Aleksandr Yurievich Brailov Engineering Graphics Theoretical Foundations of Engineering Geometry for Design 123 Aleksandr YurievichBrailov Department ofDescriptive Geometry andEngineeringGraphics Odessa Academy ofCivil Engineering andArchitecture Odessa Ukraine ISBN978-3-319-29717-0 ISBN978-3-319-29719-4 (eBook) DOI 10.1007/978-3-319-29719-4 LibraryofCongressControlNumber:2016930673 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. 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 authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Foreword One of the fundamental courses in professional engineering education is EngineeringGraphics,whichestablishesakindofengineeringlanguagetoproperly translate the design ideas into real-world parameters. The theoretical foundation of engineering graphics is engineering geometry. ThemajordifferenceandadvantageofthetextbookbyProfessorBrailovisthat eachtheoreticalnotionofengineeringgeometryisconsideredasacomplexsolution to direct and inverse problems of descriptive geometry. Each solution of basic engineering problems isaccompanied by construction of unique three-dimensional and two-dimensional models of geometrical images. Theuniversalstructureofformalalgorithmsforthesolutionofpositional,metric and axonometric problems, and also solutions of a problem of construction of development of a curvilinear surface, are developed in detail. The book introduces and explains the added laws of projective connections to facilitate the building of geometrical images in any of eight octants. Therefore, the textbook will be useful to undergraduate and graduate students and well as professors of technical universities and academies, and also for many practicing engineers. Prof. V.E. Mihajlenko President of the Ukrainian Association of Applied Geometry Honored Scientist of Ukraine Academician AS of Higher Education of Ukraine Academician AS of Building of Ukraine Dr.Sci.Tech vii Preface The necessity of writing this new textbook stems from the following facts: 1. The general level of mathematical knowledge of high-school graduates is insufficient for them to comprehend the basic concepts, and thus to study descriptive geometry independently. 2. High-schoolgraduatesdonotacquirethenecessarybackgroundingraphics.The level of many first-year students inimaginative perception,spatialimagination, andskillsforthesolutionofproblemswiththenecessarylevelofabstractionis not generally sufficient for studying modern engineering graphics. 3. Because the lecture hours assigned for Engineering Graphics are rather limited inmanyeducationalprofessionalprograms(EPP),thebasicweightoftrainingis shifted to independent work of the student (IWS). 4. The credit-modular system of training compels the teacher to spend an over- whelmingpartoflecturetimenotontheformationofknowledgeandskillsbut rather on obligatory ratings of the quality assurance of the material “not acquired” by students. 5. In the existing textbooks on Engineering Graphics, from our point of view, achievements of modern computer science and the technologies facilitating studyingofthesubjectunderconditionsnamedaboveareinsufficientlyutilized. ThereducedlecturehoursavailableforEngineeringGraphicseducationandthe developmentofcomputergraphicstechnologies,whichseeminglycansubstitutefor such education, might lead one may to ask logically “Why do we need to teach descriptive geometry at all?” This question parallels other frequently-asked similar questions:“Whydoweneedtostudyarithmeticinschoolsifwehavecalculators?” and“WhydoweneedtospendsomuchtimetolearncalculusatUniversitiesifwe have modern software programs such as MATLAB and Mathematica?” In the author’s opinion, descriptive geometry is needed, first of all, as it con- stitutes the basis for the development of the engineering geometry. The existence of practical demand for studies in descriptive geometry as the basis of engineering geometry is explained as follows. ix x Preface Although the pencil and a paper were replaced a long time ago with the computer equipped with advanced solid-modeling software packages, one should clearlyrealizethatthecomputercan’treplaceanengineer.Moreover,designersand engineers with different experience using the same graphic software can produce considerably different graphic products with the same designation. The more complicated the graphic software package is, the greater are the experience and knowledgerequiredtorunitefficiently.Inotherwords,thecomputersavesdrawing time, whereas engineers build an image of a part and/or structure in their brains. Theknowledgeofengineeringgraphicshelpshimorhertoconveytheconstructed mental image in a clear and unambiguous fashion that is readily understandable to other design/manufacturing/application professionals all over the world. For effective design, it is necessary for the engineer to know the laws of pro- jective connections and the properties of geometrical images, to possess spatial imagination and imaginative perception, and to have the skills of biunique trans- formation of two-dimensional and three-dimensional models of geometrical parts that enable the solution of direct and inverse problems of descriptive geometry. Practical expert skills indesignaresubstantiallyformedduetotheemployment of the basics of the descriptive geometry. Without these skills and abilities, the efficientdesignofdifficultparts,assemblagesandmachinesisimpossibleevenwith the use of most advanced computers because the final decisions must be selected and then accepted by the designer. Therefore,theauthorconsidersdescriptivegeometryasthebasisofengineering geometry.Thedevelopmentofengineeringgeometryisinfluencedbythetheoryof algorithms, the theory of signs (semiotics), the theory of information technologies, the theory of computer designing and other closely related branches of science. In the author’s opinion, the standard fundamental discipline “Engineering Graphics” should include three logically connected parts: 1. Engineering geometry. 2. Engineering drawing. 3. Engineering computer graphics. Descriptive geometry constitutes the theoretical basis offirst part. This new textbook provides the following advantages compared to the other existing titles: 1. Itenhancesdeeperandadequateunderstandingofthegeometricalessenceofthe studiedphenomenon.Itarguesthatthedefinitionofthetheoreticalfoundationof an engineering drawing should be carried out as a combined solution to direct and inverse problems of descriptive geometry. 2. It reveals that, to facilitate the construction of two-dimensional and three-dimensional models of geometrical parts in any of eight octants, the laws of projective connections should be formulated on the basis of a necessary and sufficient set of essential notation. Preface xi 3. It provides essential help in the development of spatial imagination and imaginative perception. It argues that the analysis of geometrical models of some images is needed for executing it is system, from uniform positions, statinginfulltheirpropertiesandfeaturesonthree-projectivecomplexdrawing. Forexample,geometricalmodelsofthemainlinesofaplaneontwoprojective complex drawings do not adequately facilitate the presentation of the solution of the inverse problem of descriptive geometry. Conditionsfortheparallelismandintersectionofstraightlinesshouldbestudied separately for geometrical images of the general and local positions. 4. Itsmethodologyofpresentationhelpsreaderstoacquiretheabilitytoadequately readdrawings.Thatisbecausecarefullydevelopsasystemofrulesofdefinition ofvisibilityofinitialgeometricalimagesandconstructiveelementsofaproduct for direct and inverse problems of descriptive geometry. 5. It presents the universal structure of algorithms for the solution to positional, metric and axonometric problems, and also solutions to a problem of con- struction of development of a curvilinear surface. These help to simplify mas- tering a course and the formation of skills for independent work by students. In the present textbook, the features just specified are realized by a statement of the laws of projective connections contributed by the author, the structured formal algorithmsforthesolutionofpositional,metricandaxonometric problems, and also by the solution of a general problem of construction of development of a curvilinear surface. Each theoretical development is considered at the solution of a basic practical problem. The solution of each basic problem is accompanied by a construction and biunique transformation of two-dimensional and three-dimensional models of geometrical parts. A system of rules of definition for the visibility of images on the basis of the method of competing points is offered. Eachstepofthealgorithmisreflectedinasign(semiotics)modelforthesolution of an engineering problem. Thestructureoftheofferedalgorithmsforthesolutionofproblemspresentedin the eighth, ninth, tenth and eleventh chapters of the textbook is sufficiently uni- versal to help students to solve various problems with no additional or with only minimum instructions. The major objective of the present textbook is to represent the course of Engineering Geometry on the basis of recent developments in the field. The textbook consolidates the author’s twenty-five-year experience of teaching at the Department “Descriptive Geometry and Engineering Graphics,” the Odessa NationalPolytechnicUniversityandattheDepartment“DescriptiveGeometryand Drawings,” the Odessa Academy of Civil Engineering and Architecture. xii Preface The textbook includes the foreword, preface, references, appendix and 11 chapters: 1. A projecting method. The methodology and basic operations of projection. 2. Types of projection. The center of projection. 3. Formation of the complex drawing. Octants. The method of Gaspard Monge. 4. Geometrical models and analytical model of a point. 5. Geometrical models and analytical models of a straight line. 6. Geometrical models and analytical models of a plane. 7. Geometrical models and analytical models of a surface. 8. Positional problems. 9. Metric problems. 10. Development of surfaces. 11. Axonometric projections. All sections are grouped in seven logical information blocks. The first, second, third, and fourth chapters are unified as the first information block. The fifth and sixth chapters are unified as the second information block. The seventh, eighth, ninth,tenth,andeleventhchaptersareaccordingtothethird,fourth,fifth,sixthand seventh information blocks. Each information block concludes with review questions. In the textbook, on the basis of the stated theoretical positions of engineering geometry,thesolutionsoftwenty-threebasicproblemsareofferedandanalyzedin great detail. Detailed explanations of application of the basic laws and use of properties of models of geometrical images in the solution of basic engineering problems better enable successful mastery of the theoretical part of Engineering Graphics courses. In the textbook, the long-term operational experience of the author, both at the theory level (lecture courses), and at the methodical level offormation of skills of performance of design documents and possession of computer technologies, is generalized. A tailored synthesis of theoretical and methodical knowledge is pre- sented to facilitate the preparation of students capable of answering the call of modern techniques and technologies. Theauthorexpressessinceregratitudeforencouragement,counselandvaluable remarks to Professors: Sukhorukov J.N., Podkorytov A.N., Mihajlenko V.E, Vanin V.V., Kovalyov S.N., Sazonov K.A., Astakhov V.P, Radzevich S.P., PereleshinaV.P,AjrikjanA.L.,DzhugurjanT.G.,DashchenkoA.F.,SemenjukV.F., Dorofeyev V.S, Kivalov S.V., Grishin A.V., Barabash I.V., Karpjuk V.M, KlimenkoE.V.,KitN.V.,MaksimovM.V,MaslovO.V,KosenkoS.I.,PetroN.N., Panchenko V.I. The author also extends his gratitude to his colleagues in the department and at theacademyanduniversityforgenerouslysharingtheirexperienceandknowledge, delicacy and tactfulness, keenness and for their attention to the solution of the illustrative problems. Theauthorwillbegratefultothebenevolentreaderforsuggestionsandremarks which will result in raising the quality of this textbook. Contents 1 Descriptive Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Subject Matter of Descriptive Geometry . . . . . . . . . . . . 1 1.2 Aims and Problems of Descriptive Geometry. . . . . . . . . . . . 2 1.3 Types of Geometric Figures and Objects (Images) . . . . . . . . 2 1.4 A Determinant of a Geometric Image (Object). . . . . . . . . . . 3 1.5 A Projecting Method. The Components and the Operations of Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Types of Projection. The Center of Projection. . . . . . . . . . . . . . . 7 2.1 Central (conical) Projection . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Parallel (cylindrical) Projection. . . . . . . . . . . . . . . . . . . . . . 7 2.3 Properties of the Central (conic) Projection . . . . . . . . . . . . . 8 2.4 Properties of Parallel (cylindrical) Oblique-Angled Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Properties of Parallel Rectangular (orthogonal) Projection . . . 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Formation of the Complex Drawing. Octants. The Method of Gaspard Monge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 The Concept of Octant . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Essence of the Method of Gaspard Monge . . . . . . . . . . 16 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4 Geometrical Models and an Analytical Model of a Point . . . . . . . 19 4.1 The Laws of Projective Connections. . . . . . . . . . . . . . . . . . 22 4.2 Classification of Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Review Questions on the First Block (Chaps. 1–4). . . . . . . . 23 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 xiii xiv Contents 5 Geometric and Analytical Models of a Straight Line . . . . . . . . . . 25 5.1 Classification of Straight Lines. . . . . . . . . . . . . . . . . . . . . . 26 5.2 Ways of Representation for a Line Segment and Determinants of a Straight Line . . . . . . . . . . . . . . . . . . 26 5.3 Geometric Model of a Straight Line of General Position . . . . 27 5.4 The Peculiarities of a Complex Drawing of a Straight Line of General Position . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.5 Geometric Models of a Level Line. . . . . . . . . . . . . . . . . . . 28 5.5.1 A Geometric Model of a Horizontal Level Line and Properties of This Model . . . . . . . . . . . . 28 5.5.2 A Geometric Model of a Frontal Level Line and Properties of This Model . . . . . . . . . . . . 29 5.5.3 A Geometric Model of a Profile Level Line and Properties of This Model . . . . . . . . . . . . 30 5.5.4 Peculiarities of the Complex Drawing of a Level Line. . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.6 Geometric Models of a Projecting Straight Line. . . . . . . . . . 32 5.6.1 A Geometric Model of a Horizontally Projecting Straight Line and Properties of the Model . . . . . . . 32 5.6.2 A Geometric Model of a Frontally Projecting Straight Line and Properties of This Model . . . . . . 33 5.6.3 A Geometric Model of a Profiled Projecting Straight Line and Properties of the Model . . . . . . . 34 5.6.4 Peculiarities of the Complex Drawing of a Projecting Straight Line. . . . . . . . . . . . . . . . . 35 5.7 Analytical Models of a Straight Line. . . . . . . . . . . . . . . . . . 36 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6 Geometric Models and Analytical Models of a Plane . . . . . . . . . . 39 6.1 Classification of Planes. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6.2 Ways of Representation of a Plane in the Complex Drawing. Plane Determinants. . . . . . . . . . . . . . . . . . . . . . . 40 6.3 A Geometric Model of a Plane of General Position. . . . . . . . 41 6.4 Peculiarities of the Complex Drawing of a Plane of General Position. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.5 Geometric Models of a Plane of Level . . . . . . . . . . . . . . . . 42 6.5.1 A Geometric Model of a Horizontal Plane of Level and Properties of This Model . . . . . . . . . 43 6.5.2 A Geometric Model of a Frontal Plane of Level and Properties of This Model . . . . . . . . . 44 6.5.3 A Geometric Model of a Profile Plane of Level and Properties of This Model . . . . . . . . . 45 6.5.4 Peculiarities of the Complex Drawing of a Plane of Level. . . . . . . . . . . . . . . . . . . . . . . 47 Contents xv 6.6 Geometric Models of a Projecting Plane . . . . . . . . . . . . . . . 47 6.6.1 A Geometric Model of a Horizontally Projecting Plane and Properties of This Model. . . . . . . . . . . . 48 6.6.2 A Geometric Model of a Frontally Projecting Plane and Properties of This Model. . . . . . . . . . . . 49 6.6.3 A Geometric Model of a Profiled Projecting Plane and Properties of This Model. . . . . . . . . . . . 50 6.6.4 Peculiarities of the Complex Drawing of a Projecting Plane. . . . . . . . . . . . . . . . . . . . . . 52 6.7 Analytical Models of a Plane. . . . . . . . . . . . . . . . . . . . . . . 52 6.8 The Main Lines of a Plane . . . . . . . . . . . . . . . . . . . . . . . . 53 6.9 Review Questions for Chap. 5 and this Chapter . . . . . . . . . . 54 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 7 Geometric and Analytical Models of a Surface . . . . . . . . . . . . . . 57 7.1 Ways of Formation, Description and Mapping, and Classification of Surfaces. . . . . . . . . . . . . . . . . . . . . . . 57 7.2 A Surface Contour and a Surface Sketch. The Way of Representing a Surface in a Complex Drawing. . . . . . . . . 60 7.3 Ruled Developable Surfaces with One Directional Line. . . . . 60 7.4 Ruled Undevelopable Surfaces with Two Directional Lines and a Plane of Parallelism. . . . . . . . . . . . . . . . . . . . . 64 7.5 Ruled Undevelopable Surfaces with Three Directional Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 7.6 Screw Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 7.7 Surfaces of Revolution and Their Analytical Models. . . . . . . 70 7.8 An Indication of a Point Belonging to a Surface. . . . . . . . . . 73 7.9 Review Questions the Third Block (This Chapter) . . . . . . . . 73 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 8 Positional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 8.1 The Concept and Classification of Positional Problems . . . . . 79 8.2 The Concept of Competing Points. The Rule to Define the Visibility of Constructive Elements of a Product. . . . . . . 79 8.3 Mutual Location, Intersection and Belonging of the Same Linear Geometric Images to Each Other . . . . . . . . . . . . . . . 80 8.3.1 Mutual Location, Intersection and Belonging of Points to Each Other. The Rule to Define the Visibility of Competing Points . . . . . . . . . . . . 80 8.3.2 Mutual Location, Intersection and Belonging of Straight Lines to Each Other . . . . . . . . . . . . . . 82 8.3.3 Mutual Location, Intersection and Belonging of Planes to Each Other. . . . . . . . . . . . . . . . . . . . 87