Encyclopedia of Analytical Surfaces S.N. Krivoshapko • V.N. Ivanov Encyclopedia of Analytical Surfaces 123 S.N. Krivoshapko Department of Strength of Materials and Structures Peoples’ Friendship University of Russia Moscow Russia V.N. Ivanov Department of Strength of Materials and Structures Peoples’ Friendship University of Russia Moscow Russia ISBN 978-3-319-11772-0 ISBN 978-3-319-11773-7 (eBook) DOI 10.1007/978-3-319-11773-7 Library of Congress Control Number: 2014953230 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Preface This book is an encyclopedic edition on analytic and differential geometry of regular analytical surfaces, which has found application in some parts of mathematics or in different branches of techniques and building. The encyclopedia was formed along the same principle as the sci- entific edition Encyclopedia of Analytical Surfaces, Krivoshapko S.N., Ivanov V.N., Pub- lishing House LIBROCOM, 2010, 560 pp. However, the present book is a supplemented and recasted edition. The main demand for this book lies in its most complete account of materials on the geometry of each surface in one or two pages. A bibliography from several titles is presented at the end of each paper. These references may help to find information for an extended study of problems connected with the geometry of a presented surface, with strength analysis of a shell in the form of this surface and their application. At the beginning or at the end of most of the book parts, a one- or two-page list of the literature on geometry, application, and strength analysis of shells with middle surfaces in the form of corresponded surfaces is given. Only the surfaces that can be pictured by means of descriptive geometry and computer graphics have been included in the encyclopedia. The material of the book is grouped into 38 classes of surfaces. The indications of forms of generatrix and directrix lines and the laws of their location concerning the base planes and lines are taken as the principle of classification. The order of equations of surfaces, total curvature, and kinematics of generation of surfaces are also taken into consideration. Before the description of surfaces belonging to the same class, a one- or two-page general charac- teristic of surfaces of this class is given. The encyclopedia contains both classical surfaces known to geometricians for several centuries and surfaces known only to a narrow range of specialists. The surfaces discovered and investigated by the authors are also included in the book. Thin-walled smooth structures are the most economical structures. From the geometrical point of view, shells are described by the form of their middle surface. However, traditionally a limited circle of these structures, such as spherical, cylindrical, conical, shallow translation shells, and some shells of revolution, constitute a small percentage in comparison to those consisting a variety of geometrical forms presented by geometricians but unknown to archi- tects and civil and mechanical engineers. The main aim of the encyclopedia is to help in exposure and decision of scientific and technical problems connected with the theory of forming of thin-walled structures on the basis of geometrical investigation of the middle surfaces of shells. The generation of nontraditional effective constructive forms of large-span space for achieving the maximum level of manufacturing resourcefulness shall favor the fulfillment of complex fundamental and applied problems raised for science in architectural and -building spheres. The availability of a wide choice of different forms and surfaces gives an opportunity to solve some problems in machine-building sphere too. The authors consider that they kept off the reiteration of some mistakes passing from one edition to another and eliminated wrong variant readings in the definition of some surfaces. The chapter with the most formulas presented in the book was tested by the authors. v vi Preface The authors tried not to include in the book questionable formulas or formulas giving rise to doubts. The materials in the encyclopedia will be interesting and useful to mathematicians, engi- neers, architects, postgraduate students, lecturers, and specialists dealing with geometry of surfaces, and for specialists working in other fields of knowledge but using geometrical images in their work. The encyclopedia also contains a dictionary of geometrical terms in Russian, English, French, and German languages. There is an Index at the end of the edition. Pages 152–155, 156–158 were written by Ph.D. Ya.S. Pul’pinskiy; pp. 145–148, 331–337 were written by Ph.D. V.A. Nikityuk, pp. 606–612 were written by Ph.D. G.S. Rachkovskaya; D.Sc. Professor V.N. Ivanov prepared pp. 19, 88, 97–98, 190–195, 197–202, 201–203. 210– 211, 240–252, 266–278, 281–290, 307–314, 339–340, 344–357, 356–357, 364–372, 376–378, 222–223, 385–386, 394–395, 400–407, 412–413, 425–426, 434–435, 441–442, 443–444, 463–469, 492–496, 501–502, 515–518, 522–526, 532–533, 541–542, 547–550, 558–559, 563, 564–565, 569–570, 570–581, 634–635, 642–646, 662. All the rest were written by D.Sc. Professor S.N. Krivoshapko. The authors express thanks to assistant I. Kushnarenko for help in preparation of this edition. Contents 1 Ruled Surfaces ............................................. 1 1.1 Ruled Surfaces of Zero Total Curvature. 2 1.1.1 Torse Surfaces (Torses) . 3 1.1.2 Cylindrical Surfaces . 37 1.1.3 Conical Surfaces . 59 1.2 Ruled Surfaces of Negative Gaussian Curvature . 72 1.2.1 Catalan Surfaces . 77 1.2.2 Twice Oblique Cylindroids. 97 2 Surfaces of Revolution ....................................... 99 2.1 Middle Surfaces of Bottoms of Shells of Revolution Made by Winding of One Family of Threads Along the Lines of Limit Deviation . 145 2.2 Middle Surfaces of Bottoms of Shells of Revolution Made by Plane Winding of Threads . 146 2.3 Middle Surface of Bottoms of Shell of Revolution Made by Winding of Threads Along Geodesic Lines . 147 2.4 Middle Surfaces of Shells of Revolution with Given Properties. 148 2.5 Surfaces of Revolution with Extreme Properties . 152 2.6 The Surfaces of Delaunay . 155 2.6.1 Nodoid and Unduloid Surfaces of Revolution . 156 3 Translation Surfaces ......................................... 159 3.1 Surfaces of Right Translation . 163 3.2 Translation Surfaces with Congruent Generatrix and Directrix Curves . 174 3.3 Surfaces of Oblique Translation . 175 3.4 Velaroidal Surfaces . 179 4 Carved Surfaces ............................................ 185 4.1 Monge Surfaces with a Circular Cylindrical Directrix Surface. 189 4.2 Monge Surfaces with a Conic Directrix Surface . 196 4.3 Carved Surfaces of General Type . 199 5 Surfaces of Congruent Sections ................................. 207 6 Continuous Topographic and Topographic Surfaces.................. 213 6.1 Aerodynamic Surfaces Given by Algebraic Plane Curves . 219 vii viii Contents 7 Helical Surfaces ............................................ 225 7.1 Ordinary Helical Surfaces . 227 7.1.1 Ruled Helical Surfaces. 229 7.1.2 Circular Helical Surfaces . 232 7.1.3 Ordinary Helical Surfaces with Arbitrary Plane Generatrix Curves . 239 7.2 Helical Surfaces of Variable Pitch . 249 8 Spiral Surfaces ............................................. 259 9 Spiral-Shaped Surfaces ....................................... 275 9.1 Spiral-Shaped Cyclic Surfaces with Circles of Variable Radius in the Planes of Pencil. 288 10 Helix-Shaped Surface ........................................ 293 10.1 Helix-Shaped Preliminarily Twisted Surfaces with Plane Generatrix Curve . 307 10.2 Helix-Shaped Twisted Surfaces with Plane Generating Curves in the Planes of Pencil . 311 11 Blutel Surfaces ............................................. 315 12 Veronese Surfaces........................................... 317 13 Tzitzéica Surfaces ........................................... 319 14 Peterson Surfaces ........................................... 325 15 Surfaces of Bézier........................................... 327 16 Quasi-ellipsoidal Surfaces ..................................... 331 16.1 Quasi-ellipsoidal Surfaces with Six Values of Semi-axes . 333 16.2 Quasi-ellipsoidal Surfaces with Cylindrical Insertions . 336 17 Cyclic Surfaces ............................................