By S.N. Krivoshapko, V.N. Ivanov
ISBN-10: 3319117726
ISBN-13: 9783319117720
ISBN-10: 3319117734
ISBN-13: 9783319117737
This encyclopedia offers an all-embracing choice of analytical floor periods. It offers concise definitions and outline for greater than 500 surfaces and categorizes them in 38 periods of analytical surfaces. All sessions are pass references to the unique literature in a superb bibliography.
The encyclopedia is of specific curiosity to structural and civil engineers and serves as helpful reference for mathematicians.
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Kuzmenko EA, Petukhova GI. Design of evolvent helicoids with the help of polyhedral helical surfaces. Mater. 34 Otchetn. nauchn. konfer. Voronezh. gos. tehnolog. akad. za 1994 god. Voronezh, December 8-13, 1994; p. 284. In any arbitrary position, the plane P touches the circle cylinder. The formed surface will be a surface of zero Gaussian curvature. This surface is a part of an open helicoid limited by lines of principal curvatures. A family of straight lines is called meridians; the orthogonal trajectories of the points of a generatrix straight line are called parallels (Fig.
2, p. 42-44. Yuhanio Marulanda Arbelais. Strength Design of Shells in the Shape of G. Monge’ s Ruled Surfaces. Ph Dissertation, Moscow, UDN. 1970; 154 p. 5π A developable conical helicoid has a conical spiral u = u1 x ¼ xðuÞ ¼ r0 sin k cos u Á eku ; y ¼ yðuÞ ¼ r0 sin k sin u Á eku ; z ¼ zðuÞ ¼ r0 cos k Á eku as the cuspidal edge (Fig. 1). This conical spiral lies on a circular cone, where λ is the angle between an axis Oz and a directrix of the cone, φ is the angle between a plane xOz and a mobile plane of the axis cross-section; k is any positive or negative constant number; r0 is a constant value.
1. The same torse but limited by the conical spirals u1 and u2 ðu1 ¼ 0:2 m u 0:5 m ¼ u2 Þ is presented in Fig. 2. 5π x u=0 ϕ=0 Fig. 1 Fig. 3 1 Ruled Surfaces 9 conical helicoid. The coordinate line u = 0 is the edge of regression. G. Kardashevskaya derived parametrical equations of the plane cuspidal edge (u = 0) after developing of a conical helicoid on a plane (Fig. 3): ! lðas þ 1Þ 1 x ¼ cosðl ln as þ 1 þ sinðl ln as þ 1 j jÞ j jÞ að1 þ l2 Þ l 1 À ; að1 þ l2 Þ ! lðas þ 1Þ 1 y ¼ sinðl lnjas þ 1jÞ À cosðl lnjas þ 1jÞ að1 þ l2 Þ l 1 ; þ að1 þ l2 Þ (3) Parametrical equations: x ¼ xðu; tÞ ¼ v sin k cos u À t sin u; y ¼ yðu; tÞ ¼ v sin k sin u þ t cos u; z ¼ zðu; tÞ ¼ v cos k; where v = kt + ceksin λφ = kt + ceau; t is a distance from the line of tangency of the cone and its tangent plane to a point on a straight directrix of the conical helicoid lying in this tangent plane; φ, t are hyperbolic coordinates of the developable conical helicoid formed with the help of a kinematic method.
Encyclopedia of Analytical Surfaces by S.N. Krivoshapko, V.N. Ivanov
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