models of the Lobachevsky plane used (in particular, in the Cayley-Klein model), the “model Euclidean” angles do not necessarily coincide with the corresponding angles of the Lobachevsky planimetry.

3) in which the following notations for partial derivatives are used: p= ∂z , ∂x q= ∂z , ∂y r= ∂2z , ∂x2 s= ∂2z , ∂x∂y t= ∂2z . 3) with respect to the function z = z(x, y) means to describe, in the Euclidean space E3 , all surfaces with a priori given curvature K according to their shape and position in space. 3) cannot be integrated exactly. Nevertheless, when the curvature of the surface is constant, important particular typical cases can be studied exhaustively. F. Minding carefully studied surfaces of constant positive curvature, as well as surfaces of constant negative curvature.

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History of Algebraic Geometry by Jean A. Dieudonne

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