Descriptive geometry 9789855836118

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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

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Hmelnitskaya, L. V. 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” / L. V. Hmelnitskaya, T. V. Matsiushynets, 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 theoretical 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 accordance with USDD. УДК 514.18(075.8) ББК 22.151.3я7

ISBN 978-985-583-611-8

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© Hmelnitskaya L. V., Matsiushynets T. V. Leshkevich A. U., 2021 © Belarusian National Technical University, 2021

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 Historical reference ........................................................................................... 7 1.2.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

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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 provide the students with knowledge and the skills necessary for performance and reading 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 construction; – employ shape description and drawing preparation techniques of multi-view orthographic projection and 3D visualization using isometric, oblique, and perspective 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 geometric 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 Strategic Action Plan on Implementation of the Major Objectives of the Educational System 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 universities 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 Geometry”. 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 V1; – the first new horizontal plane of projection – with H1; – the first new profile plane of projection – with W1. 7. Projections of points and lines are marked with the same letters like their originals 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 lowercase index of the plane of projection. Examples: – in the frontal projection V with index v – projection of the plane α – αv, projection of the angle φ – φv; – in the horizontal projection H with index H – projection of the plane α – αH, projection of the angle φ – φH; – in the profile projection W with index W – projection of the plane α – αW, projection of the angle φ – φW.

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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 information 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 geometry 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 Polytechnique 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 projection p oof a triangle ABC in a plane p of proojection

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

Fig. 1.2. Parallel projection p oof a trianglee ABC in a plane p of proojection

All pparallel prrojections are subdivvided into o the follow wing threee categories: 1. O Orthogonall projectio ons are draawn as thee multi vieew drawinggs, which show flatt representtations of principal p views v 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 subdivided into three different varieties: isometric, dimetric and trimetric. 1.3. Orthogonal projection Orthogonal projection is a special case of the parallel projection, when the direction 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 projection. 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 complex 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 perpendicular 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 profile (W) plane of projection turns around about the axis OZ to the right and the horizontal (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

Fig g. 1.3. Planees of projecttion (V, H, W): W a – a 3-dim mantional m model; b – a 2-dimantion nal drawingg

Firstt Angle Projection P n method is used by our country, c ountries from f Thee Commonnwealth off Independ dent States (CIS) an nd Europeean countrries. Another meth-od, Thirdd Angle Prrojection method, m iss used by USA, U Australia and the otherss. 2. PROJEC CTION OF F A STRA AIGHT LINE. L CL LASSIFIC CATION OF O STR RAIGHT LINES L ACCORDIING TO THEIR T LOCATIO L ON IN SPACE 2.1. Projectiion of a sttraight lin ne by the First Ang gle Projecction meth hod The straight line is the shortest ddistance between b tw wo points. The projeections off the ends oof any linee can be drawn d by uusing the principles p p n developeed for the projection of a pointt. Single projection p of the linee does nott determin ne its posittion in spaace. Theree has to bee at least tw wo projecctions for a unique definition d of the strraight linee in space.. A top vieew of twoo end poin nts of a linne it is a horizontal h projectionn of the line. Frontt view of ttwo end points of th he line it is a frontaal projectiion of the line. Both h of them m are projecctions of the t one strraight linee. 2.22. Classificcation of straight llines according to their t locattion in sp pace Linees in space can occcupy any llocation. There T are two big ggroups off them ac-cording too their loccation. 1. G General poosition. 2. P Particular (specific) ( position. A sttraight linee, which iss not paralllel to any y plane of projectionn is called a straightt line of geeneral possition or an a obliquee line. In other o word ds, this linne is incliined to alll 10

three prinncipal plannes of pro ojection. Itt cannot appear a in its i true lenngth in any of threee principal planes off projection n [5] (fig. 2.1).

Fig. 2.1. A line of geneeral position n

i on of a straaight line with the plane p of prrojection is i called a The point of intersectio l trace of a straight line. The straight liine, which h is parall el to one or o two plaanes of proojection, is i called a line of paarticular position. p When W the lline in spaace is orieented in suuch a way y, that it iss parallel too the given plane off projectioon, it is pro ojected as a true lenngth. Therre is next sub-classiification: The straight liine, which h is paralleel to one plane p projection, is called a liine of lev-el and aall its poiints have the sam me distancce to this plane, ii. e. they have thee same cooordinate. Inn other wo ords, the lline, which h is parallel to one pplane of projection, p , is seen inn an adjaccent view as a line parallel to t the fold ding (axis)) line. (OX X, OY orr OZ). If thhe line is parallel p to the horizoontal plan ne of projeection it iss called a horizontal h l level linee (fig. 2.2). The line, whhich is parrallel to tw wo planess of projeection, is called thee project-ing line. IIn other words, w thiss line is pperpendicu ular only to t one plaane of projjection. Iff the line iss perpendicular to th he horizonntal plane of projection it willl be called the hor-izontal prrojecting line. Therre is anotther namee for this line, in soome coun ntries it iss called thee frontal profile p linee. These sstraight lin nes are pro ojected as the point in the in-dicated pplane of projection and as thhe true len ngth in oth her two pplanes of projection p n (fig. 2.3)..

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Fig. 22.2. Lines off level: a – a froontal level line; l b – a hhorizontal leevel line; c – a profile leevel line 12

Fig. 2.33. Projectin ng lines: a – a frontal proojecting linee; b – a horiizontal projeecting line; c – a profille projecting g line 133

2.3. Determiining the true lengtth of a lin ne by the right triaangle Method Findding the trrue length of a segm ment is a basic b probllem in Deescriptive geometry. g . There aree several different methods oof its deteermining. The Righht trianglee Method,, Method ffor replacement pro ojection pplanes, Meethod for rotation rround projjecting orr main linee and etc. The right trianngle Meth hod: there is a right triangle where w onee leg is equ ual to anyy projectionn of the segment, s and a the seecond leg is equal to t the disttance betw ween end-points off the other projection n of the seegment. A hypotenu use of thee triangle is i the truee length off the segmeent. In otther wordss, the hypo otenuse off the right triangle t is equal to thhe true len ngth of thee segment A AB, if one of two sides s is prrojection of o this seg gment, andd another one is thee differencee between the ends of o the otheer projectio on which has h to be m measured (fig. ( 2.4).

Figg. 2.4. Deterrmining truue length of a line of geeneral positiion: a – a 3-dim mentional m model; b – a drawing off this modell

2.44. Position ns of straiight lines relative to each oth her Straiight lines can be parrallel (fig. 2.5), interrsecting (ffig. 2.6) orr crossing (fig. 2.7). 1. Iff two liness are parallel in spacce, they haave paralleel projectiions in all planes off projectionn. (In geneeral case) The convverse is alsso true. 2. Iff two liness intersectt, they orgganize a co ommon po oint whichh is the po oint of in-tersectionn. Both prrojections of this p oint lie on n the sam me connecttion line. The con-verse is aalso true annd these liines are caalled interrsecting lin nes (fig. 2 .6. point 1). 1 3. Iff two liness are crosssed, the pllace of theeir intersecction seem ms like thee commonn point in one planee of projeection andd like two o points in i anotherr projectio on. Thesee 14

points aree called coompeting points. Thhey belong g to the saame conneection linee but havee different location in space: one o under another or o one behiind anotheer. The po oint whichh is the neaarest to observer is visible v (figg. 2.7).

Fig. 22.5. Parallell lines: a – a 3-dimantional al model; b – a drawing g of them

Fig. 2.66. Intersectin ng lines: a – a 3-dimantional al model; b – a drawing g of them 155

Fig. 2 .7. Crossing g lines: a – a 3-dimantional al model; b – a drawing g of them

2.5. Projection of a straight angle – Theorem m A sttraight anggle is projected in a plane off projection n as the trrue size, iff one sidee of the strraight anggle is paraallel to thee plane off projectio on, and thhe other side is nott perpendiccular to thhis plane of o projectioon (fig. 2.8). In otheer words iif one sidee is a levell line and aanother onne is not a projectingg line a straight ang gle is projeected in its real sizee in that projection where w the level l line has a projjection as a true lenggth.

Fiig. 2.8. Projjection of a straight ang gle

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3. PROJEC CTION OF O A PLA ANE. CLA ASSIFICA ATION O OF PLANE ES AC CCORDIN NG TO TH HEIR LO OCATION N IN SPA ACE 3..1. Plane. Setting m methods of o plane in n a drawin ng Projection of a plane in ways: n a drawingg can be set s in the following f w – byy projectioon of threee points (points do d not liee on the ssame straaight line)) (fig. 3.1, a); – byy projectioon of the straight s linne and pro ojections of o the poinnt (both of them doo not lie onn the samee straight line) (fig. 33.1, b); – byy projectioon of two parallel p linnes (fig. 3.1, c); – byy projectioon of two intersectin i ng lines (fi fig. 3.1, d);; – byy projectioon of any shape s (triaangle, recttangle) (fig g. 3.1, e); – byy plane traaces (fig. 3.1, 3 f).

Fig. F 3.1. Seetting metho ods of planee: a – by proojection of thhree points (А, В, С); b – by projecction of the straight linee (AB) and projections p of the pooint; c – by projection p of two paralllel lines (AB B, CD); d – by b projectioon of two inttersecting lines ((AB, CD); e – by projeection of anyy shape (qu uadrangle AB BCD); f – bby plane tracces (P) 177

Plannes are alw ways draw wn with lim mited size. But, in principle, p tthe plane has h indef-inite exteent. Tracce of the plane p is th he straightt line of in ntersection n of the pllane with the planee of projection. The point of intersectio i on of the plane p with h axes is ccalled the vanishingg point of ttraces. P belon nging to a plane. Line L belon nging to a plane 3.2. Point Therre are twoo theoremss of the deescriptive geometry: g : 1. Iff point bellongs to th he line whhich belon ngs to the plane p it m means poin nt belongss to this plaane. 2. The line beelongs to the t plane, if it passees through h two poinnts belong ging to thee plane or iif it passess through one pointt of the ab bove planee and is paarallel to a line con-tained in the plane According to these theeorems thhe following probleems of deescriptive geometryy can be soolved: draawing a line on thee plane; co onstructing a point on the pllane; con-structing a lackingg projectio on of the point con ntained on n the planne; checkiing of thee point beloonging to the plane (fig. 3.2, a).

Fig. 3.2. Plane lines: l a – belonging a line (m) to t a plane (A ABCD), and belonging g a point (K K) to a line (m); b – main (princciple) lines of a plane (frontal ( – f, horizontal – h)

Maiin (princip ple) lines of a plan ne Therre are a loot of lines belongingg to the pllane. Thosse of them m which haave a spe-cial or paarticular poosition sho ould be diistinguisheed. Main line of a plane is a straight line of lev vel which belongs too the plan ne. Frontall line is thee line of thhe frontal level beloonging to the given n plane. Hoorizontal line is thee line of the horizonttal level belonging tto the giveen plane (ffig. 3.2, b)). 18

3.3. Classification n of planess accordin ng to their location n in spacee A pllane can have h the fo ollowing ppositions relative r to the planess of projecction: – ann inclined position p to o all planees of projeection; – a pperpendicuular positiion to the plane of projection; p ; – a pparallel poosition to the t plane oof projection. Firstt variant is called as the plaane of general position and both the other aree called as the plane of particu ular (speciific) positiion. The plane, whhich is no ot parallel and not perpendicu p ular to anny planes of o projec-tion, is caalled the plane p of general g possition. This plane iss inclined to all majjor planess of projecction. (Wee can also call this pplane as the t obliqu ue plane) ((fig. 3.3). It cannott appear inn its true siize in any principal planes off projection n.

Fig. 3.3. Plane oof general position (AB BCD): a – a 3-dimantion 3 nal model; b – a drawin ng of it

The plane, whhich is peerpendicullar to one plane of projectionn, is called d the pro-jecting pllane (fig. 3.4). If a plane p is p erpendicu ular to the frontal pllane of pro ojection itt is a fronttal projectiing plane.. It has froontal projeection as the t straighht line. Th his projec-tion can bbe also coonsidered as a trace of the plaane. Otherr projectioons are projected inn non-true size. Thee followin ng figure (fig. 3.4)) shows all a three ppassible projecting p g planes: a frontal projecting plane (a),, a horizon ntal projecting planne (b) and d a profilee projectingg plane (cc). Therre is an im mportant property p o f the projeecting planes, whichh is called d as a col-lective onne: if a point, a line or a figurre are conttained in a plane, w which is peerpendicu-lar to the projectionn plane, th heir projecctions on the t above plane coinncide with h the tracee of the proojecting pllane. 199

Fig. 3.44. Projecting g planes: a – a frontal prrojecting plaane; b – a horizontal h prrojecting plaane; c – a proofile projectting plane 20

A pllane, whichh is paralleel to the pllane of proj ojection, is called a leevel plane (fig. 3.5).

Fig. 33.5. Level planes: p a – a fronttal level plane; b – a hoorizontal lev vel plane; c – a profile llevel plane 21

Anyy line or figure contaained in thhe level pllane paralllel to a proojection plane, p pro-jects to thhe last planne in true shape. positions of two pla anes 3.4. Mutual p osition of planes theere are parallel plannes and in ntersectingg According to mutual po planes. I the partticular casse when th he line off In ggeneral casse planes usually inntersect. In intersectiion is infinnity, the planes p becoome paralllel. The parallel p plaanes coinccide whenn the distannce betweeen them iss shortened to zero. The planes aree considerred to be pparallel if two interssecting linnes of one plane aree parallel too two inteersecting lines l of annother plan ne. It meaans in a drrawing tw wo parallell planes haave two paarallel projections off two interssecting lin nes with thhe same naame layingg in each prrojection. This is a parallelism p m property y of the planes. Thiss property is used too solve probblems requuiring the constructioon of a parrallel plane (fig. 3.7)).

Fig. 3.7 P Parallel plan nes α and β

A coommon ellement of two interssecting plaanes is a line of inteersection which w be-longs to bboth planes. As it was w mentiioned above, planess can be oof several types ac-cording tto their mutual m locaation withh the plan nes of projjection. S So there arre severall cases of pplane interrsection. In thhe first casse when both b planes are the planes p of particular p position the t line off intersectiion is a prrojecting line l whichh is projected as a point in a drawing g. In otherr words, booth planess of particcular posittion are projected p in i the linees, so their point off intersectiion is the line l of intersection (which is projected in the poiint accord ding to thee property of projectting line) (fig. ( 3.8). 22

Fig. 3.8 Two interseecting fronttal projectin ng planes: a – a 3-dimantional al model; b – a drawing g of them

In thhe secondd case, wh hen only oone plane is the plane of partticular possition andd another is the planne of geneeral positioon. In thiss case, plaane of part rticular po osition hass one projeection whiich is a lin ne (accordding to thee propertiees of liness of particcular posi-tion); at tthe same time t the plane p of geeneral possition has all projecttions with h distortedd and non-ttrue size images i of the originnal object. So one projection p of the linee of inter-section coincides with w projeection of tthe plane of particu ular positiion which h is a linee and anothher projecttion line of o intersecction can be b construccted (fig. 33.9, a). The third casee is that when w both planes arre the plan nes of the general position. p Itt means that they doo not havee any projeection as a line. Alll their projjections arre distort-ed and thhey are proojected in non-true ssize. If three t plan nes interseect, they have h onlyy one com mmon poin nt, it meanns they do o not havee any linee of interssection, thhey have only thee point of intersection (fig. 3 .10). So, accordingg to this in nformation n, for solvving the taask by thee third casse it is neccessary to introducee an auxil-iary plaane (third plane) ((intermediary) andd draw the lines of intersecction of this t planee with two o given pllanes (fig.. 3.9, b). The T inter-section of o the draw wn lines sshows thee commonn point off the planes. In ordder to find anotherr common n point it is necesssary to usse anotherr auxiliary y plane. Fig. 3.10. Intersecction of threee planes

233

Fig. 3.9 . Plane intersection: a – iintersection of a frontall projecting plane (α) with w a plane of general position (A ABC); b – intersectio on of two plaanes of gen neral position (ABC, DE EF)

3.5. Rela ative posiitions of a line with h a plane o interseccting lines with a plane p is thhe point (iin generall The common element of case). Acccording too the classsification oof the linees and classsificationn of the plaanes theree are 3 casees. The first casee that is wh hen both tthe line an nd the plaane are eleements of particularr ne of partiicular position has the projecction in a line. Thee position. In this caase a plan point of intersectioon can bee found w without ad dditional action a onlly with th he help off connectioon lines (ffig. 3.11, a). a The second caase that iss when onnly one eleement – a line or a pplane is an n elementt of particuular positiion. In oth her words another element e iss an elemeent of gen neral posi-tion. (fig. 3.11, b) In this caase the pooint of inteersection has h to be found by y auxiliaryy line whicch must beelong to th he given pllane in accordance with the ffollowing theorems: belongingg of the pooint to thee plane, beelonging of o the straight line too the planee. The third casee that is when w both a line and d a plane have the ggeneral po osition. Inn this case a line lyinng on the plane andd intersectting the giiven line m may be ob btained ass the line oof intersecction of an n auxiliarry plane passed p thro ough the lline with the givenn plane (figg. 3.12). 24

Fig. 3.11 1. Relative ppositions off a line with h a plane: a – a projectiing line (m) with a projjecting plan ne (ABCD); b – a projeecting line (m m) with a plaane of generral position

Fiig. 3.12. Inteersection off a line of geeneral posittion with a plane p of genneral positio on: a – a 3-dimantional al model; b – a drawing g of them 255

3.6. Line L and plane perrpendicullarity Therre are twoo basic theeorems forr solving taasks in case of perppendicularity: – if a line is perpendicu p ular to twoo intersectting lines which bellong to some plane,, this line iis perpenddicular to this t plane;; – theeorem of a straight angle projjection (it has been describedd above). So pprojection of a perp pendicular can be drrawn as th he true sizze only fro om frontall line of thhe level annd horizon ntal line off the levell, in other words froom two in ntersectingg lines of thhe level, which w can be specifi fied in any y plane. 1. The frontall projectio on of the liine m’’ wh hich is perrpendiculaar to a plaane is per-pendiculaar to the frrontal projjection of the frontaal level lin ne f’’ (m’’  f’’); 2. The horizontal projeection of tthe line m’ which iss perpenddicular to a plane iss perpendiccular to thhe horizontal projecttion of thee horizontaal level linne h’ (m’  h’). All ttasks of peerpendicu ularity can be divideed into threee big grou oups. The first grouup sets co onstructionn of a perrpendiculaar line from m a planee to spacee (fig. 3.13) (draw a perpendiccular).

Fig. 3.13. Drawing a parallel plaane to the given plane 26

Therre is next algorithm m for these tasks: 1. D Draw projeection of the t frontall level lin ne f (f’’, f’’) and proj ojection off the hori-zontal levvel line h (h’’, ( h’) in n the givenn plane off general position p α ( ABC) in the fol-lowing w way: f’ ∥ x, f’’ by thee additionnal point 1; h’’ ∥ x, h’ by the additionaal point 2. 2. D Draw projeections off the line m from an ny point (for ( exampple A) of the givenn plane α too space byy followin ng actions: m’’ f’’; m’ h’. 3. Seet requiredd distancee on projecction of th he line m according a to the giv ven condi-tions (sett additionaal point D with disttance in 15 1 mm). As A it is obbvious linee m is thee line of a general position, p so s firstly iit has to be b limited d as a segm ment. Theen its truee length shhould be found f by any methood (only method of the righht triangle has beenn considereed above).. 4. The requireed plane β with diistance 20 0 mm from m the givven plane α can bee drawn byy two interrsecting liines (k annd l). Thesse two linees have too be parallel to twoo given inteersecting lines l (sides of the giiven triang gle). l’’ ∥ A’’B’’, A n’’’ ∥ A’’C’’, l’ ∥ A’B’,, n’ ∥ A’C’’. β (l ∩ n)) ∥ α (( AB BC). The second group g setss drawing of a perp pendicularr from a ppoint in space s to a plane (om mit a perpeendicular)). The casee, which is used mo ore often, iis drawing g an inter-section pooint of thiis perpend dicular witth the plan ne (fig. 3.1 14).

Fig. 3.14. 3 Drawinng a perpen ndicular to a plane 277

Grapphical algorithm forr these tassks is the following: f 1. D Draw projeection of the t frontaal line f (ff’’, f’) and d projectioon of the horizontal h l line h (h’’, h’) in the t given plane of tthe generaal position n α ( ABC C) in the followingg way: f’ ∥ x, f’’ by the additiional poinnt 1; h’’ ∥ x, h’ by the additioonal pointt. 2. D Draw a perrpendiculaar line m thhrough thee given po oint (K). m m’’ f’’; m’ m h’. 3. Seet an auxiiliary projecting plaane β thro ough the liine m (theese steps have h beenn considereed above).. 4. D Draw an addditional liine 3–4. 5. Sppecify thee point of intersecti on O by intersectio i on of the aadditional line withh the perpeendicular. 6. Iff it is neceessary to determinee the real distance between b tthe given point andd the givenn plane it can c be fou und by the method of o the rightt triangle. By tthe third group g of tasks of thhe perpend dicularity it is necesssary to draw d somee auxiliary plane (geeometrical locus) whhich is perrpendiculaar to the liine of gen neral posi-tion (fig. 3.15). Geeometricall locus is set only by b two inteersecting lines (a frrontal linee and a horrizontal linne (lines of o level)).

Fig. 3..15. Drawin ng a perpenddicular plan ne to a line of o general pposition

Therre are fourr steps forr solving th the task. 1. D Draw an auuxiliary plaane β whicch is perpeendicular to o the givenn line l by two inter-secting linnes – a froontal line m and a hoorizontal line n. m’ ∥ x, m’’ l’’; n’’ ∥ xx, n’ l’. 2. Sppecify point O as a point off intersecttion of thee given liine l with auxiliaryy plane β bby the wayy which is given aboove. 28

3. C Connect projections of the poinnts O and K with eaach other. 4. D Determine the true siize of the rreceived segment s of general pposition OK. O 4. METHOD M S OF DRA RAWING TRANSF FORMAT TION Diffferent methhods of orrthogonal projection ns transforrmation arre used to make thee solution oof metric and posittional probblems sim mpler. After such trransformattions new w projectionns help to solve the problem bby minimu um graphiic means. 4 Metho 4.1. od for Reeplacemen nt Projecttion planees The Method for f replaceement of pplanes of projection n is made up of thee substitu-tion of a pplane withh a new on ne. The neew plane should s be perpendiccular to th he remain-ing one. The posittion in spaace of thee geometriic figure remains r un unchanged in space.. The planee should be b position ned in succh a way that t the geeometric fi figure has a particu-lar positiion to it which w is conveniennt for sollving the problem. This man nipulationn makes it possible by b changiing generaal position n of lines and plannes into a particularr one (fig. 4.1).

Fig. 4.1. 3-dimention 3 nal model of a plane rep placement

Fou ur principal problems solved d by repla acement of o projecti tion planees 1. Transformaation of a line of ggeneral po osition into a line of the level. Suchh transform mation helpps to determine the true size of the line-segmentt and its in nclinationn angles to the projecction plan nes (fig. 4.22). 299

Fig. 4.2. Transformat T tion of a linne of generaal position in nto a frontall level line

In oorder to soolve the problem it is necesssary to draaw a new plane V1, which iss parallel to the segm ment. In this t case tthe new coordinate axis passses paralleelly to thee horizontaal projectiion of thee given lline. It is necessary to draw w connecttion liness through tthe horizonntal projecctions A’ and B’ wh hich are perpendicuular to the new axis.. Then z-cooordinatess of the po oints (distaance from m the x-axiis to the fr frontal pro ojection off the pointss) are laidd off on th hem. The nnew projeection A′1B′ B 1 is equaal to the trrue lengthh of the segment, annd the ang gle φH is eequal to the t inclinaation anglle contain ned by thee segment aand the hoorizontal plane p of prrojection H. H 2. Transformaation of a line of thhe level into a projecting linee, i. e. set it perpen-dicular too the projeection plan ne and prooject the line as a po oint in thiss plane (fig g. 4.3).

Fig. 4.3. Trransformatiion of a horiizontal leveel line into a frontal proojecting linee 30

As tthe given line l is parallel to thee horizonttal plane, it i is necesssary to traansform itt into a prrojecting line l by replacement nt of the frontal f plaane V withh a new V1, whichh should bee perpenddicular to A’B’. As a result of o it the given g line is projectted in thee plane V1 aas a point (B’’1 ≡ A A1’’). In order to traansform th he general position line l AB in nto a projeecting one,, it is nec-essary to make twoo replacem ments, i. ee. solve bo oth probleems, the ffirst and th he secondd ones, succcessively.. Firstly we w transforrm a generral positio on line intoo a line off the levell and then it is transfformed intto a projeccting one. 3. Transformaation of a plane of tthe generaal position n α (∆ABC C) into a projecting p g one (fig. 4.4), i. e. positioned p d perpendiicular to one o of the projectionn planes.

Figg. 4.4. Transsformation of a plane oof general position into a frontal prrojecting pllane

It iss necessarry to replaace the pllane of prrojection with a neew plane V1 or H1, which is perpendiccular to the plane α. An ad dditional plane p is peerpendicu ular to thee plane α if it is draawn perpen ndicular tto one of the t main lines of thhe plane (frontal ( orr horizontaal line). Soo firstly on ne of thos e lines has to be draawn (h or ff). The seecond stepp is to draw w a new axis x wh hich is peerpendicullar to the chosen m main line. If it is a frontal linne f, x1 ff’’ and it gives us a new horiizontal plaane of proojection H1, if it is a horizontaal line h, x1 h’ and it gives us a new w frontal pllane of prrojection V1. By thee same wayy we speccify all poiints of thee given plaane. All of them shoould lie on n one linee (the planee had to be b transforrmed into a line). An ngle betw ween new pprojection n, which iss the line nnow and an a axis x1 is the anggle of incliination off the givenn plane to the planee of projecttion H or V (which depends oon previou us chosen steps). plane intoo a level plane. 4. Transform the plane α (∆ABC)) from a projecting p p Thee true size of the planne is deterrmined in this case (fig. 4.5). The first stepp is to set an additiional axis x1 paralleel to the pprojection n which iss generatedd in a line. Then wee draw anoother projeection of all a points bby measurring coor-31

dinates inn previouss plane of projectionn. These actions a giv ve us the ttrue size of o the giv-en plane α at the ennd.

F Fig. 4.5. Traansformation n of a frontaal projecting g plane into o a horizontaal level plan ne

4.2. Meth hod for rootation arround pro ojecting aaxis According to this meth hod objectt has to bee rotated around soome projecting axiss for changging its property p from f geneeral into particular p . In otherr words the t objectt changes iits positioon and plaanes of proojection are a static in i contrastt with thee previouss method. A As it is knnown linees of the llevel are parallel p to one plane ne of projeection andd they are pprojected in i this plaane in truee size. Acccording to the propeerty of linees of levell one projeection is innclined to the t axis annd the otheer two prin nciple projjections arre parallel.. Thereforee line of the t generaal positionn, which has h all projections as an obllique line,, should rootate in thee followin ng way: onne of its prrojection should s beccome paraallel to thee axis x. Fiirstly, a projecting p axis i shoould be seet for this rotation. Axis i haas to passs through aany end point of the given seegment. Then T it is necessary n to rotate the givenn projectionn to the horizontal h position ((parallel to o the axis x) by a ccompass. We W finishh the task bby specify fying anoth her projecction of th hat segmeent by connnection lines. l Thee particularr case of the t metho od of rotattion aroun nd a projeccting axis,, which iss called ass the methood of planne-parallel transfer, aand it is not n consideered here.

32

F Fig. 4.6. Dettermining th he true lenggth of a line by rotation n around proojecting axis: a – a 3-dimantion 3 nal model; b – a drawin ng of it

44.3. Method for rottation aroound main line (fro ontal or h horizontall) So aas it is obbvious from m the nam me of the method, firstly thee main lin ne (f or h)) should bee drawn (ffig. 4.7).

Figg. 4.7. Deteermining a plane p true siize by rotatiion around its main line ne (horizontaal): a – a 3-dimantion 3 nal model; b – a drawin ng of it 333

Thiss line will be used as a an axis line of rottation. Then we draaw through h vertexess of the givven plane traces of auxiliary planes β1 and β2 wh hich are pperpendicu ular to thee main linee. These planes p are planes whhere vertex xes will be b rotated. The third d vertex iss fixed beccause it liees on the main m line ((axis of ro otation). Points P of inntersection n of thosee perpendicculars withh the main n line are centres off rotation. So, as it iis shown in i a draw-ing, a raddius of rotaation is the line of ggeneral po osition. It means m that at, firstly, we w shouldd find its trrue size by any kno own methood (three methods for determ mination of o the truee size havee been alrready disccussed aboove). In order o to finish soluttion of the task wee have to uuse a comppass as it is shown inn an exam mple. 5. S SURFAC CES A suurface is a set of all successivve position ns of the movement m t of a line.. This linee is called a generatrrix; the mo ovement m may retain n or chang ge a shapee. The mov vement off the generratrix can be subjeccted to a llaw or it can c have an a arbitrarry charactter. In thee first case, the surfaace will be legitimaate, and in n the secon nd one it w will be raandom (ir-regularityy). The laaw of the generatriix movem ment is usu ually deteermined by anotherr line, whicch is calleed a guidee line on w which thee generatriix slides in its mov vement. Inn some casses, one of o the guid de lines caan be con nverted to a point ((vertex co onical sur-face), or iin the infinity (cylin ndrical surrface). 5.1. Facceted surffaces. Priism and pyramid p A faaceted surface is thee surface, which is formed by y the movvement of a rectilin-ear generratrix on a broken line, forr example, pyramid dal (with top) and prismaticc (without top) surfaaces (fig. 5.1). 5

Fig. 5.1. Form mation of faaceted surfacces 34

Polyyhedron iss a solid shape s withh four or more flat surfaces ((prism (4 – faces),, pentagonn (5 – facees), hexago on (6 – faaces), etc.)). Polyhed dron outlinne is draw wn as pro-jections oof its facess and ribs in a drawiing A prrismatic suurface is the t surfacce, which is i formed by the m movement of o a recti-linear genneratrix byy a broken n line, duee to this we w obtain faces f withh the help of paral-lel movem ment in acccordance with the ggiven direection. A prrism is thee polyhedrron that hhas two bases that arre the sam me and theey are par-allel, andd some facces are teetragons (qquadrangles). Prism m is calledd straight if its ribss (lines of intersectioon of the adjacent faces) aree perpendiicular to tthe base, and is in-clined if it is not [77]. Prism is i called aas a regulaar prism iff its basess are a regu ular poly-gon (polyygon whicch is inscriibed into tthe circle and a has eq qual sides)) (fig. 5.2)).

F 5.2. A drawing Fig. d of a prism with h a regular triangle basse

A pyyramidal surface s is the surfacce, which is formed d by the m movement of a recti-linear genneratrix byy a broken n line withh one fixed d end of th he generattrix. Pyraamid is thee polyhedrron havingg one basee, top, faces and ribbs (lines off intersec-tion of thhe adjacentt faces) which inter sect at onee point (to op of pyram mid) in geeneral [7].. if its top is cut and Pyramid is called truncated t d as a resu ult of it thhe pyramid d has twoo 355

bases. Pyyramid is called c as a regular ppyramid iff its base is i the reguular polygon, whichh is inscribed into thee circle an nd has equual sides (ffig. 5.3).

Fig. 5.3. A drawing of a pyramid with w a regulaar square baase

5.2. Surfa aces of revvolution. Cylinderr and conee Surfface of reevolution is the surrface whicch is form med by m movement of a linee (generatrix) aroundd another fixed straiight line. This T fixed d straight lline is callled an ax-is line of rotation. The T surfacce is called ruled if the generaatrix is a sstraight lin ne and thee surface iss called cuurved wheen the geneeratrix is not n a straiight line. A All points of gener-atrix rotaate aroundd the axis on circless with releevant radiiuses whicch are called paral-lels. Paraallel planess are alwaays perpenndicular to o the axis of rotationn. The parrallel withh the smallest diameter is calleed the necck, and witth the larg gest as thee equator. A meridi-an is an ooutline whhich is forrmed by iintersectio on of surfaace with tthe plane of level – frontal orr profile. The T merid dian, whichh is paralllel to the frontal f plaane of projjection, iss called thee main meeridian (fig g. 5.4). 36

Fig. 5.4. A surface of rrevolution and a its speciial elementss

Cyliindrical suurface of revolution r n is a ruled d surface formed f byy parallel movement m t of a straigght line (ggeneratrix) on circlee or ellipsse line. Cy ylinder is a body bo ordered byy cylindricaal surface of revolu ution (sidee surface) with two parallel baases which are per-pendiculaar to the axxis of rotaation. Cyliinder is caalled circu ular if its bases are circles, and a ellipticcal if the bases aree ellipses. Cyliinder is caalled straig ght if the aaxis of rotation is peerpendicullar to its bases. b Therre are threee types off cutting ccylinder. 1. Iff cutting plane p is paarallel to th the bases it i cuts cuts a shape which is similar too the base ((fig. 5.5, a). a 2. Iff cutting plane is paraallel to the axis of rottation it cuts some reectangle (fiig. 5.5, b). 3. Iff cutting plane p is in nclined to the axis of o rotation n (angle iss not 90o) it cuts ann ellipse (fi fig. 5.5, c)..

Fig. F 5.5. A s traight circu ular cylindeer: a – by a parallel plaane to its basse; b – by a perpendicu ular plane too its base; c – byy an inclined d plane 377

Fig. F 5.6. Poossible cylin nder sections

Cyliinder has two equaal rectanguular projeections and d the third rd projectiion as thee shape of tthe base (circle or ellipse) e in general caase. A coonical surrface is prroduced bby rotation n of a linee (generattrix) round d the axiss (directrixx) interseccting it. In n other woords one end of thee generatrrix is fixeed and re-ferred to as a verteex (top). Cone C is a body bord dered by conical c suurface of revolution r n (side surfface) withh one basee which is perpendiccular to th he axis off rotation. The pointt with axis is fixed an of interseection of generatrix g nd is referrred to as a vertex. Conne is calledd as a straight conee if the ax xis of rotation is pperpendicu ular to thee base and the cone is called as an incclined wheen the axis is not pperpendicu ular to thee base (fig.. 5.7). 38

Fig. 5.7. A straight circular cone

Trunncated conne is form med by a ccutting plaane parallel to the bbase of co one and itt has two bbases. Therre are fivve types of o conical sections which aree sub-diviided in acccordancee with direcction of a cutting pllane. 1. A As a result of interseection of a cone with h a plane perpendicu p ular to thee cone ax-is, a circle is obtainned(fig. 5.8, a) (In ccase if con ne has a circle as a bbase). 2. Iff a cuttingg plane passses throuugh the verrtex of a cone, c it cuuts a pair of o genera-trix lines,, which foorm a trian ngle (fig. 55.8, b). 3. Iff the cuttinng plane is inclinedd to the rottation axiss of a conee and does not passs through its vertex then t it is possible p too obtain th hree types of lines inn the section. 4. Iff a cuttingg plane is not paralllel to any generatrix x, then ann ellipse iss obtainedd in its secttion (fig. 5.8, 5 c). 5. Iff a cuttingg plane is parallel too one gen neratrix, th hen a paraabola is ob btained inn its sectionn (fig. 5.8, d). 6. Iff a cuttingg plane is parallel p too two geneeratrixes, then a hyyperbola iss obtainedd in its secttion (fig. 5.8, 5 e). 399

Fig. 5.8. P Possible con ne sections: a – by a paarallel planee to its basee; b – by a plane p passed d through thhe cone top; c – by an incclined planee to all geneeratrixes; d – by a parallel plane too a generatriix; e – by a perpeendicular pllane to its baase

5.3. Points an nd lines on o the surface In oorder to finnd missing g projectioons that belong b to some surfface, it is necessaryy to build aan auxiliarry line on a given suurface, passing thro ough a giveen point of o the pro-jection. F Firstly, it is necessaary to connstruct thee projectio on of this auxiliary line, andd then to ddraw the reequired prrojection of the poiint on it. For this ppurpose we w can usee various liines: lines of generaatrix, paralllels, meriidians, etc. In some casess, if the surface is the projeccting surfa face (fig. 55.2, 5.5), (i. e. per-pendiculaar to the plane p of projection) p ) missing projection n of the ppoints can be foundd without aadditional constructiions, becaause this kind k of surfaces has collected property. AP Prism and a cylinder can be ssuch kind d of surfacces. Thereefore, all the t pointss on the suurfaces of the t prism and the cyylinder, being in a horizontal h l projectio on are dis-tributed oon the lines of the base (polyyhedron or o circle, or o ellipse)) accordin ng to theirr visibility.. We consider the examp ples of coonstruction n of pointts on a priism and a cylinder.. The pointts and linees lying on n the surfface canno ot enter intto the surfface and go g beyondd it. The viisible part of the fro ontal projeection of the surfacee is locateed below th he axis off symmetryy in the hoorizontal plane p of pprojection,, the invisible part iis located above thee axis of syymmetry. 40

A point in space has three coordinates – x, y, z. So it is possible to use axis lines x, y and z by taking some imagine base line when they are even absent. The base line enables draftsmen to draw the profile projection more comfortable. If the horizontal projection of a surface has an axis of symmetry (circle, square and other), a base line should be drawn through this axis. If the horizontal projection of a surface has no axis of symmetry (triangle, pentagon and other), base line should be drawn through the widest place of the polygon. The widest place can be a point or a side (line). If points belong to ribs (specific elements of surface), their projections can be found by connection line. These connection lines have to be perpendicular to base line from the frontal projection of points to the profile projection of necessary ribs. If points belong to faces, distance between the base line and the projection of a point in the horizontal projection should be measured and put aside in the profile plane of the projection. In order to finish projection all points should be connected in the given or taken order according to their visibility. Pay attention, when two planes intersect, they always organize line of intersection. This kind of lines have to be drawn according to their visibility. In order to solve projection points on the surface of a pyramid or a cone, one of the methods described below can be used. 5.4. Generatrix Method This method is used on the basis of the theorem described above: if a point belongs to the line, projection of this point belongs to the projection of this line. Line m’’ as the one of the uncountable numbers of generatrixes is drawn from the top of a pyramid or a cone up to its base through the given point K’’ (fig. 5.9, 5.10). A line is the shortest distance between two points. In order to draw any line we should draw its two end points. Both these points of the line m belong to some specific elements of the surface and their projections can be found by connection lines in accordance with their belonging and without any auxiliary actions. One of them is on the top (vertex) and another one is on the base of pyramid or cone). Two projections of a point are always connected between each other by a connection line. Connection lines have to be perpendicular to the axes lines of the planes of projection according to the properties of orthogonal parallel projection. So, for solution of the horizontal projection of the ends of the line m’ it is necessary to do the following actions. The first point is the top of the pyramid or the cone. The second point 2′ can be found by the intersection of the base outline of a pyramid or a cone with a connection line in the horizontal projection. Then the connection line is omitted from frontal projection of the given point K’’ to the intersection with horizontal projection of the line m’. The point of this intersection is the required point.

41

Figg. 5.9. Soluttion a pointt on a pyram mid by Geneeratrix Methhod an nd Method oof auxiliary cutting plan nes

5.5. Method M off auxiliary y cutting planes p Wheen a planee of the lev vel interseects with surface, s it formes a polygon or o a circlee in cross-ssection (inn case if base b of surrface is a polygon or o a circlee and it is the planee of level tooo). It deppends on the t type off surfaces.. The projection of nnecessary point liess on the sidde/sides off the polyg gon or on the circle. Firsttly, an imaging cuttting plane of the lev vel should d pass throough the projection p n of the point P’’. Thhis imagin ng plane α has to bee parallel to t the bas e of a pyrramid or a cone. As it has beeen mentioned abovee if a cuttiing plane is paralleel to the baase it cutss the same shape wiith the basse but sm maller. So the next step s is to draw this auxiliaryy section pllane. If it is a polyggon (pyram mid case)), an addittional poin nt on any rib (excep pt verticall projectionn of a ribb, because in this caase a conn nection lin ne from tthe additio onal pointt and rib linne will bee overlapped) has too be taken.. This poin nt is the itnntersection point off the additiional planne with th he rib of ppyramid. The T conneection linee has to be b omittedd 42

from the projectionn of addittional poinnt to the projection p of the rellevant rib. Then ann auxiliary figure is drawn, according too all information wh hich was ggiven earllier, it fol-lows thatt each line of the aux xiliary figgure has to o be parallel to each relevant line l of thee base. Thee last actioon is to drraw a connnection lin ne from th he given ppoint to itss intersec-tion withh the liness of the au uxiliary fiigure. Thiis point off intersecttion is thee requiredd point. If a surface has h a line of symmeetry, the so olution can n have twoo symmettry points. If itt is a circlle (cone and a torus cases), we w can just measuree a radius (distancee between tthe center line (axis line) and tthe generaatrix line (surface booundary lin ne)). Thenn we draw a circle inn the horizzontal projection by a compas and omit the conneection linee from the projectionn of the giv ven point P P’’ to the intersectio on with thhe horizonttal projec-tion of ouur additionnal circle. The T point oof their inttersection is the requuired point.

F 5.10. So Fig. olution a poiint on a con ne by Generratrix Methood an nd Method oof auxiliary cutting plan nes

5.6. Surffaces of reevolution.. Sphere and a toruss Spheerical surfface is a surface s forrmed by rotation r off a circle round its diameter.. This surfface is a seet of equaally-spacedd points from fr the ceenter. Sphhere is thee only onee surface w which has an infinity y numberss of axis passed p thro ough the ccenter. Th his proper-ty is usedd in solvinng points on o a surfacce and man ny other descriptive d e tasks. 433

Spheere is drawn as a circle c in aall three prrojectionss with the equal diaameter. Inn other worrds all its outlines are a circless. Every point p on th he sphere is rotated d on somee circles whhich are called paraallels. A frrontal outlline is the main fronntal merid dian. It is projected p as a horizzontal linee on the hoorizontal axis a line in n the horizzontal pro ojection an nd as a veertical linee the samee with the vvertical axxis line in the profilee projectio on. A horizontal outline iss the equaator of sph here. It takes the saame placee with thee horizontaal axis in the t frontal and the pprofile projjections. A prrofile outliine is the main m profille meridian n and it is combined c with the vertical v ax-is line in tthe horizonntal projecction and w with the vertical axis line in thee frontal prrojection. A ball is a boody bordeered by a sphericall surface. Method oof auxiliarry cuttingg planes is used for solving s a point p on thhe ball. Therre is onlyy one type of sectio n for a baall it is a circle. Buut this circcle can bee projectedd as three different images inn the draw wing. It deepends onn the direcction of a cutting pllane. 1. Iff a cuttingg plane is a horizonttal plane of o level, th his sectionn is drawn n as a linee in the frontal andd profile projection p ns and as a circle in the hoorizontal projection p n (fig. 5.11, a). 2. Iff a cuttingg plane is a profile pplane of th he level, th his sectionn is drawn n as a linee in the frontal andd horizonttal projecttions and as a circcle in thee profile projection p n (fig. 5.11, a). 3. Iff a cuttingg plane is a projectiing plane, this sectiion is draw wn as a liine in thiss plane of pprojectionn and as ellipses in tw two other planes p (fig g. 5.11, b)).

Fig. 5.11. Possible caases of projeection for baall section: a – by y level planees; b – by a projecting plane p 44

Takiing more additional a l points is recommended for drawing d off the ellipse. A spphere has unique prroperty, it has only one possib ble type oof section and it hass simple shhape – circcle, which h needs onnly specify fying its ceenter and knowing its radius.. So accordding to thiis all poin nts on its ssurface can n be solveed by the m method off auxiliaryy cutting pllanes in alll planes of o projectioons. See th he followiing exampple (fig. 5.12).

Fig. 5.12. 5 Solvinng a point on a circle su urface

Toruus surfacee is formeed by rotaation of fo orming cirrcle on thee guidex round thee axis whicch belonggs to the same planee as the circle c but it does noot pass through thee center of this circlee. The guid dex is a ciircle with a radius. Toruus is a boddy bordereed by a to rus surfacce. Torus is i a body bbordered by b a toruss surface. T Torus is open o if thee outlines of the gen neratrix ciircle do noot touch each e otherr and do noot intersecct (fig. 5.1 13, b). In oother words if R is more thann r (R > r)) where R is the raddius of guide circle and a r is th e radius of o the geneerating circcle. Toruus is closeed if R is equal e to r (R = r), what w meanss that outllines of thee generat-ing circlee touches each otheer (fig. 5.13, c). To orus is self-crossingg if outlin nes of thee generatinng circle inntersect. In I this casse R is lesss than r (R ( < r). T This kind of o surfacee formationn can produ duce severaal types off surfaces such s as torroid and glloboid (fig g. 5.13, a),, and toroiid (fig. 5.113, d). If torus has the frontaal projectiing axis I it is draw wn as it iss shown inn an exampple below.. 455

Fig. 5.13.. Torus surfface types: a – a globoid; b – an openn torus; c – a closed torrus; d – a torroid

Therre are sixx types off sections for torus. Two of them are circles iff a cuttingg plane passses througgh the axis of rotatioon. Metthod of cuutting plan nes is usedd for soluttion of thee projectioon points on o a toruss surface. W When a guidex g circcle is paraallel to th he frontal projectionn, auxiliarry cuttingg planes arre planes of o frontal level whicch are pro ojected as circles inn the fronttal projec-tion and as lines inn the horizontal proojection. The T follow wing figurre (fig. 5.1 14) showss c plaanes. an exampple of solvving pointss on torus by the Meethod of cutting 1. Iff a cutting plane is perpendicu p ular to thee axis line, the sectioon is a plaane whichh is bordereed by two circles witth a big raadius (Rbc)) and a sm mall radius (Rsc) (fig. 5.15, a). 2. Iff a cuttingg plane co ombines (ppasses thrrough the axis line)) with the axis line,, the sectioon is twoo circles which w havve the saame radius with thhe generattrix circlee (fig. 5.155, b). Distaance t is zero z in thiis case. Th his distancce t is takken for exp planation.. It is a distance whiich describ be locationn of the cu utting plan ne. 3. W When R ≤ t < R2, wh here R is thhe radius of the guiide circle aand R2 is the radiuss of the bigggest circle of a torus, the seection is an a ellipse with twoo axes of symmetry s y (vertical aand horizoontal). 4. W When R1 < t < R, wh here R1 is the radiuss of the sm mallest cirrcle of thee torus thee section iss a wavy curve. c 5. W When t = R1, the secction is a ddouble pettal curve. This sectiion is callled a lem-niscate off Bernoullli. 6. W When t < R1, the secttion is twoo ellipses with w one line l of sym mmetry. 46

Fig. 5.14. Solvinng points on n a torus su urface

Fig g. 5.15. Posssible cases torus sectio ons: a – two t circles which w are eexactly the same s with generatrix g ciircle; b – a pllane bordereed by two ccircles (with h a big radiu us and with a small) 477

All tthese fourr sections are a called curves off Perseus (fig. ( 5.16) .

Fig. 5.16. Posssible cases torus sectio ons

48

6. IN NTERSEC CTION OF SURFA ACES The simplest body is a cone, a ssphere, etcc. But a co omplex boody which h containss several boodies is ussed more often. o The line of inntersection n of two ssurfaces iss a set of the pointss belongin ng to bothh surfaces. So, findinng these points is thhe way to solve the line of inttersection.. Shape off the interssection linne dependss on mutuual location n of interssecting suurfaces and d their lo-cations acccording to t the projection plaanes. ossible casses of surfface interrsection 6.1. Po Therre are fourr types of mutual loocation. 1. Full penetrration. All generati ting lines of o the firsst surface (cylinder)) intersectt another ssurface, buut not all generatrixxes of the second su urface inteersect the first one.. In this caase the inttersection line of thhe surfacees decomp poses intoo two closed curvess (fig. 6.1, a). 2. Partial cuttting-in. Not N all genneratrixess of both surfaces s inntersect eaach other.. In this caase the inteersection line l is onee closed cu urve (line)) (fig. 6.1, b).

Fig. F 6.1. Typpes surface intersection ns: a – Full penetraation; b – Partial P cuttin ng-in

3. U Unilateral or one-sid de contactt. All geneerating lines of the oone surfacee intersectt the other surface, but b not alll generatriixes of thee second surface s inttersect thee first one.. There is a commonn tangent plane in oone point of the surrfaces (figg. 6.2, a). The T inter-section linne decompposes into two closeed curves which w meeet in the pooint of con ntact. 4. B Bilateral or o doublee contact. All geneerating lines of bothh surfacess intersectt each otheer. The inntersecting g surfaces have two o common n tangent pplanes. In n this casee the interssection linne decomp poses into two planee curves meeting m inn the poin nts of con-tact (fig. 6.2, b). 499

Fig. F 6.2. Typpes surface intersection ns: a – Unilateral U or o one-side ccontact; b – Bilateral or double coontact

Therre are twoo big grou ups of inteersection surfaces s according a to their lo ocation inn space. In other worrds there are a two vaariants acccording to o relative llocation of surfacess and planees of projeection: priv vate casess and geneeral cases. Fourr private cases: c 1. Inntersectionn of some surfaces, s w where side faces are the projectting faces (fig. 6.3).

Fig. 6.3. Intersection n of a straigght circular cylinder c witth a hexagoonal prism 50

2. Inntersectionn of some surfaces w when only one surfacce is projeecting (fig.. 6.4, а). 3. Inntersectionn of some surfaces of revoluttion with the same axis of ro otation forr them (figg. 6.4, b). Coaxial C su urfaces inntersect in a circle, the t plane oof which is i perpen-dicular too the axis of o rotation n surfaces . At that, if i the axiss of rotatioon surfaces is paral-lel to thee projectioon plane, the interssection lin ne projectts onto thhis plane as a line-segment.

Fig. 6.4. Inntersection of o surfaces: a – Interrsection of a straight ciircular conee with a triangular prism m; b – Interrsection of a straight circullar cone, a sphere s and a straight circular cylin nder with a ssphere

4. Inntersectionn of some surfaces of revoluttion of thee second oorder whicch are de-scribed arround sphhere. (The Monge’s theorem, see fig. 6.8). All these taskks from prrivate casees which were conssidered beefore can be solvedd on only bby surface property.. All otheer tasks arre generall without aany additiional actio cases. In order to solve s thesse tasks it is necessary to usee some auuxiliary meediators – planes orr spheres. This depeends on thhe type of surfaces and a their m mutual loccation andd location aaccording to the plaanes of proojection. Therre are threee method ds of solvinng task fro om generaal cases. 1. M Method of auxiliary cutting plaanes 2. M Method of auxiliary concentricc spheres 3. M Method of auxiliary eccentric spheres 6.2. Metthod of au uxiliary cu utting level planes The followingg examplee illustratees how to apply thee method oof auxiliary cuttingg planes for construcction of the intersecttion line of o a spheree with a coone (fig. 6.5). 6 51

Figg. 6.5. Determining of an intersecttion line of a straight ciircular conee with a sph here by the Method off auxiliary cu utting level planes

In oorder to coonstruct th he intersecction line of the giv ven surfacees it is ad dvisable too introducee the fronttal plane and a a num mber of ho orizontal planes p (αVV1, αV2, αV3 s V , αV4) as the auxiliiary planees. Characcteristic pooints of projectionss should bbe identified firstly.. Then fronntal projecctions of th he highestt and loweest points (1’’ and 55’’) are fou und as thee points off intersectiion of outtlines. Hoorizontal projections p s of the ppoints 1’ and a 5’ aree determineed due to the connection liness and surfface properties. Auxxiliary horrizontal planes cut tthe spheree and the cone c in cirrcles. Find d the pro-jection 4’’’ of the points p lyin ng on the ssphere equ uator with h the help of horizon ntal planee αV3, whicch passes through the t centerr of the sp phere. Thee plane inntersects the spheree along thee equator and a the co one in the circle of radius r r. The T horizoontal projeections in-tersectionn of the laatest yield ds the horiizontal pro ojection 4’. The horrizontal projectionss of points 4’ are thee points of visibilityy bounds of the intersection line on th his projec-tion. The passing points p (2, 3) 3 are deteermined by b means of o auxiliarry horizon ntal planess αV1 and αV2. After all pointss are connnected in accordance a e with thee given or taken or-der and oobtain a sm mooth curv ve line in aaccordancce with vissibility. 6.3. Metthod of au uxiliary concentricc spheres Thiss method is i widely used u for ssolution off problemss on consttruction off intersec-tion liness of rotatioon surfaces with inteersecting axes. a 52

Befoore studyiing this method m we consider a particular case foor of intersection off rotation ssurfaces, the t axes of o which ccoincide, i. i e. the caase when coaxial su urfaces off rotation intersect. Thiss propertyy is used for fo construucting a lin ne of muttual interseection of two revo-lution surrfaces by means m of auxiliary a sspheres. In n this casee we can uuse both, concentric c c (construccted from one centeer) and ecccentric (d drawn from m differennt centers) spheres.. We are ggoing to coonsider ap pplication of auxiliaary concen ntric spherres, those with w havee a constannt center. Noote: if a plane p of revolution r n surface axes is not paralleel to the projection p n plane, the circles of their in ntersectionn are proj ojected as ellipses aand this makes m thee problem solution more m complicated. F For this reeason the method oof auxiliarry spheress should bee used undder the following coonditions: a) inntersectingg surfaces are the suurfaces of revolution n; b) suurfaces axxes interseect and an intersection point is taken foor the centter of aux-iliary sphheres; c) thhe plane produced by b the surffaces axess (plane off symmetry ry) is paralllel to onee of the proojection pllanes. Usinng this meethod, it iss possiblee to constrruct the lin ne of inter ersection of o the sur-faces in oone projecction. The applicatioon of this method m is describedd below (ffig. 6.6).

Fig. 6.66. Determinning of an in ntersection lline of a straaight circulaar cone withh a straight circular cylinder by y the Methood of auxiliary concenttric spheres 533

The points 1’, 3’, 4’, 10’ are determined as the points of level generatrixes of the surfaces belonging to the plane of axes intersection (the plane of symmetry). Find the other points by the Method of auxiliary spheres. It is necessary to draw an auxiliary sphere of an arbitrary radius from the intersection point of the axes for the given surfaces (point O’’). This sphere is simultaneously coaxial to the cone and the cylinder and cuts them along the circles the planes which are perpendicular to the corresponding rotation axes. The frontal projections of those circles are line-segments. The constructed sphere intersects the cone along the circle of diameter, and the cylinder – along the circles (represented by color in fig. 6.6) in the intersection points of the horizontal line-circle with the vertical lines-circles obtain points 2’’ and 5’’–9’’ respectively, which belong to the intersection line. In such a way it is possible to construct a certain amount of points of the desired intersection line. Students should note that not all the spheres may be used for the problem solution. So the limits of the auxiliary spheres usage should be considered. The maximum radius of a cutting sphere is equal to the distance from the centre O to the farthest intersection point of the level generatrixes (from O’’ to 4’’ and 10’’). The minimum cutting sphere is a sphere, which contacts one surface (the larger one) and cuts another (the smaller one). If an auxiliary sphere cuts only one given surface, this sphere is not proper for the problem solution. Rmax > Raux > Rmin. In order to construct the second projection of the intersection line one may use the circles obtained while cutting the cone by auxiliary spheres or drawing additional sections of the surface. Points 13’’≡14’’ and 15’’≡16’’ lying on the level generatrixes of the cylinder are the points of visibility bounds of the intersection line on the horizontal projection. 6.4. Method of auxiliary eccentric spheres All auxiliary eccentric spheres, which are used in this method, have different centers. Therefore, a center for all spheres should be specifying before their drawing. It is necessary to set any circular cross-section for the one surface from two given in order to find a center for an auxiliary sphere. The next part of the solving algorithm is the same as for the Method of auxiliary concentric spheres There are three graphical conditions for applying the method of auxiliary eccentric spheres: a) when revolution surfaces of fourth order (open or closed torus) or surfaces of elliptical cylinder intersect; b) there is a common plane of symmetry for both surfaces and it is a level plane; c) axes of surfaces intersect or cross.

54

F Fig. 6.7. Deetermining of o an interseection line of o a straightt truncated ccircular con ne witth half of to orus by the M Method of auxiliary a eccentric spheeres

6.5. Moonge’s Th heorem If tw wo surfacees of the seecond ordeer are desccribed arou und a spheere, the lin ne of theirr mutual inntersectionn decompo oses into tw wo plane curves. c Th he planes oof these cu urves passs through a straight liine conneccting the in intersection points of the tangeent lines. Fig. 6.8 preseents an exaample witth a constrruction off surface inntersection n lines onn the basis of Mongge’s theoreem, wheree two cyliinders, a cylinder c aand a conee and twoo cones aree describedd around a sphere.

Figg. 6.8. Privaate intersection case. D etermining of an interssection line of two straiight circular coones which are circumsscribed arou und a spheree by Mongee’s theorem 555

7. UNFOLDING (DEVELOPMENT) OF SURFACES Unfolding is a flat plane as a result of combining surface of a body with the plane. This surface is a flexible inextensible film which is not torn by this kind of transformation (involute). These kinds of surfaces are called unfoldable. They can be polygons, ruler surfaces (cylindrical, conical) and surfaces with edge of regression (torso). There are three types of unfolding. 1. Accurate unfoldings can be constructed for faced surfaces of prism and pyramid, circle cylinders and cones. 2. Approximate unfoldings can be constructed by replacing (approximation) given surfaces with segments of developable prismatic and pyramidal surfaces. This is true for circle inclined cones, elliptical cylinders with circle sections and complex surfaces. 3. Conditional unfoldings are used for undeveloped surfaces of a sphere, torus surfaces, toroid etc. There is only one point on the unfolding for each point of surfaces. There are several properties of inter-relation between a surface and an unfolding: 1. Lengths of the relevant lines of surfaces and unfolding are equal. 2. Lines, which are parallel on the surface, are parallel on the unfolding. 3. Angels between relevant intersecting lines on the surface and on the unfolding are equal. 4. Areas of the relevant shapes on the surface and on the unfolding, which are border by closed lines, are equal. 7.1. Unfolding of a polyhedron. 1. Determination of a true size for side faces or edges is a key point of unfolding in such case. 2. There are three methods for construction of prism unfolding: 3. Method for normal section. 4. Method for unrolling 5. Method for triangulation 7.1.1. Method for normal section This method for unfolding lateral surface of a prism can be applied if in the drawing: – edges of the prism are straight lines of the level, in other words they have the true size in one of the given projections; – there are no images of the prism bases with the true size in the projections. NB: If in the drawing edges of the prism are straight lines of general position, then we should change the position of the prism relative to the projection planes, transforming the edges into straight lines of the level, for example, by replacement of the projection planes. 56

The constructtion of an unfoldingg of the laateral surfaace of a pprism whille using a method foor normal section is performedd according g to the following grraphical allgorithm: 1. D Drawing a plane of normal n se ction which is perp pendicularr to the edges in thee projectionn where thhey have the t true sizze. 2. D Determinattion of thee true size of a polyg gon of the normal seection plaane by anyy method (ffor exampple using method m forr replacem ment of pro ojection pplanes) 3. Expansion of a naturral polygon of the seection into o a straighht line in a free areaa of the drrawing annd drawin ng perpenndicular sttraight lin nes througgh the po oint of itss vertices – directionns of the edges. 4. Seetting asidde true len ngth of thhe corresp ponding ed dges on thhe directio ons of thee edges on both sides of the no ormal secttion line. 5. C Connectionn of endpo oints of thhe edges as a straightt line segm ments and d comple-tion of a fflat figuree as the sid de surface of the priism. 6. C Completionn of the drrawing in accordan nce with GOSTs G – ffold lines should bee drawn byy a long daashed doub ble dottedd thin line

Fig. 7.1. Constructioon a surface of a triangu ular prism hile using m method for normal n sectiion wh

7.1.2. Meethod for unrolling g when in a drawing: d Thiss unfolding method is used w – eddges of a prism p are straight s linnes of the level; – baases of a prism (or one of thhe bases) are planees of the level and d have thee true size. The essence of o this meethod is thhat by "cu utting" thee prism suurface at one of itss edges, thhe nearest prism face is com mbined wiith the plaane of unnfolding (tthis planee should bee parallel to the edg ges). Thenn, by rotatting the prism sequuentially around a thee followingg edges, alll other faces of the prism aree also align ned with tthe plane of o unfold-ing, one bby one (figg. 7.2). 577

Fig. F 7.2. Rollling a prism matic surfacce

7.1.3. 7 Meth hod of tria angulation Unroolling of Pyramid P Surface S The developm ment of thee side surfface of thee pyramid accordingg to the tru ue size off its edges is carried out using g the follow wing graphical algo orithm. 1. Sppecify thee true size for all pyyramid edg ges by any y method ((method of o rotationn around prrojecting line, meth hod of repplacement planes off projectioon and etcc.) and forr its base. ((If the basse is the plane p of thhe level, th hen it’s tru ue size is given in one o of thee projectionns). 2. D Draw the faces f of th he pyramiid successsively in the t free fiield of thee drawingg accordingg to their true sizes with com mpass by some s addiitional tickk arcs, so o that theyy have a cirrcular verttex S and adjoin eacch other. 3. Fiinish the drawing d in n accordaance with GOSTs – fold liness have to be drawnn by Long dashed doouble dotteed thin linne. 58

Fig. 7.3.T The unrollinng of a triang gular pyram mid surface

7.2. App proximatee unfoldin ng of cylin ndrical an nd conicall surfaces Appproximate unfolds cy ylindrical and coniccal surfacees Cyliindrical annd conicall surfaces are develo oped in th he same w way as prismatic andd pyramidaal surfacess. In this case, c a cyylindrical surface is replaced (approxim mated) byy an inline polygonaal prismatiic surface (usually 12-angle/faces), annd a conical surfacee is replaceed by an innscribed polygonal p ppyramidall surface. Circcular Cylinnder surfaces. The surface off a right ciircular cyllinder can be unrolleed by the ffollowing methods: m – noormal secttion metho od in the free field of the drrawing, iff a generattrixes is a straight liine of the level, and d the basess are not perpendicu p ular to the generatrix xes; – meethod of roolling undeer the sam me conditions (unrolliing follow ws after pro ojection). The rolling method is used for unnrolling off an ellipticcal cylindeer (normall section – ellipse) iff a generattrixes is a straight liine of the level and there is a circular base b in thee projectionns. Graphhical algoriithms for constructiing surfacce developpment are similar too the abovee graphicaal algorithm ms for connstructing prism unrrolling.

599

8. AXONOMETRIC PROJECTIONS 8.1. Method for axonometric projection A complex drawing as a two-dimensional image is rather simple and easily measured, but it is not representable as a three-dimensional object in space. It is often necessary to have in addition to it a drawing of pictorial view, which may be obtained by projecting an object and its coordinate axes onto one plane. Then, the one projection will provide a visual and metrically distinguished image of the object. Such types of an object representation are called axonometric projections. Depending on the direction of projecting relative to the axonometric projection plane Q, axonometric projections are classified as rectangular (projection angle is equal to 90 degrees  90) and oblique ( 90). Method for axonometric projection consists in the following: a given figure and axes of rectangular co-ordinates to which the figure is related in space are projected in some axonometric projection planes. Axonometric projections as parallel projections have some properties: – axonometric projection of a straight line is also straight; – if lines are parallel on the object, they are also parallel in its axonometric projection; – axonometric projection of a circle in axonometric projection is an ellipse (in general case). Axonometric axes are projections of coordinate axes x, y and z. Therefore the projection direction is not parallel to any axes of the orthogonal coordinate system, true dimensions of the segments in the axonometric projection are distorted. There are coefficients of distortion for determination of distortion degree for each axis. If we take some equal x, y and z coordinates, we shall obtain some distorted coordinates. The ratio of length of the axonometric projection segment to its true size is referred to as distortion coefficients for axes, as reflected in formulas (8.1)–(8.3).

Kx  Kx  Kx 

ex1 ex ex1 ex ex1 ex

;

(8.1)

;

(8.2)

,

(8.3)

where K x , K y , K z – a distortion coefficients for axes x, y and z; ex , e y , ez – x, y and z coordinates;

ex1 , e y1 , ez1 – distortion x, y and z coordinates. 60

They have fractional values (0.87, 0.47 etc.), so there are reduced factors as 1 or 0.5 which are used in drawings. There are next types of axonometric projections according to the mutual ratio of these factors: 1. Isometric – when three factors are equal K x  K x  K x . 2. Dimetric – when only two of three factors are equal K x  K x  K x . 3. Trimetric – when all factors are different K x  K x  K x . There are orthogonal and oblique axonometric projections what depend on an angle of projection ray inclination. Pohlke's theorem is a fundamental theorem of axonometric projection. It was established in 1853 by Karl Wilhelm Pohlke, German painter and teacher of Descriptive Geometry. The first proof of the theorem was published in 1864 by Hermann Amandus: 1. Orthogonal isometric projection. 2. Orthogonal dimetric projection. 3. Oblique frontal dimetric projection. 4. Oblique frontal izometric projection. 5. Oblique horizontal izometric projection. First three types are used more often in the practice of technical drawing. 8.2. Orthogonal isometric projection

By this type of axonometric projections angles among axis lines are equal to 120. A formula (8.4) is a formula for determination of distortion coefficients.

K x2  K 2y  K z2  2.

(8.4)

Therefore K x  K y  K z  0.82. As it has been mentioned above this value should be rounded to the nearest whole number for simplifying work with axonometric. So K x  K y  K z  1. In this case, the obtained representation is enlarged by 1.22. A circle in axonometric projection is generally projected in an ellipse. When constructing an ellipse, it is necessary to know the direction of its axes and their dimensions. Note: a minor axis of an ellipse is always perpendicular to the major one. When a circle projection is constructed (a circle lies in one of the co-ordinate planes), the minor axis of the ellipse is directed parallel to the axonometric axis which does not participate in the formation of the plane the drawing is in. The following example of isometry of a complex body (fig. 8.1).

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Fig. 8.1. An isometrric projectio on of a comp plex body

Therre are sevveral meth hods for ddrawing a circle in isometricc projectio on. All off them are considereed clearly at any stuudy book. The T follow wing exam mple is thee only onee of them. The biggest axis of ellipse iis AB = 1.22d, the smallest aaxis is CE E = 0.71dd where d iis the diam meter of th he given ccircle. Axo onometricc projectioon of a cirrcle whichh lies in thee frontal projection p has the biiggest axis AB perpendicular to the axo onometricc axis y andd the smaallest CE has h the sam me directiion with th his axis y . Axonom metric pro-jection off circle which w lies in the fronntal projeection has the biggeest axis AB B perpen-dicular too the axonnometric axis a y andd the smallest CE with w the ssame direcction withh this axis yy. When a circle beelongs to tthe profilee plane off projectionn ellipse axis a AB iss perpendiccular to thhe axis x and ellipsse axis CE E is combiined with the axis x. x For thee third casee, AB is perpendicu p ular to thee axis z an nd the axis CE has the same vector ass axis z (figg. 8.2).

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Fig. 8.2. An isom metric projeection of a circle c

8.3. Orthogon nal dimettric projection For the orthoggonal dim metric projeection two o factors are a equal aand the th hird one iss A their simplificaation theyy a half of one of theem. So K x  K z  00.94 K y  0.47 . After are: K x  K z  1 and a K y  0.5 0 . The aaxis z is vertical, v the axis x haas angle 7o10’ withh the horizoontal leveel and the axis y hass angle 41o25’. These angles can be sp pecified inn the follow wing way (fig. 8.3).

Fig. 8.3. 8 Angles between axxes for an orrthogonal diimetric projjection 633

A pyyramid wiith some cutting c plaanes was drawn in the orthoggonal dim metric pro-jection ass an exampple (fig. 8.4).

Fig. 8.4. An A orthogon nal dimetricc projection of a straigh ht pentagonaal pyramid withh some secttions

As ffor a circlee in the orrthogonal ddimetric projection p it has all tthree projections ass an ellipsee. Two off them are the samee and the third t one is i differennt. The big ggest axiss AB for alll three elllipses is 1.06d. 1 Forr circles in i the horiizontal annd profile planes off projectionn the smalllest axis CE C has thee same dirrection wiith axes x and z, respectively,, and is equual to 0.355d. For a circle in th the frontall plane of projectionn it is 0.95 5d and thee biggest aaxis AB is perpendiccular to thhe axis y. The biggeest axis annd the sm mallest onee are mutuaally perpeendicular (fig. ( 8.5).

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Fig. 8.5. A an orthogoonal dimetriic projection n of a circlee

8.4. Oblique froontal dim metric projjection As for all diimetric projection distortion ns coefficcients are:: K x  K z  1 andd K y  0.5 (their givven valuess have beeen mentioned abovee). But forr the fronttal type off dimetric projectionn the axis position is specificc. The axiis z is verrtical, the angle be-tween axes z and x is 90o an nd the anggle of the axis a y is 45 4 o from thhe horizon ntal level.. Accordinng to the known k coeefficient annd angle of o axis linees the circcle which lies in thee frontal pllane of proojection iss not distoorted and saves s its true repressentation as a a circlee with the ggiven diam meter (fig.. 8.6). 655

F 8.6. An Fig. n oblique froontal dimetrric projectio on of a circlle

Bothh other projections are the saame and have h the fo ollowing pparameterss: the ma-jor axis A AB is equaal to 1.07d d and the m minor CE E is equal to 0.33d. The majo or axis hass o an inclinaation in 7 14’ to thee axes z aand x for the t projecttion of a ccircle of the t profilee level andd for the prrojection of o a circlee of the ho orizontal leevel, respeectively. According A g to previouus inform mation this kind of ddimetric prrojection is used forr bodies which w con-sist from circles baasically beecause theyy are not distorted d in the fronntal view (fig. 8.7).

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Fig. 8.7. An obliique frontall dimetric prrojection off a complexx body

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APPENDIXES UNIFIED SYSTEM FOR DESIGN DOCUMENTATION (USDD)

“Standardization is the process of formulating and applying rules for an orderly approach to a specific activity for the benefit and with the cooperation of all concerned, and in particular for the promotion of optimum overall economy taking due account of functional conditions and safety requirements.” (ISO – International Organization for Standardization) These Complex rules are called Standard. Standard is a full complex of recommendations relevant to the preparation of technical product specifications. Standards are set of different levels. There are local standards, national standards, regional standards and international standards. Typical examples of the different levels of standards may include: – Local Standards: SUA formats for writing various academic reports; – National Standards: DIN (Deutsches Institut für Normung) (German), BS (UK), GOST (CIS), etc.; – Regional Standards: Standards set for the East African Community, etc.; – International Standards: ISO (International Organization for Standardization) DIN – an internationally accepted German national standard, BS – an internationally accepted British national standard. The Republic of Belarus and countries from The Commonwealth of Independent States (CIS) use Unified System for Design Documentation (USDD). The goal of USDD standards is to adopt unified regulations for completion, presentation and legalisation of engineering papers. This is a set of the State Standards which are called GOST (it is abbreviation from Russian ГОСТ – State Standard). GOST 2.301–68 Formats 2 – class by USDD 3.. group of standard which are relate to design drawings -68 – year when standard was established Formats – The name of GOST GOST 2.301–68 Formats

There are basic and additional formats. This state standard such as the other international standards recommends an area of the largest sheet of paper as one square meter with sides in the ratio of 1:2. This format is called A0. The dimensions of the sheet of paper are 841 mm × 1189 mm. The size of A1 paper is a half the size of A0 while A2 is a half of A1 and so forth. Note that a higher order paper size (which is always smaller in size) is obtained by simply halving the preceding size along its longer side.

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For technical drawingss A4 is cconsidered d to be th he smallesst basic paper size.. Format A A4 is onlyy used in an uprighht position n (fig. A.1). The othher formaats (whichh are lager in size thhan A4) can c be useed in uprig ght (verticcal) positiion or in horizontal h l position.

Fig. A.1. A4 and A33 formats with w their dim mensions Table A.1

Sizes off the basicc formats Formats F A0 A1 A2 A3 A4

mm) Size (m 11898841 5948441 4205994 2974220 2102997

Therre are threee basic componen c nts of Form mats. Thee external frame of format iss the line w which bordders size of o format.. This fram me is used d if sheet oof paper has h biggerr size than standard format f and is drawnn by thin continuous c s line. By another case this iss the bordeer of paperr or it can n be considdered as a cutting liine for takking away y unneces-sary part of paper. The interrnal formaat frame iss a line wh hich is donne with th he follow-ing indennts from thhe externall format fr frame: The border haas to be 20 0 mm widde on the leeft edge. In I this plac ace we are making a hole withh a hole puunch and file f drawinngs togeth her. All the other borders have h to bee 5 mm wide. w (righ ht, upper, bbottom with w 5 mm m spacing. P Please, paay attention n – left wiith 20 mm m spacing). The interrnal framee is drawnn by the thiick continnuous line. 699

Andd the thirdd componeent is the ttitle block k or the baasic inscriiptions. It is a tablee with som me informaation in th he right bbottom corrner of th he format. There is the Statee Standard for it – GOST G 2.10 04–2006 B Basic inscrriptions. GOST T 2.104–2 2006 Basi c inscripttions (Title Block – UK) The basic inscriptions are the reectangular frame thaat is locateed at the right r handd bottom corner of sheet. s It in ncludes seeveral secttions for providing p administrrative andd technical informatiion. This table t has ffollowing size (fig. A.2) and all its item ms, whichh are usuallly filled, are a numbeered and giiven below w.

Fig. A.22. A basic in nscription

1. N Name of thhe object, product, p ppart of macchine. 2. D Document mark. m 3. M Material ussed to man nufacture tthis part. 4. Sccale. 5. N Number off a group (N Name of aan organizzation). 6. Shheet numbber. 7. Total numbber of sheeets. 8. Sttudent’s name n (Draughtsmann’s name) (after ( Разр раб. in russsian). 9. Siignature of o a studen nt (Draughhtsman’s signatures) s ) (only thiis elementt has to bee written byy pen, all the other by b pencil only). 10. D Date of crreation. 11. N Name of the t person n who hass to check k this draw wing (Lect cturer’s Naame, afterr Пров. in Russian. 12. S Signature of a lectu urer (This rrow is filled after ch hecking byy lecturer)). 13. D Date of chhecking (Th his row is filled at th he same tim me with chhecker’s siignature).

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Fig. A.3 3. An exampple of filling g a basic description

GOST 2.302–68 8 Scales The objects which w are encounter e red in day--to-day liffe are usuaally eitherr too largee or too sm mall to be drawn to their true size. For instance, a car or a building,, is drawnn in its truee size if, and a only if, a piecee of paperr is large enough too accomm modate thee true dimeensions off that car or buildinng. But as it was mentioned m d above, th he largestt basic sizee of paperr is A0 (8411189)), which iss a lot sm maller thann these objjects. Thee discrepanncy betweeen the actu ual sizes oof objects and the siize of papeer we use for draw-ing necesssitates uss to preparre drawinggs that are either sm maller or bigger in size thann the actuall objects. This is on nly possiblle through h the use of scales. Scalle is simplly the ratio of the liinear dimeension app pearing onn the draw wing com-pared to tthe correspponding liinear dimeension on the object. The scaale has no units as itt is simplyy a ratio (i.e. dimension on draawing: dim mension on object) Therre are incrrease and reduced r sccales. Table T A.22 Recomm mended scaales for En ngineering g Drawingg Reeduced scaales 1:2 1:2.5 1:4 1:5 1:10 …

Truue size (Reeal) 1:1

Inccrease scales 2:1 2.5::1 4:1 5:1 10:1 …

GOST T 2.303–68 8 Lines Diffferent linees are useed to convvey the neeeded sen nse in Enggineering Drawing.. These linnes can bee of differrent thicknness and style. s Two o line thickknesses are recom-mended ffor drawinng by Statee Standardd. A thick k line and a thin linee in the rattio of 2:1.. (next worrds such as a wide an nd narrow have got the same meaning with thick k and thinn and are uused by UK K standard ds) Statee Standardd recomm mends thickkness S in n the rangee from 0.55 to 1.4 mm m. This iss depends oon the com mplexity and a size o f a drawin ng. Thickn ness S for student’s drawingss is recomm mended ass 1 mm. 71

Table T A.3 Lines andd their chaaracteristicc Typ pe of line

Dessignation

Recomm mended line thicckness [mm m]

Appplications

1. Continuuous thick linne (wide line,, mane line)

m S = 1 mm

1. Visible edgges and outlin nes (visible lin ne) 2. Lines of thee internal fraame 3. Some lines of the title block b

2. Continuuous thin (naarrow) line

nge in the ran from S/3 to S/2

1. 2. 3. 4. 5. 6. 7.

0.5 mm

Dimension lines Extension llines Connectionn lines Reference llines Imaginary ooutlines of in ntersection Short centeer lines Hatching linnes (Shading g)

3. Continuuous irregulaar line (wavyy)

in the ran nge from S/3 to S/2 0.5 mm

1. Limits of ppartial views or sections provided the liine is not an axis

4. Dashedd thin line

in the ran nge from S/2 to S/3 0.5 mm

1. Hidden out utlines and edges e (Line off an invisible counter)

5. Long daash-dot thin line

S/2..S/3

1. 2. 3. 4.

0.5 mm 6. Long daashed doublee dotted thinn line (with double dasshed)

S/2..S/3

7. Continuuous straightt thin line w with zig zags

S/2..S/3

1. Bending liines (Folding g edges in deevelopments))

0.5 mm

0.5 mm 8. Open liine

Center linees. Lines of syymmetry Pitch circlee for gears Pitch circlee for holes

1.5 S

1. Limits of partial or interrupted vieews; 2. Suitable foor CAD draw wings provid ded the line iis not an axis 2. Noting of ccutting planes

1.5 mm

GOST 2..304–81 L Letters forr drawing gs (Fonts)) Therre are fouur types off fonts: tw wo italic and a two reegular. Buut only on ne type off font shouuld be usedd in case of o the one drawing. Fourr font typees: 1. Type: Reguular A (d = h/14). 2. Type: Italicc A, with angle a to 7 5о (d = h/14). 3. Type: Reguular B (d = h/10). 4. Type: Italicc B, with angle a to 755о (d = h/1 10). 72

Fig g. A.4. Gridds for differrent font typ pes: a – for regularr types; b – for italic ty ypes

For capital letters l (fiig. A.5) g = 6d o r 0.6h. Except E foor: I – 1d d; J – 4d; C, E, F, L – 5d; A A, M, Q, V, X, Y – 7d; W – 9d.

Fig. A.5. Some chharacteristiccs of capital letters: a–w width; b – hight h

For lowercasee letters (fig. ( A.5)) g = 5d. Except for: f I – 1 d; l – 2d d; j – 3d;; c, f, r, t – 4d; a – 6dd; w, m – 7d.

Fig. A.5. Some charracteristics of lowercasse letters: a – hight; b – thiknesss; c – width

The followingg table (tab ble A.4) ccontains paarameters of letters (options) as height,, space bettween letteers, etc. Table T A.44 Param meters of letters l Parameeter (Typees B) Uppercaase height Lowercaase heightt Letter sppacing Minimum m word sppacing Thickneess

Desiggnation h c a e d

Size 0.7h 0.2h 0.6h 0.1h

5 5 3.5 1 3 0.5

7 7 5 1.4 4.2 0.7

10 10 7 2 6 1 733

REFERENCE

1. Belyakova, Е. I., Descriptive geometry. Short course for the practical classes: student book / Е. I. Belyakova, P. V. Zelen; cor. P. V. Zeleni. – Minsk: BNTU, 2010. – 229 p. (In Russian). 2. Bogolyubov, S. K. Engineering Drawing / S. K. Bogolyubov, A. Voinov; transl. from the Russian by Leonid Levant. – Moscow: Mir 1968. – 352 p. 3. Descriptive geometry [Electronic resource]: Server provides a place for members of the Carnegie Mellon community to publish course. – Mode of access: https://www.andrew.cmu.edu/user/ramesh/teaching. – Date of access: 20.12.2019. 4. Design Handbook: Engineering Drawing and Sketching [Electronic resource]: Massachusetts Institute of Technology. – Mode of access: https://ocw.mit.edu/courses/mechanical-engineering/2-007-design-and-manufacturing-i-spring-2009/relatedresources/drawing_and_sketching/ – Date of access: 20.12.2019. 5. Engineering Graphics.Textbook. [Electronic resource] / Department of Engineering Graphics and Industrial Design. – Tomsk: Tomsk Polytechnic University, 2014. – Mode of access: http://portal.tpu.ru:7777/SHARED/d/DOLOTOVA/in_st/archiv_1/1.pdf. – Date of access: 20.12.2019. 6. Gordon, F. Engineering graphics and descriptive geometry in 3-d / F. Gordon. – Canada: Macmillan Company of Canada, 1977. – 368 p. 7. Kasperov, G. I. Engineering and computer drawing: Texts of lectures for students / G. I. Kasperov, A. L. Kaltygin and V. I. Gil. – Minsk: BSTU, 2014. – 74 p. 8. Krylov, N. Descriptive geometry / N. Krylov, P. Lobandievsky, S. Men; transl. from the Russian by G. Yankovsky, Moscow: Mir, 1974. – 383 p.: ill. 9. Kutepov, A. Problems in Geometry / A. Kutepov, A. Rubanov; transl. from the Russian by Oleg Meshkov; Moscow: Mir publishers, 1978. – 208 p. 10. Maguire, D. Engineering Drawing from First Principles / D. Maguire. – Oxford: Butterworth-Heinemann, 2012. – 268 p. 11. Simmons, C. Manual of Engineering Drawing / C. Simmons, D. Maguire, N. Phelps. – 5th edition. – Oxford: Butterworth-Heinemann, 2019. – 638 p. 12. Strategic action plan on implementation of the major objectives of the education system development in line with the EHEA principles and tools: approved by Ministry of Education 01.06.2018 [Electronic recourse]. – Mode of access: https://edu.gov.by/sistema-obrazovaniya/glavnoe-upravlenie-professionalnogo-obrazovaniya/vysshee-obrazovanie/strategicheskiy-plan-deystviy/strategic%20plan.pdf. – Date of access: 10.05.2020. 13. Walsh, C. J. Engineering drawing and descriptive geometry / C. J. Walsh. – Cambridge: Harvard Univ. Press, 2014. – 247 p. 14. Projection drawing: student book for the practical classes of a discipline “Descriptive geometry. Engineering drawing”/ P. V. Zeleni [and oth.]. – Minsk: BSPA, 2002. – 61 p. (In Russian). 15. Zeleni, P. V. Engineering drawing. Practical workbook for the projection drawing: student book for the students of university with technical specifications / P. V. Zeleni and Е. I. Belyakova; cor. P. V. Zeleni; Belorussian national technical 74

university, department of "Engineering mechanical drawing". – Minsk: BNTU, 2014. – 199 p. (In Russian). 16. Zhurik, Iran. A trilateral cooperation agreement with AIC Campus in Sri Lanka and the ESIGELEC Higher Engineering School in France were signed at BNTU. [Electronic recourse] / I. Zhurik. News portal of BNTU – Times. BNTU. – 2020. – Mode of access: https://times.bntu.by/events/575-trehstoronnee-soglashenie#leftmenu. – Date of access: 10.05.2020.

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Учебное издание ХМЕЛЬНИЦКАЯ Людмила Васильевна МАТЮШЕНЕЦ Тимур Владимирович ЛЕШКЕВИЧ Александр Юрьевич

НАЧЕРТАТЕЛЬНАЯ ГЕОМЕТРИЯ Учебное пособие На английском языке Редактор В. И. Акулёнок Компьютерная верстка Н. А. Школьниковой Подписано в печать 01.02.2021. Формат 6084 1/8. Бумага офсетная. Цифровая печать. Усл. печ. л. 8,84. Уч.-изд. л. 3,45. Тираж 100. Заказ 773. Издатель и полиграфическое исполнение: Белорусский национальный технический университет. Свидетельство о государственной регистрации издателя, изготовителя, распространителя печатных изданий № 1/173 от 12.02.2014. Пр. Независимости, 65. 220013, г. Минск.

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