Engineering Graphics. Theoretical Foundations of Engineering Geometry for Design

343 Pages·2016·11.64 MB·english

by  Aleksandr Yurievich Brailov

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
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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