Descriptive geometry

76 Pages·95.24 MB·Russian

by  Hmelnitskaya, L. V.

Ministry of Education of the Republic of Belarus

Belarusian National Technical University

DESCRIPTIVE GEOMETRY

Teaching guide for students of the following specialities:
1-53 01 01 “Automation of technological processes and production
(in areas)”, 1-36 01 01 “Machine Building Technology”

Approved by the Academic Association in the
field of ingineering equipment and technology

Minsk 2021

1
УДК 514.18(075.8)
ББК 22.151.3я7
Х65

R e v i e w e r s:
Department of Engineering Graphics
Belarusian state Technological University;
P. V. Avramenko

Hmelnitskaya, L. V.
Х65 Descriptive geometry : teaching guide for students of the following specia-
lities 1-53 01 01 “Automation of technological processes and production (in areas)”
1-36 01 01 “Machine Building Technology” / L. V. Hmelnitskaya, T. V. Matsiu-
shynets, A. U. Leshkevich. – Минск, БНТУ, 2021. – 75 с.
ISBN 978-985-583-611-8.

This textbook is developed for first-year students getting technical specialisations in
order to improve their skills of independence work and to provide their study process by
additional material. It also can be intended for distance learning case as it contains theoret-
ical material on the descriptive geometry as the first and the fundamental chapter for the
course “Engineering graphics”. In addition, the algorithms given here enable students to
solve similar tasks by the case of their independence work.
The appendix provides general information about designing of drawings in accord-
ance with USDD.

УДК 514.18(075.8)
ББК 22.151.3я7

ISBN 978-985-583-611-8 © Hmelnitskaya L. V., Matsiushynets T. V.
Leshkevich A. U., 2021
© Belarusian National
Technical University, 2021
2
CONTENTS

INTRODUCTION ......................................................................................................... 5
LEGEND ....................................................................................................................... 6
1. INTRODUCTION TO THE SUBJECT ................................................................. 7
1.1. Historical reference ........................................................................................... 7
1.2. Projection Method ............................................................................................. 7
1.3. Orthogonal projection ....................................................................................... 9
1.4. Monge′s Method ............................................................................................... 9
2. PROJECTION OF A STRAIGHT LINE. CLASSIFICATION OF
STRAIGHT LINES ACCORDING TO THEIR LOCATION IN SPACE ................ 10
2.1. Projection of a straight line by the First Angle Projection method ................ 10
2.2. Classification of straight lines according to their location in space ............... 10
2.3. Determining the true length of a line by the Right Triangle Method ............. 14
2.4. Positions of straight lines relative to each other. ............................................ 14
2.5. Projection of a straight angle – Theorem. ....................................................... 16
3. PROJECTION OF A PLANE. CLASSIFICATION OF PLANES
ACCORDING TO THEIR LOCATION IN SPACE .................................................. 17
3.1. Plane. Setting methods of plane in a drawing ................................................ 17
3.2. Point belonging to a plane. Line belonging to a plane ................................... 18
3.3. Classification of planes according to their location in space ......................... 19
3.4. Mutual positions of two planes ....................................................................... 22
3.5. Relative positions of a line with a plane ......................................................... 24
3.6. Line and plane perpendicularity ..................................................................... 26
4. METHODS OF DRAWING TRANSFORMATION ........................................... 29
4.1. Method for Replacement Projection planes .................................................... 29
4.2. Method for rotation around projecting axis. ................................................... 32
4.3. Method for rotation around main line (frontal or horizontal) ........................ 33
5. SURFACES. .......................................................................................................... 34
5.1. Faceted surfaces. Prism and pyramid ............................................................. 34
5.2. Surfaces of revolution. Cylinder and cone ..................................................... 36
5.3. Points and lines on the surface ........................................................................ 40
5.4. Generatrix Method. ......................................................................................... 41
5.5. Method of auxiliary cutting planes ................................................................. 42
5.6. Surfaces of revolution. Sphere and torus ........................................................ 43
6. INTERSECTION OF SURFACES ....................................................................... 49
6.1. Possible cases of surface intersection ............................................................. 49
3
6.2. Method of auxiliary cutting level planes ........................................................ 51
6.3. Method of auxiliary concentric spheres .......................................................... 52
6.4. Method of auxiliary eccentric spheres ............................................................ 54
6.5. Monge’s Theorem. .......................................................................................... 55
7. UNFOLDING (DEVELOPMENT) OF SURFACES ........................................... 56
7.1. Unfolding of a polyhedron. ............................................................................. 56
7.1.1. Method for normal section. ...................................................................... 56
7.1.2. Method for unrolling. ............................................................................... 57
7.1.3. Method of triangulation ............................................................................ 58
7.2. Approximate unfolding of cylindrical and conical surfaces ........................... 59
8. AXONOMETRIC PROJECTIONS ...................................................................... 60
8.1. Method for axonometric projection ................................................................ 60
8.2. Orthogonal isometric projection .................................................................... 61
8.3. Orthogonal dimetric projection ...................................................................... 63
8.4. Oblique frontal dimetric projection ............................................................... 65
APPENDIXES ............................................................................................................ 68
UNIFIED SYSTEM FOR DESIGN DOCUMENTATION (USDD) .................... 68
GOST 2.301-68 Formats ......................................................................................... 68
GOST 2.104-2006 Basic inscriptions (Title Block – UK) ..................................... 70
GOST 2.302-68 Scales ............................................................................................ 71
GOST 2.303-68 Lines ............................................................................................. 71
GOST 2.304-81 Letters for drawings (Fonts) ......................................................... 72
REFERENCE .............................................................................................................. 74

4
INTRODUCTION

“Engineering drawing” is the purpose of studying subject matter development of
spatial representation and imagination, constructive and geometrical, abstract and
logical thinking, abilities to the analysis and synthesis of spatial forms and relations
on the basis of the graphic models of space which are almost realized in the form of
drawings of concrete spatial objects and dependences.
The main objective of teaching subject matter "Engineering Drawing" is to pro-
vide the students with knowledge and the skills necessary for performance and read-
ing drawings of different function and taking decisions in drawings of geometrical
and technical tasks.
Knowledge and abilities received by the students when studying this discipline
are necessary for development of the subsequent special disciplines.
Upon completion of the course, students will be able to:
– explain and apply the basics of blueprint reading, preparation and detailing of
technical drawings, drawing scale, title block, revision block, additional notes, etc.;
– utilize free-hand sketching and basic drafting instruments in geometric con-
struction;
– employ shape description and drawing preparation techniques of multi-view
orthographic projection and 3D visualization using isometric, oblique, and perspec-
tive views created via instrumental drafting techniques;
– apply shape description and drawing preparation techniques through creation
of parametric 3D solid models using AutoCAD software in order to prepare detailed
drawings which contain all necessary dimensions and annotations, including geomet-
ric dimensioning;
– use additional shape description tools of sectioning, auxiliary, detail, break, and
broken-out views to complete shape description in order to create the assembly of many
components of the design object and generate exploded assembly and bill of materials.
The Republic of Belarus has joined to the Bologna process after several attempts
since 2015. As a result of the Bologna Agreement there were developed Strate-
gic Action Plan on Implementation of the Major Objectives of the Educational Sys-
tem and Work Plan for Implementing European High Education Area (EHEA) Tools
in Belarus. Since that our system of education has been changing. We have already
started reorganization with cutting of Bachelor training program from five years to
four. There was developed a multi-level system of higher education with a certain
number of credit units (lending) for each level [11]. All these events have made it
possible to increase number of international agreements among Belarusian universi-
ties and some foreign universities.
It has become necessary to start revision of curricula in general and in case of
engineering drawing especially as one of the base subject. This revision includes
translation of lectures and creation individual tasks in English.
The proposed textbook is prepared for students with technic major as a first step
for studying “Engineering Drawing” and discover its basic branch – “Descriptive Ge-
ometry”.
5
LEGEND

1. Planes of projection are marked as:
– frontal – V;
– horizontal – Н;
– profile – W.
2. Points are marked with Latin capital letters: A, B, C, D... or numbers: 1, 2, 3...
3. Straight and curved lines are marked with lowercase Latin letters: a, b, c, d...
4. Planes and surfaces are marked with Latin letters: α, β...
5. Angles are marked with the following Latin letters: φ, γ, …
6. New projection planes (different from indicated above) are marked with the
additional indexes 1, 2, 3, 4... Examples:
– the first new frontal plane of projection is marked with V ;
1
– the first new horizontal plane of projection – with H ;
1
– the first new profile plane of projection – with W .
1
7. Projections of points and lines are marked with the same letters like their orig-
inals in space with the relevant to the plane of projection number of dashes and if it is
needed with additional index. Examples:
– in the frontal projection V with two dashes – projection of the point A – A’’,
projection of the line a – a’’;
– in the horizontal projection H with one dash – projection of the point A – A’,
projection of the line a – a’;
– in the profile projection W with three dashes – projection of the point A –
A’’’, projection of the line a – a’’’.
8. Projections of the planes and the angles are marked with a relevant letter low-
ercase index of the plane of projection. Examples:
– in the frontal projection V with index – projection of the plane α – α , projec-
v v
tion of the angle φ – φ ;
v
– in the horizontal projection H with index – projection of the plane α – α ,
H H
projection of the angle φ – φ ;
H
– in the profile projection W with index – projection of the plane α – α , pro-
W W
jection of the angle φ – φ .
W

6
1. INTRODUCTION TO THE SUBJECT

1.1. Historical reference

Artistic or engineering drawings are a graphic representation of an object, or a
part of it, and are a result of creative thoughts by an engineer or a technician. In other
words, it is a language – the graphical language that communicates ideas and infor-
mation from one mind to another [3].
Theoretical basis of the engineering drawing is descriptive geometry, which
once allowed creating one of the most genial inventions of the human mind – the
drawing. Briefly, Descriptive Geometry is a graphical solution of the point, line and
plane problems in space and deals with solving problems in three-dimensional geom-
etry by transformation them to two dimensional views.
Descriptive geometry has been formed such as a science by the end of the 18th
century. Basically, it happened thanks to Gaspard Monge (1746–1818) [4]. He was a
public figure and a genius French engineer. His work “Descriptive Geometry” was
published in 1798 and was first course of lectures for the students of the Ecole Poly-
technique in Paris. A particular branch of geometry – descriptive geometry, has played
a major part in engineering education since that. Descriptive Geometry as a separate
discipline, was included for the first time in a curriculum in Russia in 1810 [14].

1.2. Projection Method

The drawings which are used in Descriptive geometry, are called projections.
The projection is a drawing or a representation of an object in imaginary plane or
planes of projection. These projection planes serve the same purpose in Descriptive
Geometry as it is served by a movie screen.
A projection involves next three components:
1. An actual object which the drawing or projection represents (a point is the
simplest object of projection).
2. Projecting rays are directed from the viewer to the plane of projection through
the object.
3. One or several planes of projection.
The point of projection is the point of intersection of a projecting ray with the
plane of projection.
There are two types of projections with several subclassifications according to
the direction of projecting rays:
1. Central (conic) projection.
2. Parallel projection.
According to the central projection the projection a point is built by conducting
through the given point and point S (the center of projections) rays SA, SB, SC, to the
intersection with the plane of projection (fig. 1.1) [1].

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Fig. 1..1. Central pprojection oof a triangle ABC in a pplane of proojection

Deppiction of tthe objectts with thee help of tthe centrall projectioon has greeat visibil--
ity, but itt significanntly distorrts the shaape and dimmensions of the oriiginal. Theerefore, inn
practice tthe method of paralllel projecction (in pparticular, the orthoggonal projjection) iss
used morre often.
Paraallel projecction is a type of prrojection wwhere the lines fromm a vieweer (projec--
ting rays)) are mutuually parallel (fig. 1..2).

Fig. 1.2. Parallel pprojection oof a trianglee ABC in a pplane of proojection

All pparallel prrojections are subdivvided intoo the followwing threee categories:
1. OOrthogonall projectioons are draawn as thee multi vieew drawinggs, which show flatt
representtations of pprincipal vviews of thhe subjectt.
8
2. Oblique projections actually show the full size of the one view only.
3. Axonometric projections are three-dimensional drawings, which are subdivid-
ed into three different varieties: isometric, dimetric and trimetric.

1.3. Orthogonal projection

Orthogonal projection is a special case of the parallel projection, when the direc-
tion of a projecting ray S is perpendicular to the plane of projections α, it simplifies
the construction of the drawing (fig. 1.2). All rays of projection pass along parallel of
this direction S.
All other cases of parallel projection are called oblique parallel projections. In
other words, when the angle between projecting rays and the plane of projection is
90˚ this is orthogonal projection. And if this angle is not 90˚ – this is oblique projec-
tion. But in both cases all projecting rays are mutually parallel.
Some properties of the parallel projection:
1. Projection of a point is a point.
2. Projection of a straight line is a straight line (in common case).
3. If a point belongs to a straight line, its projection belongs to the projection
of this straight line.
4. If a point divides a straight line into some ratio, its projection divides the
projection of this straight line with the same ratio.
5. Parallel straight lines in space are always projected into parallel projections
in the plan of projection (in common case).

1.4. Monge′s Method

Two or more interconnected orthogonal projections of the original are mostly
used in an engineering practice drawing. Such kind of the drawing is called a com-
plex drawing in orthogonal projections or a complex drawing.
Orthogonal projection in two orthogonal planes is the basis of the method that
was developed by Gaspard Monge [4]. He was the first one who proposed to use the
Orthogonal projection in mutually perpendicular planes of projection.
The position of any point is defined by 3 coordinates: OX, OY and OZ in space
according to Descartes’ system. The axis lines of this system organize 3 mutually per-
pendicular planes of projection – horizontal (H), frontal (V), profile (W). The object is
projected orthogonally in three mutually perpendicular planes of projection which are
then combined into the one drawing. It becomes possible by the next actions.
The axis OY is divided (it should be cut along the OY-coordinate) and the pro-
file (W) plane of projection turns around about the axis OZ to the right and the hori-
zontal (H) plane of projection turns down around the axis OX to combine with the
frontal (V) plane of projection (fig. 1.3)
Two mutual perpendicular planes divide the space on 4 parts which are called:
First, Second, Third and Fourth angle projection respectively (fig. 1.3) [14]. Three
plains of projection divide space on 8 parts – octants.
9
a b

Figg. 1.3. Planees of projecttion (V, H, WW):
a – a 3-dimmantional mmodel; b – a 2-dimantionnal drawingg

Firstt Angle PProjectionn method is used by our country, countries ffrom Thee
Commonnwealth off Independdent States (CIS) annd Europeean countrries. Another meth--
od, Thirdd Angle Prrojection mmethod, iss used by UUSA, Australia and the otherss.

2. PROJECCTION OFF A STRAAIGHT LLINE. CLLASSIFICCATION OOF
STRRAIGHT LLINES ACCORDIING TO TTHEIR LLOCATIOON IN SPACE

2.1. Projectiion of a sttraight linne by the First Anggle Projecction methhod

The straight line is the shortest ddistance bbetween twwo points. The projeections off
the ends oof any linee can be ddrawn by uusing the pprinciples developeed for the pprojectionn
of a pointt. Single pprojection of the linee does nott determinne its posittion in spaace. Theree
has to bee at least twwo projecctions for a unique ddefinition of the strraight linee in space..
A top vieew of twoo end poinnts of a linne it is a hhorizontal projectionn of the line. Frontt
view of ttwo end points of thhe line it is a frontaal projectiion of the line. Bothh of themm
are projecctions of tthe one strraight linee.

2.22. Classificcation of straight llines according to ttheir locattion in sppace

Linees in space can occcupy any llocation. TThere are two big ggroups off them ac--
cording too their loccation.
1. GGeneral poosition.
2. PParticular ((specific) position.
A sttraight linee, which iss not paralllel to anyy plane of projectionn is called a straightt
line of geeneral possition or aan obliquee line. In oother wordds, this linne is incliined to alll
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