Download Encyclopedia of Analytical Surfaces by S.N. Krivoshapko, V.N. Ivanov PDF

By S.N. Krivoshapko, V.N. Ivanov

This encyclopedia provides an all-embracing choice of analytical floor periods. It presents concise definitions and outline for greater than 500 surfaces and categorizes them in 38 sessions of analytical surfaces. All periods 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 worthy reference for mathematicians.

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2, p. 25-28. 1 Torse Surfaces (Torses) Torse or torsal surfaces are nondegenerated ruled developable surfaces with cuspidal edges and they can be developed on a plane without any lap fold or break. All developable surfaces have only parabolic points in which k1k2 = 0. So, an equality of Gaussian curvature to zero (K = 0) is the sufficient and necessary condition for a developable surface. 3 Every single-parametric system of planes (with the exception of a pencil of planes passing through any axis or parallel to each other) has an envelope surface, which is a developable surface.

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.

If we shall take x = x(u, φ), y = y(u, φ), x = x(u, φ) but s = const, then we shall construct a ruled surface of trajectory of the chosen rectilinear generatrix with s = const. ■ Monge’s Ruled Surface with the Circular Cylindrical Directing Surface Monge’s ruled surface with the circular cylindrical directing surface is formed by a straight line belonging to a plane P when rounding without slip on a directing circular cylinder with the radius r (Fig. 1). z If we intersect an open evolvent helicoid by circular cylinders with radii Rint and Rext, the axes of which coincide with the z axis of the open evolvent helicoid then their intersections will be cylindrical helical lines.

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