By S.N. Krivoshapko, V.N. Ivanov
This encyclopedia offers an all-embracing number of analytical floor sessions. It presents concise definitions and outline for greater than 500 surfaces and categorizes them in 38 sessions of analytical surfaces. All sessions are move references to the unique literature in a superb bibliography.
The encyclopedia is of specific curiosity to structural and civil engineers and serves as important reference for mathematicians.
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Extra resources for Encyclopedia of Analytical Surfaces
59-71. 1 Ruled Surfaces of Zero Total Curvature Cylindrical, conical, torse surfaces, a plane, and surfaces of polyhedrons are called surfaces of zero Gaussian curvature or developable surfaces. A surface is called a developable surface if it can be developed on a plane without any lap fold or break. During this process, the length of the curves and the angles between two curves belonging to the developable surface remain unchanged. Cylinders and cones are the simplest developable surfaces, being degenerate.
2 helical line. The length of the arc of the cuspidal edge between points 0 and t is pﬃﬃﬃ s ¼ a 2 sinh t: Curvature k and torsion κ of the hyperbolic helical line can be determined with the help of the following formula: 1 a : ¼ 2 k¼j¼ 2 2acosh t 2a þ s2 Forms of deﬁnition of the studied developable surface (1) Parametric form of deﬁnition (Fig. 1): sht u; x ¼ xðu; tÞ ¼ a cosh t þ pﬃﬃﬃ 2 cosh t u y ¼ yðu; tÞ ¼ a sinh t þ pﬃﬃﬃ ; 2 u ; z ¼ zðu; tÞ ¼ at þ pﬃﬃﬃ 2cocht where u is the length of the rectilinear generatrix of the torse taken from the cuspidal edge until arbitrary point; À1 u 1; À1 t 1: Coefﬁcients of the fundamental forms of the surface and its curvatures: pﬃﬃﬃ A ¼ 1; F ¼ a 2 cosh t; B2 ¼ F 2 þ u2 ; 2cosh2 t u2 ; 2cosh2 t u ; L ¼ M ¼ 0; N ¼ 2cosh2 t u ku ¼ k1 ¼ 0; kt ¼ 2 ; 4a cosh4 t þ u2 1 1 k2 ¼ ; K ¼ 0; H ¼ : u 2u A2 B2 À F 2 ¼ ■ Torse with a Given Line of Curvature in the Form of the Second-Order Parabola Assume a line of principal curvature of a torse surface in the form of the second-order parabola x ¼ 0; y ¼ v; z ¼ Àav2 : pﬃﬃﬃ u ¼ uðtÞ ¼ ðz0 À atÞ 2cosh t are lying.
18-23 June, 1973, Kielce (Poland), 1973; p. 115-124. Finikov SP. A Course of Differential Geometry. Moscow: GITTL, 1952; 343 p. Monge G. Application of Analysis to Geometry. : ONTI, 1936; 699 p. The Literature on Geometry and Strength Analysis of Shells in the Form of Developable Surfaces Babenko VI. Stability of three-layered developable shells. Dokl. AN USSR. 1986; A, No. 11, p. 19-22. Amirov M. Graphical-and-analytical method of construction of a torse development. : Voprocy geometrii, Samarkand: SGU, 1979; p.