Engineering and Computer Drawing

Home , Projection plane

Educational Institution «BELARUSIAN STATE TECHNOLOGICAL UNIVERSITY»

G. I. Kasperov, A. L. Kaltygin, V. I. Gil

ENGINEERING AND COMPUTER DRAWING

Texts of lectures for students majoring in 1-40 01 02-03 «Information Systems and Technologies»

Мinsk 2014

73 Учреждение образования «БЕЛОРУССКИЙ ГОСУДАРСТВЕННЫЙ ТЕХНОЛОГИЧЕСКИЙ УНИВЕРСИТЕТ»

Г. И. Касперов, А. Л. Калтыгин, В. И. Гиль

ИНЖЕНЕРНАЯ И МАШИННАЯ ГРАФИКА

Тексты лекций для студентов специальности 1-40 01 02-03 «Информационные системы и технологии»

Минск 2014

2 УДК [004.92+744](075.8)=111 ББК 30.11я73 K22

Reviewed and approved by the publishing board of Belarusian State Technological University.

Peer-review by: S. E. Belsky, head of the department of machine elements and hoisting and conveying equipment, educational institution «Belarusian State Technological University» PhD (Engineering), assistant professor; V. A. Stoler, head of the department of engineering graphics,educational institution «Belarusian State University of Informatics and Radio Electronics» PhD (Engineering), assistant professor

Kasperov, G. I. K22 Engineering and computer graphics : texts of lectures for students of Information Systems and Technologies programme» / G. I. Kas- perov, А. L. Kaltygin, V. I. Gil. – Мinsк : BSTU, 2014. – 76 p.

In the texts of lectures in accordance with the programs outlined projection method, allowing to build the image of spatial geometric images on the plane, considered how to solve basic problems in the drawing and right images on the drawing details. Examples are given in order to facilitate independent graphic works by students.

УДК [004.92+744](075.8)=111 ББК 30.11я73

3 INTRODUCTION

Engineering and computer graphics are among the disciplines that form the basis of overall engineering training of specialists. However, amount of hours to study this discipline is different for BSTU departments. Textbooks and methodical literature are developed and published for departments scheduled with maximum amount of training hours, thus making it difficult for non-technical departments’ students to study the discipline. The theoretical basis of the engineering and computer graphics is a descriptive geometry, which once allowed creating one of the most genial inventions of the human mind - the drawing. The drawing is a kind of graphic language. With the help of just points, lines, geometrical signs, letters and numbers a variety of surfaces, machines, appa- ratus, engineering structures are pictured. This language is international and can be understood by any technically trained person whatever language he or she speaks. The role of descriptive geometry is important in the process of studying natural sciences, when studied or analyzed properties are accompanied by accessible to the human perception visual geometric models, which allow developing logical thinking. The lectures ‘Engineering and machine graphics’ have been written for training the students of the Faculty of Economic Engineering and the Faculty of Publishing and Printing. Each of the lectures (nine in total) is a separate chapter of the descriptive geometry with necessary theoretical and engineering support. This manual will help students of different forms of training to learn the basics of descriptive geometry and the projection of the drawing, to create the foundation of knowledge of engineering from these disciplines.

4 Lecture 1. THE BASIC PRINCIPLES OF THE ORTHOGONAL PROJECTION

The Subject and the Method of Descriptive Geometry

Descriptive geometry is one of the branches of geometry. Its objective is the same as of geometry in general, namely: studying the forms of objects around us and the relationship between them, the establishment of ap- propriate laws and the use of them in solving specific tasks. Descriptive geometry is highlighted by using a graphical way in which the geometric properties of figures are studied directly by drawing to make decisions in general geometrical tasks. In other branches of geometry drawing is an auxiliary means (it is drawing that makes it possible to illustrate the properties of figures). Certain geometrical laws are to be used to bring a drawing to a geo- metrically equivalent image of the object (figure). In descriptive geometry the drawing is built with the use of projection, so the drawings used in descriptive geometry, are called projective drawings. Thus, the descriptive geometry comprises: – the objective of the methods of construction of projective drawings; – the solution of geometric tasks, related to three-dimensional shapes; – the application of methods of descriptive geometry to the studying of theoretical and practical issues of science and technology.

Brief history of the descriptive geometry

Descriptive geometry arose from the needs of the practical activities of mankind. The construction models of various facilities, fortress fortifica- tions, habitation, temples, and so on were required to be drawn firstly. From primitive paintings, transmitted approximate geometric forms of structures the transition to the compilation of projective drawings was gradually accomplished, reflecting the geometric properties objects de- picted on them. French geometry and engineer Gaspard Monge played an outstand- ing role in the development of descriptive geometry. In his work «De- scriptive geometry», published in 1798, Str. Monge gave the first scien- tific report on principles of representing three-dimensional objects in a two-dimensional plane.

5 Descriptive geometry course was firstly taught in Russia by French engineer K. Pottier (former student of G. Monge) in St. Petersburg in 1810. He published the Descriptive Geometry Course in French in 1816. From 1818 the descriptive geometry course teaching was continued by Professor Yakov Alexandrovich Sevastianov. He translated K. Pottier’s Descriptive Geometry Work in Russian. In 1821 Professor Sevastyanov wrote his own course of lectures on descriptive geometry. Followers of Professor Sevastyanov – Makarova N. I., Kurdyu- mov V. I., Fedorova E. S., Chetvertuhina N. F., Gordon V. O. and others – made great contributions into descriptive geometry teaching development in Russia.

Legend

1. Points of space are marked with Latin capital letters: A, B, C, D... or numbers: 1, 2, 3... . 2. Straight and curved lines of the space are marked with lowercase Latin letters: a, b, c, d... . 3. Plane and the surface are marked with Latin capital letters: P, Q, F, V, W... . 4. Plane of projection and the field of projections are marked with π (lowercase letter of the Greek alphabet). 5. In the formation of complex drawing plane projections and the field of projections are marked with letter π with the addition of a sub- script 1, 2, 3, 4...: – horizontal plane projections are marked with π1; – front plane of projection – with π2; – profile plane projections – with π3. The new plane projections (different from indicated above) are marked with π4, π5, π6... . 6. The projection of the points, lines and planes are marked with the same letters like their originals with the additional index and the corresponding index of plane projections. Thus, the projection of point A, line a and plane Q is respectively marked: – on plane π1 – A′a′Q′; – on plane π2 – A″a″Q″; – on plane π3 – А′′′, а′′′, Q′′′.

6 7. To specify the method of the task of plane next to a letter of plane designations of those elements by which they are set are written in paren- theses, for example: Q (А, В, С), P (а // b), V (m × n). 8. For some lines and planes the permanent marks are developed Depending on line position in space: – horizontal – h; – frontal – f; – profile – p. Depending on plane position in space: – horizontal – H; – frontal – F; – profile – P. 9. Angles are marked the following lowercase letters: α, β, γ, δ… 10. Basic operations are marked: the coincidence of two geometric elements – ≡; membership of a geometry element to another – or ; the intersection of two elements – ×; the result of the geometric operations – =. 11. Plane projections: – horizontal – Ph; – frontal – Pv; – profile – Pw.

The basic properties of projection

The central projection (vista) is to build an image (projection) point A′ of the point by conducting through the point A and point S (the center of projections) line SA, called projective straight up to the intersection with plane π1, called plane of projections (Fig. 1). The method of the central projection of points of space on plane projections π1, can be written using the following symbolic equality: А′ = π1 × SА, А′ – the point of intersection of plane π1 with the direct SА. Fig. 1 shows the construction of the projections of points A, B, C and D varies located on plane of projection π1 and Fig. 1 the center of the project tion S.

7 Projection can be run for any point of the space, except for the points lying in plane passing through the center of the projections and parallel to plane projections π1 (non-proprietary point). The depiction of the objects with the help of the central projection has great visibility, but it significantly distorts the shape and dimensions of the original, so as it doesn’t remain parallel direct and rela- tions segments. Therefore, in practice the method of parallel projection (in particular, the orthogonal projection) is often used. Parallel projection assumes a given plane projections π1 and the direction of the projection S, not parallel to plane of projection (Fig. 2). In the building of any point A in the projection A′ it is necessary to carry through the point and projective line parallel to the direction of projection S, up to the intersection with plane π1.

The basic properties of parallel projection

The projection of a point is a point. The projection of a straight line is a straight line. All direct, that project points A, B, C to the line l (Fig. 2) lie in the same plane (called to projective plane)passing through the straight line l and a parallel to the direction of the projection S. This plane intersects plane of projection π1 on the straight line, which, according to the determination of the projection of figures as sum of total projections of all its points, is a projection of a line l. We will call these properties the properties of straightness. The projection of a point that lies on a straight line, is a point, which lies on the projection of a given straight line. This property, called the property of accessories, immediately follows from the determination of the projection figures as a sum of the projections of all points. The considered three properties have a place in the central projection. The projections of parallel lines are parallel straight lines. If direct l and m are parallel, and their projected planes will be parallel as containing a pair of in- Fig. 2

8 tersecting respectively parallel lines (l // m, AA′ // MM′) (Fig. 2). It follows that l′ // m′ as direct intersection of the parallel planes by the third plane. This property is called the saving of preservation of parallelism. The attitude of the projection of segments, lying on parallel lines or in one and the same straight line, is equal to the attitude segments. Let AB and MN – segments, lying on the parallel lines l and m, and

А′В′ and M′N′ – their projection on plane π1 (Fig. 2). ∗ Let’s build segments in projective planes АВ and MN* equally paral- ∗ lel to the segments А′В′ and M′N′. It is obvious that the triangles АВВ and ∗ MNN are similar, as well as their equal sides are parallel (А′В′ / M′N′ = ∗ ∗ АВ / MN = АВ / MN). The projection of the figures does not change in the parallel transfer of planes projections. We will take the triangle ABC as a projective figure and will project it in the direc- Fig. 3 tion to S on plane

π1 and π1, parallel to each other (Fig. 3). As segments А′ A′, В′ B′, С′C ′ parallel and equal to each other, then the quads А′В′ B′ A′, В′С′C ′ B′ and С′А′ A′ C ′ are parallelograms. Therefore, triangles А′В′С′ and A′ B′ C ′ have equal sides, the same size and, therefore, these triangles are equal among themselves. The orthogonal projection is a special case of parallel projection, when the direction of projection S is perpendicular to plane of projections π1, it simplifies the construction of

Fig.4 the drawing (Fig. 4).

9 In the orthogonal projection it is not difficult to establish a correla- tion between the length of the natural length of AB and the length of its projection А′В′: А′В′ = АВ · cosα. The orthogonal projection has received the greatest application in technical drawings. The considered methods of projection allow you to solve the direct problem uniquely, to build the projection drawing of the original, and not allow solving the inverse task – to reproduce the original from the only projective drawing.

Comprehensive drawing of the point

The greatest application in technical practice a drawing composed of two or more interconnected orthogonal projections of the reflections of the original received. Such drawing is called a complex drawing in the orthogonal projections or a complex drawing. The principle of the formation of such drawing is that the original is being projected orthogonal to the two mutually perpendicular to plane of the projections that will be combined after with plane of the drawing. One of planes projections π1 is horizontal and is called the horizontal plane of the projections, the other π2 – vertical and is called the front (Fig. 5). The direct line of intersection of planes is called the axis of projections and is indicated by the letter x. Let’s project some point A or- thogonally to planes π1 and π2 and we will get two projections of it: А′ – the horizontal to plane π1 and A″ – the frontal to plane π2. Projective directs АА′ and АА″ determine the projective plane АА′Аx А″, perpendicular to both planes of projections and to ′ Fig. 5 the x axis. Directs Аx А and Аx А″, which are projections to projective plane on plane of projections π1 and π2, are also perpendicular to the x-axis. To get a flat drawing, let’s compatible plane π1 with plane π2, rotate plane π1 around the x-axis in the direction shown in Fig. 5. As a result we obtain a complex drawing (Fig. 6), consisted of two projections А′ and А″ point Аx, lying on the same straight line.

10 The images obtained with a combination of planes of projections to a plane of the drawing, is called a diagram (from the French word epure – drawing). There is a distance from the point A to plane π1 on the diagram А″Аx, А′Аx – the distance from the point A to plane π2 that is the witness confirming that the projection of a point on two mutually perpendicular planes of projections are determining its position in space. Fig. 7 shows the direct of the general position of l we can find the projection of the line l (l′ and l″) using the projections Fig. 6 of points A and B, that are lying on it. So on the complex drawing (Fig. 8), any direct l can be specified by projections of points А′, А″, В′ and В″, that belong to it. However, any parallel projection has the properties of straightness and accessories, the direct l on a complex drawing can be set by its projections l′ and l″, passing through the points А′, В′ and А″, В″.

Fig. 7 Fig. 8

The projections of the rising direct are oriented in the same way on the complex drawing, and downward – the opposite. For the division of the segment AB in the given ratio it is enough to divide one of the projections of the segment in this ratio, and then to project sharing points to the other projection of the segment.

11 In Fig. 9 for the division of the segment AB in the ratio

of 2 : 5, the random line А′В0 is carried out, on which seven equal segments are set. The end point of the line В0 is connected with the projection В′,

and from the point М0′ which is situated in two divisions from the point А′, the direct М М′ is set, which is parallel Fig. 9 0 to the segment В′В0 . Let’s put a vertical line of the connection to the intersection with the projection А″В″ and find the projection M″.

A complex drawing consisting of the three orthogonal projections

A complex drawing consisting of two projections, is a reversible drawing, you can reproduce the original from this drawing. However, the reproduction of the original, which has profiled elements, and in particular the profile directs or planes, becomes easier, when in addition to the two main projections there is another projection on the third plane. In the role of such plane of the projections plane perpendicular to both main planes π1 and π2 is applied, called a profile plane projection; it is designated as π3 (Fig. 10). The line (x-axis) of the intersection of planes of projections π1 and π2 is called the abscissa, of planes π1 and π3 (y-axis) – ordinate, of planes π2 and π3 (z-axis) – applicate axis. In fig. 10 some point A that is held in the space showed and its projections on planes of projections π1 (А′), π2 (А″) and π3 (А′′′). The point А′′′ is called the profile projection of the point A. After combining plane projections with the turning of planes π1 and π3 at an angle of 90 degrees, we will get a diagram of the point A in the system π1, π2 and π3 (Fig. 11). The y-axis as if it is splitting: one part of it with plane π1 falls down, and the other with plane π3 goes to the right. You should pay attention to the fact that the frontal and horizontal projections on the diagram always lie on the same perpendicular to the

12 axis of x (of the communication line А′А″), frontal and profile projections of the point – are on the same perpendicular to the axis z (the line of the connection А″А′′′), at the same time the projection of the point А′′′ is placed at the same distance from the z-axis, as the projection of the point А′ from the x-axis.

Fig. 10 Fig. 11

While building a profile projection of the point you can use the constant of the reflection (the Monge’s constant, Fig. 11), which is the bisec- tor of the right angle and is inclined to the vertical and horizontal lines of a projection of connection at an angle of 45 degrees.

The system of rectangular coordinates

The position of the point in space can be determined also with the help of its rectangular (Cartesian) coordinates. The coordinates of the point – it is numbers, that express the distance from it to planes of the projections, called planes of coordinates. We know coordinates of the point (X, Y, Z) , we can construct the diagram of the point on the specified coordinates by taking the axis coordinate as if it is the axis of the projections (Fig. 11). From the beginning of coordinates we can set the X coordinate (positive – to the left, negative – to the right). Through the set point Аx the vertical line of the communication can be set, on which the Y coordinate lies (the positive – down, the negative – up), and the horizontal projection of the point А′ can be determined, and then the Z coordinate (the positive – up, the negative – down) and the frontal projection of the point A″ can be found. The profile projection of the point А′′′ can be found with the use of the connection line and the Monge’s constant.

13 Methods of setting of plane on a drawing

Plane on the diagram can be set by projections of geometric elements, determining it (Fig. 12): – by projections of three points (А, В, С), which don’t lie on the same direct; – by projections of the direct (EF) and the point (D), which don’t not belong to a given direct; – by intersecting straights (m, n); – by two parallel straights (k, l); – by shows of plane Ph and Pv.

Fig. 12

The location of the direct and plane comparatively to the projections of planes

Directs and planes, that are inclined to all major planes of the projections (π1, π2, π3), are called directs and planes of the general position. Fig. 13 shows an example of the diagram of the general position, and Fig. 14 and 15 – planes of the general position, that are set by the triangle ABC.

Fig. 13 Fig. 14

Directs and planes, that are perpendicular or parallel to plane of projections, are called directs and planes of the private position. Directs and planes of the private provision are divided into the projective direct and planes that are perpendicular to plane of projection, and into directs and planes of the level, that are parallel to plane of projections.

Fig. 15 Fig. 16

The straight that is perpendicular to the horizontal plane of projections π1, is called the horizontally projective direct (Fig. 16). It projects all its points on the horizontal plane of the projections in one point, this point is its horizontal projection. The frontal and the profile projections of the direct are parallel to the axis of the applicate z. The segment AB of the horizontally to projective direct is parallel to planes π2 and π3 and is being projected on plane without a distortion (АВ = А′В′ = А′′′В′′′).

15 The straight that is perpendicular to the frontal plane of projections π2, is called the front projective (Fig. 17). The frontal projection of this line is being projected in the point, and the horizontal and the profile projection are parallel to the y-axis and are being projected on planes π1 and π3 without a distortion (АВ = = А′В′ = А′′′В′′′). Fig. 17 The straight, that is perpendicular to the profile plane of the projections π3, is being projected on this plane in the form of the point and is called the profile projective direct (Fig. 18). The horizontal and the frontal projections of this direct are located parallel to the x-axis and are being projected on planes π1 and π2 without a distortion (АВ = Fig. 18 = А′В′ = А″В″). Plane that is perpendicular to the horizontal plane of the projections, is called the horizontally projective plane (Fig. 19). This plane projects all its points on the horizontal plane of the projections in one direct line, which is its projection. The angles β and γ, which are formed by the horizontal projection of the horizontal projective plane with the horizontal and vertical directs of the level, determine the inclina- Fig. 19 tion to planes π2 and π3.

16 The plane that is perpendicular to the frontal plane of projections is called the frontally projective plane (Fig. 20). The frontal projection of the used plane is a direct line and angles α and γ determine the inclination of plane to the planes of projection π1 and π3. The plane that is perpendicular to the profile Fig. 20 plane of the projections is called the profile projective plane (Fig. 21). The profile projection of plane is a straight line and the angles α and β determine the inclination of plane to plane of projections π1 and π2.

Fig. 21 Fig. 22

The direct parallel to any plane of projections, is called the direct of the level. The direct of the level h parallel to the horizontal plane of projection π1 (Fig. 22), is called the horizontal. The direct level f, parallel to the frontal plane π2, is called the frontal (Fig. 23). The profile of the direct р is also a direct of the level in ratio to plane of the projection π3, to which it is parallel (Fig. 24).

Fig. 23 Fig. 24

The frontal and the profile projections of the horizontal on the complex drawing are equal with one and the same horizontal line of the connection. The horizontal and the profile projections of the frontal on the complex drawing are perpendicular respectively to the vertical and horizontal lines of the connection. The horizontal and the frontal projections coincide with one and the same vertical line on the profile direct of the level p. Let’s note that projective directs can also be directs of the level. So, the horizontal projected direct is at the same time the frontal and the profile direct, the frontally projecting direct – is the horizontal and the profile direct, and profile projecting – is the horizontal and the frontal. Directs of the level are being projected without a distortion on the plane of projections parallel to them. That’s why the horizontals on plane of projection π1 are not distorted, on plane π2 – frontals, and on plane π3 – profile directs.

Simultaneously with it on field π1 angles β and γ of the inclination of the horizontal to planes of projections π2 and π3 can be measured, in the field of π2 – angles α and γ of the inclination of the frontal to planes π1 and π3, and the field of π3 – angles α and β of the inclination of the profile direct to plane of projections π1 and π2. You can set countless number of horizontal planes, frontals and profile directs in plane of the general position, while all the horizontals will be parallel to each other as frontals and profile directs will be parallel to each other too. Plane parallel to any plane of projections, is called plane of the level, as all the points of plane are equally removed from plane of the projections.

Fig. 25 Fig. 26

The plane ABC, that is set by the triangle and is parallel to the horizontal plane of projections π1, is called the horizontal plane of the level (Fig. 25). The same plane that is parallel to the frontal plane of the projections π2, is called the frontal plane of the level (Fig. 26). The plane that is parallel to the profile plane π3, is called the profile plane of the level (Fig. 27). Each plane of the level is a project plane at the same time. For example, a horizontal plane is both the frontal and profile projective plane, the frontal plane of the level is horizontally and profile projective and the profile of plane of the level is horizontally and frontally projective plane. Planes of the level on a complex drawing are made by one follow: the horizontal – by the frontal; the frontal – by the horizontal; the profile – by the horizontal or by the frontal. Fig. 27 All the figures that are lying in plane level are projected without distortion on plane, to which they are parallel.

19 Lecture 2. DETERMINING THE ACTUAL SIZE OF A LINE SEGMENT

Traces a straight line

Following a straight line called the point of intersection of this line with the plane of projection. To find the trail of the front of the line AB (Fig. 28), it is necessary to continue its horizontal projection of the А′В′ to the intersection with the x-axis at the point N′, and then from the point of N′ draw a vertical line of communication to the intersection with a line which is a continua- tion front projection A"B" the line AB at point N". Point N is the front track of the line AB. To find the horizontal trace is necessary to continue the frontal projection of A"B" to the intersection with the x-axis at point M". From the point M" is held a projection of the А′В′ line AB at point M'. The point M is the horizontal trace of the line AB. To build the traces of the line AB on the orthographic (Fig. 29), you must do the same construction (Fig. 28) to extend the horizontal projection of the straight line to the intersection with the x-axis at point N. From the point N′ restore perpendicular to the junction with the frontal projection of the line at point N″.

Fig. 28 Fig. 29

Similarly, for the construction of the horizontal trace line АВ must continue to the intersection with the x-axis is not the frontal projection (point M″). From the point of intersection perpendicular to restore to the intersection with the extension of the horizontal projection of the line (point M′).

20 Keep in mind that a line parallel to a plane of projection, the trace on that plane can not have, because it does not intersect this plane.

Traces of the plane

The straight, in which the plane P intersects the plane of projection, is called the trace of the plane (Fig. 30). Fig. 31 is a diagram in planes P, π1, π2, π3, which confirms the possibility of any reference plane on the orthographic its tracks. It should be borne in mind that the front frontal projection plane coincides with the track by the track, while the horizontal projection of the – with the axis x. Similarly, the horizontal projection of the horizontal trace coincides with itself after it, and a front view of a horizontal track – with the axis of x.

Fig. 30 Fig. 31

Direct and to the point of the plane

The direct belongs to the plane in the following cases: – if it passes through two points belonging to a given plane; – if it passes through the point belonging to a given plane, and is parallel to a line which is in the plane or parallel to it. The direct belongs to the plane defined by the following if: a) direct traces of the same name are on the next plane (Fig. 32, 33);

Fig. 32 Fig. 33

21 b) one is parallel to the plane of the tracks and the other track has a common point (Fig. 34, 35).

Fig. 34 Fig. 35

A line in the plane can be constructed not only when the plane is set tracks, but also any other form of a reference plane. As an example, Fig. 36 is built directly into the plane defined by two parallel lines m and n. Initial- ly perform arbitrary horizontal or front view of the line AB, which crosses the parallel lines at points 1 and 2. If you hold a straight front view, we get projections of 1" and 2" in which we find the horizontal projections of points 1′ and 2′, and through them to draw a horizontal projection of the line AB. If construction begins with a horizontal projection of a line, then over the projections of points 1′ and 2′ projections are 1" and 2". The point belongs to the plane if it lies on the line belonging to this plane. To test that the point belongs to a plane through one of its projections hold a straight line belonging to a given plane, and then the above methods based projection of the second line. If the second point of view is on the newly constructed projection line, then it belongs

to the plane. Otherwise, the point does Fig. 36 not belong to a predetermined plane.

Determining the actual size of a line segment

The need to determine the actual values of line segments is faced in most of the metrical problems. The actual size of the segment can be defined in the following ways: – method of a right triangle; – rotation about an axis perpendicular to the plane of projection; – change the projection planes.

22 The method of a right triangle. The distance between two points A and B is determined by the length of a line segment, concluded between these points (Fig. 37). From the known properties of the rectangular projection, the projection will be equal to the original length only when it is parallel to the projection plane: Fig. 37

([АВ] // π1) ↔ [А′В′] = [АВ]; ([АВ ] // π2) ↔ [А′′В′′] = [АВ]. In all other cases, the segment is projected onto the plane with distortion. The projection interval is always less than its length. To establish the relationship between the length of a line segment and the length of its projection, we consider Fig. 37. In a right-angled tetrahedron АВВ′А′ (angles at А′ and В′ lines) are the sides of the segment AB and its horizontal projection of the А′В′ and the bases – sections АА′ and ВВ′, the magnitude of the equal distance all A and B segment of the H plane π1. ∗ Draw a rectangular plane tetrahedron АВВ′А′ through A line АА , parallel to the horizontal projection of the segment A'B'. Obtain a right triangle AA*B, whose leg AA* is equal to А′В′, and the leg is equal to the difference between ВА∗ applicate endpoints: [ВА*] = [ВВ′] – [АА′]. The hypotenuse of the triangle is equal to the length of the segment AB: [АВ]2 = [А′В′]2 + ([ВВ′] – [АА′])2 . The relationship between the length of the segment and its frontal projection can be set using the triangle ABB, in which the hypotenuse is equal to the length of the segment, one of the other two sides – front projection of the segment, and the other – the difference between the remote end of the line from the front plane of projections: [АВ]2 = [В″А″]2 + ([АА″] – [ВВ″])2.

23 Fig. 38 line segment AB is shown in orthographic projections of AB and A "B". To determine the actual size of each segment a point (A or B) is carried out at any projection lines perpendicular to the latter and are deposited intervals Δz (if the construction is performed on a horizontal projection) or Δy (if the construction is a front view). Fig. 38 shows lines are drawn from the projections of the points А′ and B".

А0В′ and А″В0 – the actual values of Fig. 38 the segment AB. A method of rotation around an axis perpendicular to the plane of projection is that the given point, line or plane figure is disposed in front of projection plane π1, π2 and π3, rotating around an axis perpendicular to a plane of projection to a desired position relative to one of them. When rotating every point will move along the corresponding trajectory. Consider the rotation of the simplest geometric elements - point A (Fig. 39). The axis of rotation L is perpendicular to plane π1. When rotated around the axis point A moves along a circle lying in a plane perpendicular to the axis of rotation. The intersection of this plane with the axis of rotation of said center of rotation.

Since the circle on which Fig. 39 the moving point A is located in a plane parallel to plane π1, then the horizontal projection of this circle is its real view and a front view – a line segment parallel to axis x (Fig. 40). The rotation around the axis of a line segment perpendicular to the plane of projection can be viewed as two rotation points of the segment. Building on the complex drawing is simplified if the axis of rotation is pushed through any endpoint of the rotating line segment. In this case, it suffices to return only a single point of the line, as another point situated on the axis of rotation remains stationary.

24 Let it be required to determine the method of rotation of the actual length of the segment AB line in general position (Fig. 41). Through the end of the segment A we draw the axis of rotation l, perpendicular to the plane π1. Rotate around this axis of the second end of the segment – point B. To get the actual length of the segment to the complex drawing, you have to turn it so that it is parallel to plane π2. After rotating the horizontal projection of the segment occupies a position parallel to the axis of x. ′ ′ ′ From point А radius А В describe an arc of a circle to its intersection with a line Fig. 40 drawn from point А′ parallel to the axis of x. The point of intersection B′ – new horizontal projection of the point B. Front view of B′′ is found by the projection lines of communication, drawn from the point B′ , at its intersection with a line drawn from point В′′ parallel to the axis of x. Connecting points А″ and B′′ we get the actual length of the segment AB. The way to change the projection planes. On a complex drawing (Fig. 42) new plane π4 is conducted in parallel to the horizontal projection of the segment А′В′, or it can be undertaken in parallel to the front projection А′′В′′. Plane π4 in the intersection with the plane of projection π1 form a new axis projections x1. From the points А′ and В′ a new line connection projection perpendicular to the axis x1 will be drawn, which crosses the latter in the points Ax1 and Bх1. From the data points on the lines held postpone cuts АxА′′ and ВxВ′′ (see front view) and get points АIV and ВIV. АIVВIV segment is the actual value of the segment AB.

Fig. 41 Fig. 42

25 Lecture 3. SURFACE

Surface. Methods of specifying the surface

Surface – this is the set of all the successive positions of the moving line. This line, called the generator, the motion may retain or change their shape. The motion generator may be subject to a law or is arbitrary. In the first case, the surface will be legitimate, and in the second – random (irregularity). The law of motion generator is usually determined by other lines, called the guide on which a generator slides in its motion, and the motion of the character generator. For example, the surface Q in Fig. 43 – the surface formed by moving the l forming the fixed guide lines m. In some cases, one of the guides can be converted to a point (vertex conical surface), or in the infinity (cylindrical surface).

Fig. 43 Fig. 44

One and the same surface can be produced in various ways. For example, the cylindrical surface (Fig. 44) can be obtained as a result of movement of rectilinear generator l on a curve parallel to the guide to some preassigned position (axis O1, O2), or the movement of the guide curve m on a straight-line generator. There may be other methods of forming a cylindrical surface. In practice, of all the possible ways to form a surface for the main take the most simple. For example, a cylindrical surface for forming accepts a straight line.

26 As a result, for each surface it is necessary to know some set of data that uniquely define it. The data include both geometric surface features (shape, the shape of the guide) and the law of displacement generatrix. The set of geometric elements for defining the surface is called the determinant of the surface, given that the law of displacement determines the name of the surface. Depending on the shape of the generator and the law of its motion in space of a surface can be divided into the following groups: – ruled – the surface, the image of which is a straight line; – nonruled – forming a curved surface; – a surface of revolution formed by rotating an arbitrary generatrix around a fixed axis; – a surface formed by reciprocating manner, for example, prismatic and cylindrical surfaces also called migration. Is called translational motion in which all the points of the move object moving parallel to a given direction and with the same speed.

Sided surfaces and polyhedra

Sided surface is a surface formed by the movement of rectilinear generator in a broken line, for example, pyramidal and prismatic surface.

Fig. 45 Fig. 46

27 Pyramid is a polyhedron having a base, faces and edges (lines crossing the lateral faces) that intersect at one point (the top of the pyramid). The pyramid-shaped surface - the surface formed by the movement of rectilinear generator in a broken rail, with one point of the image is fixed (Fig. 45). The elements of the surface of the pyramid: the generator, the guide, top, face and ribs (the intersection of adjacent faces). The determinant of the pyramid includes a top surface and a guide. Knowing their position, you can hold any surface forming a pyramid. Prism is a polyhedron that has two faces (bases) that are the same and are parallel, and the other side (lateral) – parallelograms. Prism is called direct if its edges are perpendicular to the plane of the base, and inclined if not perpendicular. The prismatic surface - the surface formed by the movement of a rectilinear generator in a broken rail, thus forming some moves parallel to any given direction (Fig. 46). The elements of the knife edge: the generator, the guide, the faces and edges (the intersection of faces). The determinant of a prismatic surface includes forming and directing. Knowing their situation, you can spend any additional generator. If the prismatic surfaces are perpendicular to the plane of projection, then such a surface is called projecting. Of faceted surfaces emit a group of polyhedra – closed surfaces formed by a number of facets.

Surfaces of revolution

Surface of revolution is the surface formed by rotating a line (the image-ing) around the line (axis of rotation). In the formation of a surface of revolution describes any point in the space of a circle. These circles are called parallels. Always parallel planes are perpendicular to the axis of rotation. The parallel smallest diameter is called the throat, and the largest the equator. The lines of intersection of the rotation plane of the passing through the axis of rotation are called meridians. If the surface of the rotation of the image a straight line, we obtain a ruled surface of revolution, such as conical, and if the curve is not ruled like a sphere. The cylindrical surface of revolution (Fig. 47) is the surface formed by the rotation of the rectilinear generator around a parallel line – axis. The

28 conical surface of revolution (Fig. 48) the surface formed by the rotation of a rectilinear generator around intersecting it straight – axis. Sphere (Fig. 49) – the surface formed by rotating a circle around its diameter.

Dots and lines on the surface

To find the missing projections of belonging to a polyhedron or a curved surface, it is necessary to build any line on a given surface, passing through a given point of the projection, the projection construct an auxiliary of the line, and then build the desired projection point. As these lines can be selected form, parallels, meridians, etc. In some cases, if the surface of the projecting body, i.e. perpendicular to the plane of one of the projections missing drawing projection points can be found without additional constructions, since the surface has col- lected property. Consider the examples of construction of points on the pyramid prism, cylindrical and conical surfaces, as well as industry. Agree that the surface has no thickness, and the points and lines lying on the surface, can not enter into the surface and go beyond it.

Fig. 47 Fig. 48 Fig. 49

29 Therefore, all the points on the surfaces of the prism and the cylinder, on a horizontal projection surface are distributed around the base (polyhedron or circle) with their visibility. The visible part of the frontal projection of the surface located in a horizontal plane below the axis of symmetry, the invisible – above the axis of symmetry. In view of the likely drop the projections of the points on the front line of the projection due to the intersection with the lines of the base of the prism or cylinder on a horizontal plane of projection. In finding the projection points on the surface of a pyramid or cone, you must use one of the methods described below. The method of forming. We carry the top of the pyramid or cone forming a projection of the point А″ to its intersection with the base of the pyramid or cone on the front projection of 1″ (Fig. 46, 48). Then, considering visibility we find the horizontal projection of point 1′ in connection with the intersection of the base of the pyramid or cone in the horizontal projection. Through the projection of the vertices S′ and 1′ point draw a horizontal projection of the image S′1′. From point А″ we omit the line of projection due to the intersection with line S′1′. The intersection point А′ the communication link with the projection of image S′1′ is required point. The method of planes level. Depending on the type of surface (pyramid – Fig. 46 cone – Fig. or sphere 48 – Fig. 49) in cross-section or a circle is formed by a polyhedron. The projection of the desired point lies on one side of a polyhedron or a circle (case-factor view of a point). Consider the example of finding the horizontal projection of the point В on the three listed surfaces. In the section of the pyramid through the projection of В″ a pentagon is formed. The sides of the figure on the horizontal projection are parallel to the sides of the pyramid base and the top (and hence the size of the polygon in cross section) is defined as the point of intersection of vertical link drawn from point 2" (the point of intersection with the plane face of the pyramid level), and the ribs pyramid. From point 2′ we draw a line parallel to the base of the pyramid, and from the point В″ – vertical line of communication. The intersection of these lines is the required projection of В′. If point B is invisible, the line parallel to the base of the pyramid, is drawn from point 21′. At the intersection of the cone (Fig. 48) and the sphere (Fig. 49) planes of the level of cross-sectional image of the circle. The radius of the circle is the distance from the axis of a cone or sphere to the point of intersection of

30 the plane with the generator (point K). The horizontal surface of the said surfaces is carried radius arc (full circle plot is not necessary) above or below the horizontal (with the visibility points). From the front of the projections of point В″ vertical lines are held due to the intersection with arcs. The point of intersection of the arc and the line of communication is the desired projection of В′. Invisible projection points are enclosed in parenthes- es (В″ – on the cone, С′ – on the field). The straight lines on the surface of polyhedra are line segments in all of its projections. Finding the projections of these points is to find the projections of the extreme points of the interval. Straight lines drawn on a surface of revolution of the projections on the other projections are transformed into curves. Commonly used for the construction of the intermediate points (the more, the more accurate will be constructed projection). The points of projection points are connected by smooth lines patterns.

31 Lecture 4. THE INTERSECTION OF POLYHEDRA PLANES OF STRAIGHT LINE

The intersection of the plane of the polyhedra of private provision

The projections of the cross-section plane of the polyhedron are constructed in the following ways: – finding the points of intersection with the plane of the edges of a polyhedron, i. e., finding the vertices of the polyhedron; – finding the lines of intersection with the faces of the cutting plane, that is, finding the sides of the polygon. The intersection of the plane of the prism of private provision. Geo- metric cross-section of the body position of the private plane is a plane figure bound by lines, all points of which belong to a cutting plane and the surface of the body. Fig. 50 shows is a straight five-sided prism, dissected frontally projecting plane (indicated by the front track PV).

Fig. 50

32 To construct the projection figures section is to find the projections of the points of intersection of the plane with the edges and connect them with straight lines. Front projection required points are at the intersections of the front edges of the projections of trace PV (point 1″, 2″, 3″, 4″ and 5″). The horizontal projection of the intersection points (1′, 2′, 3′, 4′ and 5′) are horizontal ribs and projections are determined by means of lines. With the lines of communication we are building a profile projection points. Combining consistently found the projections of points, building a profile projection section. The actual dimensions of the cross section shapes are determined by one of the known methods – change the projection planes or rotation. Used to solve the problem the method of changes the plane of projection. The frontal plane projection π2 will replace a new plane π4, parallel frontal following PV. Plane π4 makes with the plane π1 axis x1. To find the actual size of the cross section, the following construction: – projection of the front points 1″, 2″, 3″, 4″ and 5″ hold the line perpendicular to the axis x1; – is set aside from the axis x1 segment equal to the distance from the x-axis to the horizontal axis of symmetry of the base of the prism, through the data points and draw a line – the axis of symmetry of the natural cross-section; – from the point of intersection of lines with the axis of symmetry of natural-delaying section with horizontal projections are symmetrical segments 2′5′ and 3′4′; – find point 2IV, 5IV, 3IV and 4IV (1IV point lies on the axis of symmetry). Combining series points 1IV, 2IV, 3IV, 4IV and 5IV, get life-size section. Scan called a plane figure, obtained by combining the geometric body with a single plane without gaps and overlap edges or other surface elements on each other (Fig. 51). The scan prism side surface of the base section and the natural shape is done in the sequence (Fig. 51). – draw a horizontal line on which an arbitrary point 60 is postponed five segments of equal size of the sides of the base of the prism, – 60 100, 10 090, 9080, 8070, 7060; – points 60, 100, 90, 80, 70 are held up vertical lines on which was taken to be, respectively, the segments equal to the length of the edges (measured at the front or profile projections), and the point marked 10, 50, 40, 30, 20 and 10;

33 – scanning through the middle of the side surface held vertical axial line and the known methods based on the bottom of the base scanner prism (point 90, 100, 60, 70 and 80) and the upper part – sectional original value (the points 40, 50, 10, 20 and 30); – сonnect the found points contour line and points 50 and 100, 40 and 90, 30 and 80, 20 and 70, 40 and 30, 90 and 80 by chain lines with two dots (fold line – the location of edges of the prism).

Fig. 51

Section of the pyramid private plane position. The correct hex pyramid (Fig. 52) crossed the frontal projecting plane P, defined front after PV. Note the frontal projection of the pyramid with the point of intersection of the plane of the pyramid edges 1″, 2″, 3″ (visible projections of the points), and 4″, 5″, 6″ (invisible projection points). The horizontal projection of the mentioned points are at the intersection of vertical lines to the horizontal projection of the corresponding rib - 1′, 2′, 3′, 4′, 5′ and 6′. Finding the core projections of the points is obvious and needs no fur- ther explanation.

Fig. 52

Connecting the serial projections of straight lines we obtain the projection of the horizontal and cross-sections of the pyramid. To find the actual size of the cross section we use the method of rotation. During the pivot the point is taken Gx″ (the point of intersection trace cutting plane with the axis x). The radius is equal to the distance from point to point Gx 1″ (6″), 2″ (5″) and 3″ (4″) to perform an arc crossing the axis x – point 1″, 2 ″, 3 ″, 4 ″, 5 ″ and 6 ″. From these points of a vertical link which, at the intersection with the horizontal communication lines drawn from points 1′, 2′, 3′, 4′, 5′ and 6′, the horizontal projection of the cross section, size and shape of the natural cross section of the pyramid – 1′, 2 ′, 3 ′, 4 ′, 5 ′ and 6 ′. To construct a surface scan of the pyramid (Fig. 53), se- lect an arbitrary point S0 (conditional cross top of the pyramid). The position of point S0 is determined only Fig.53

35 by rational arrangement sweep on the format of the drawing. From the point of S0 the radius equal to the length of its natural edges parallel to a plane (in this example – at the extreme edges of the profile projection), we draw an arc on which of the randomly selected point А0 postpone the six segments of equal length of the base of the pyramid. The resulting points А0, B0, C0, D0, E0, F0 connect straight line segments, and thin lines with the top S0. The cut side of the pyramid was done by the edge SA. It was noted above that the ribs are sized S′′′Е′′′ and S′′′B′′′, therefore, the construction of the projections of all scan points with the horizontal lines are transferred to one of these edges. Thus, the position of point 10 is found by delaying the point А0 segment equal to the length of the segment Е′′′11′′′ (Е′′′61′′′). Points 2 and 5 are on the edges having full length, so the scan surface side of postponing points В0 and Е0, respectively, the segments В′′′2′′′ and Е′′′5′′′. Point 3′′′ and 4′′′ of the communication line is also drawn over the edge S′′′Е′′′ and lie off on the unfolding of the points of С0 and D0 segment Е′′′31′′′ ≡ Е′′′41′′′. Connect dots 10, 20, 30, 40, 50 and 60 with straight lines. Then the vertices of С0 and D0 sweep build the base figure (hexagon), and from points 10 and 60 – life-size section. The places of inflection (edges and lines С0D0 and 1060) are drawn with dashed lines with two points.

The intersection of the prism and pyramid straight lines

Fig. 54 shows a rectangular prism crossed the line AB. You must define the entry and exit straight line and its visibility. Face of the prism on the horizontal projection is projected in straight lines. Therefore, using a horizontally projecting plane P, we find the point of intersection of the projection of the straight line А′В′ with the faces of the prism – the point М′ and N′. According to the vertical lines of communication we find the frontal projections of M and N – M″ and N″. From the horizontal projection it is seen that at the point M there is a straight prism, while the point N is from it. Therefore, the segment MN AB line will be invisible. Fig. 55 shows an example of a three-sided pyramid crossing straight DE. To find the entry and exit points straight in the face of the pyramid through a front view of a straight line D″Е″ spend frontally projecting the

36 plane P (frontal trace of the plane PV), which will cut the edges of the pyramid at 1", 2" and 3".

Fig. 54 Fig. 55 According to the vertical lines of the projection due find the position of the horizontal projections of the points of intersection – 1', 2' and 3'. Com- bining the horizontal projections of the points we obtain a cross-sectional triangle in which the plane P cuts pyramid. The projections of the points of entry and exit in a straight edge of the pyramid М′ and N′ located at the intersection of the projection line D'E' with the sides of the triangle 1'2'3'. Along the lines of communication we find frontal projection projections of M" and N". The plot line MN is inside the pyramid, so it’s invisible.

37 Lecture 5. CROSSING OF SURFACES OF ROTATION BY THE PLANE AND STRAIGHT LINE

Crossing of surfaces of rotation Planes of private position

The line of intersection of a curve surface and a plane represents a flat curve. In case of intersection a plane with a lining surface on its forming the intersection line presents a straight line. Usually the construction of this line is made on its separate points. The points of a line of crossing of a surface of rotation with a plane is the way of the auxiliary secants planes crossing the given surface on some lines which should be graphically simple – straight lines or circles. The points of intersection of these lines, being general for a surface of rotation and a secant plane, will be points of a required line of crossing. As intersection lines of each of auxiliary intersecting planes with the given surface of rotation and a plane crossing it are competing lines, the construction of points of intersection lines of surfaces of rotation with a plane is made by the same ways which are used at a finding of projections of points and straight lines on a surface. Intersection of a cylindrical surface by a plane of private position. The form of a cylindrical surface section depends on the intersecting plane position. At crossing the circular cylinder by a plane it can be obtained the following figures in section: – a circle, if the intersecting plane is perpendicular to a cylinder axis; – an ellipse, if the intersecting plane is inclined to a cylinder axis; – a rectangle, if the intersecting plane is parallel to a cylinder axis. The construction of flat section of the direct circular cylinder to the similar construction of flat section of a prism as the direct circular cylinder can be considered as a direct prism with uncountable quantity of edges – the forming cylinder. In Fig. 56 three projections of the right circular cylinder are brought, crossed by frontal projecting plane P. Plane P is set by a frontal trace. From the complex drawing it is clear that plane P crosses not only a lateral surface, but also the top basis of the cylinder. As it is known, the plane located at an angle to an axis of the cylinder, crosses it on an ellipse. Hence, the section figure in this case represents an

38 ellipse part. The frontal projection of a figure of section coincides with a frontal trace of plane PV. The horizontal projection of section coincides with a horizontal projection of the basis of the cylinder. For the construction of a profile projection of section it is necessary first of all to find the projections of its points. For example, to find the horizontal projection of point 6 we build a joint line through its frontal projection 6″ before crossing with a horizontal projection of a circle of the basis (point 6′). By means of a joint line on two available projections 6′ and 6″ a profile projection 6′′′ it is found. The profile projections of points of a section figure have been obtained thus we connect a curve line by a curve instrument. The true size of the section is found by the way of the projection planes change. A new axis of projections x1 we build to parallel frontal trace PV on any distance.

Fig. 56

Fig. 57

From the intersection points of the cylinder with a plane (1″, 2″ ≡ 21″, 3″ ≡ 31″, 4″ ≡ 41″, 5″ ≡ 51″ and 6″ ≡ 61″) we draw lines perpendicular to new axis x1. Then the distances equal to distance of listed points of section from axis x on a horizontal projection of the cylinder (coordinate y) is built from axis x1 on these lines. IV IV IV IV IV IV IV IV Connecting the found points 1 , 2 , 3 , 4 , 5 , 6 , 61 , 51 , IV IV IV 41 , 31 , 21 on a curve smooth line, we obtain the true size of the section of the cylinder cutting off with plane P. The development of the lateral cylinder surface execute as follows: the horizontal line is drawn (anywhere on a free place) on which from point I0 (Fig. 57) 12 identical lines are drawn, equal to the distances (chords) between the points of division (are designated by the Roman figures), located on a circle.

40 From points of division I0, II0, III0 and etc. vertical lines are drawn for which the distances from the cylinder basis to section points 1″ (1′′′), 2" (2′′′), 3" (3′′′) and etc. are drawn. Obtained points 10, 20, 30, 40, 50 and 60, and symmetric points 210, 310, 410, 510 and 610 are connected with a curve smooth line. Points 60 and 610 are connected by a straight line. The cylinder basis is situated on an axis of symmetry of development and concerns a horizontal line in point VII0. The true size of section is situated in the top part of the development symmetrically rather its axes. Intersection of a conic surface by a private position plane. At various position of the intersecting plane α in relation to an axis of a right circular cone various figures of section are obtained: – a circle if the intersecting plane is perpendicular to a cone axis; – the ellipse if the intersecting plane is inclined to an axis and crosses forming lines of a cone; – a triangle if the intersecting plane passes through cone top; – a parabola if the intersecting plane is parallel to one of the forming; – a hyperbole if the intersecting plane is parallel to two forming. Construction of flat section of a cone to similarly construction of section of a pyramid as the cone can be considered as a pyramid with uncountable quantity of edges – forming lines of a cone (Fig. 58). The circle (cone base) on a horizontal projection we divide into 12 equal parts and designate the division points by the Roman figures (I′, II′, III′ and etc.). We connect the top of cone S′ (the circle centre) and division points thin lines. From points I′, II′, III′... we draw vertical communication lines before crossing with the cone basis on a frontal projection. In Fig. 58 the construction order as an example of point II ′ is shown. The vertical projecting line crosses the cone basis on a frontal projection in point A". We connect point A′′ to a projection of top of cone S". The line A"S" crosses a trace of plane PV and a frontal projection of section of a cone in point 2". Similar constructions are carried out for all other points of the division. From the obtained frontal projections of points 1", 2", 3" … vertical communication lines are drawn before crossing with the lines of the same name I′S ′, II′S ′, III′S′ … and the horizontal projections of section points of a cone 1′, 2′, 3′ … are found. Then it is jointed the obtained points a smooth line. Profile projections of points of section are by means of projecting lines. We find the true size of section by means of replacement of projection planes. We replace a plane of projections π1 with plane π4, which is drawn

41 to a position parallel to the frontal trace PV so, also parallel to the cone section, and forms with plane π2 axis x1. From points 1", 2" ≡ 21 ", 3" ≡ 31 ", 4" ≡ 41"… we drawn lines which are perpendicular to axes x1. On the distance equal to the distance from an axis x to a horizontal axis of the cone basis on its horizontal projection (coordinate y), we draw a line which is parallel to an axis x1 (an axis of symmetry of natural section). Then we transfer a horizontal projection distance from an axis of symmetry to points IV IV IV IV 2′ (21′), 3′ (3′) and 4′ (41′) … on an axis x1. We find points 1 , 2 , 3 , 4 IV IV IV … and points symmetric to them 21 , 31 , 41 … Connecting the found points, we find the true size of section.

Fig 58

The development of a cone surface is given in Fig. 59. We do it in the following way. We draw an axial line and it is marked on it as point S0. From point S0 in the radius equal to the length of the cone forming, we draw an arch.

42 The development corner ϕ can be defined using the formula: ϕ = πd / L, where d – diameter of the cone basis, mm; L – true length of the cone forming, mm. The development construction is carried out as follows. From point VII0 of intersection of the arch with the axial line we draw six lines to the right side and six lines to the left side too, equal to the distance between the two next points (I′II′, II′III′ and etc., Fig. 58), and then we designate these points. Then we join the obtained points I0, II0, III0 … with thin lines to point S0. To find point 10 the line equal to the length forming line from the basis of a cone to the projection of point 1″ on frontal projections is drawn on lines S0I0. To find true lengths of forming to points 2", 3", 4"… it is transferable these points on horizontal projecting lines to forming line, having true size (in Fig. 58 it is the left forming on a profile cone projection). After that these values carrying over on development. The cone basis is drawn in the lower part of development, and the natural section – in its top part is symmetric to the general axis of the symmetry of this development.

Fig. 59

43 The intersection of cylindrical, conic and spherical surfaces by a straight line

The intersection of a cylindrical surface by a straight line. In Fig. 60 the right circular cylinder crossed by a straight line АВ of general location is presented. The horizontal projection of the circular cylinder is a circle, therefore the horizontal projections of all the points located on a cylindrical surface including two required points of intersection, will be located also on this circle (points M′ and N′). Frontal projections of required points M" and N" are found on vertical communication lines. Line MN, which is in the cylinder, is invisible. The intersection of a conic surface by a straight line. In Fig. 61 one of the ways to find the points of intersection of straight line АВ with a surface of a right circular cone is shown. On a frontal projection of straight line АВ (A"B") any point K (K") which incorporates to the top of cone S (S") with straight line S"K" is selected. The horizontal traces of points F′ and M ′ of intersection straight lines (АВ and SK) are drawn by the known way – continue frontal projections S"K" and A"B" before crossing with an axis x in points F" and M". From the last points join lines are drawn before crossing with the horizontal projections of lines (S′K′ and А′В′) in points F′ and M′. Then points F′ and M′ are joined with a straight line, which crosses the basis of a cone (circle) at points Е′ and D′. To connect points Е′ and D′ to top point S′ straight lines which intersect a horizontal projection А′В′ in points 1′ and 2′ – the horizontal projections an input and exit of a straight line are required. On vertical join lines find frontal projections of points 1"and 2". The part of line АВ, located between points 1 and 2, is invisible (is in a cone). Crossing of a spherical surface by a straight line. The points of the intersection of a straight line АВ with a sphere surface (Fig. 62) we find using the frontal projecting plane P (trace Ph) passing through the given straight line. Auxiliary plane P intersects sphere on a circle, which is projected on plane π1 in the form of an ellipse that complicates the construction. There- fore, in this case it is desirable to apply, for example, the method of replac- ing projection planes. A new plane of projections we choose so that auxiliary plane P is situated parallel to it, i. e. it is necessary to build a new axis of projections x1 so that it were parallel to a frontal projection A"B" – to frontal trace PV, (in Fig. 62 they coincide).

Fig. 60 Fig. 61

Then we build the new horizontal projection АIVВIV of the straight line АВ and a new horizontal projection of a circle with a diameter on which plane P intersects the sphere. On crossing new horizontal projections of a straight line and circle new horizontal projections of two required points MIV and NIV lie. By return construction it is defined frontal (M" and N") and horizontal (M′ and N′) projections of points M and N crossing of straight line АВ with a sphere surface.

Fig. 62

45 Lecture 6. AXONOMETRIC PROJECTIONS

Axonometric projections

Rectangular (orthogonal) projections do not give the spatial image of a subject. So to imitate its kind on orthogonal projections of a detail, it is necessary to «read» the drawing. Sometimes at drawing up of technical drawings, there is a necessity of evident representation of a subject when the subject is represented on the drawing in three measurements, instead of two as it happens in rectangular projections. A method of axonometric displaying (axon – an axis, metric – measurement) is applied to such images. The essence of the method of axonometric measurement states that the given subject together with axes of rectangular coordinates to which it is carried in space, is projected on some plane so that any of its coordinate axes is not projected on it in a point. Consequently, a subject is projected on this plane in three measurements. A system of coordinates being in space x, y, z is projected on some plane Р (Fig. 63). Projections xР, yР, zР of coordinate axes on plane Р are called axonometric axes. On axes of coordinates in space equal sections /0A/ = /0V/ = /0S/ are built. Apparently, from the drawing their projections on plane Р generally are not equal to lines and are not equal among themselves. It means that the sizes of a subject in axonometric projections on all three axes are deformed. Factors of distortions:

Kx = 0PАР / 0А; Ky = 0РВР / 0В; Kz = 0РСР / 0С. These factors reflect the distortion of lines on axes. The size of indicators of distortion and a ratio between them depend on an arrangement of a plane of projections and on a displaying direction. Three variants of distortion indicators ratio of the sizes on axes are possible: – distortion indicators on all three axes are identical – an isometric axonometry; – distortion indicators on two axes are equal among themselves, and the third is not equal – a dimetric projection; – distortion indicators on all three axes are not equal among themselves – a trimetric axonometry.

Fig. 63 Depending on a displaying direction in relation to a projection plane axonometric projections are subdivided in: – Rectangular – projecting beams make the right angle with a projection plane; – Oblique-angled – a direction of projecting beams any way. The indicators of distortion and displaying direction are connected among themselves by following dependence: 2 2 2 Kx + Ky + Kz = 2 + ctgα, where α – an angle of slope of projecting beams to a projection plane.

Fig. 64 In case of rectangular displaying when α = 90° and ctgα = 0, we have the following equation: 2 2 2 Kx + Ky + Kz = 2.

47 For oblique-angled and rectangular displaying the sum of squares of two any indicators of distortion cannot be less than one. The above said proves to be true according to Polke-Shvarts theorem. It proves that any three straight lines located in one plane and passing through one point, can be accepted as axonometric axes on which any indicators of distortion Kх, Kу, Kz can be chosen if only the sum of their squares was not less than two, and the sum of squares of two any of them was not less than one. The standard axonometric projections applied in drawings of all indus- tries are established by GOST 2.317–69. Rectangular axonometric projections. Rectangular isometric projection. An isometric projection is carried out with equal distortions on axes x, y, z, i. e. accept distortion factors the equal:

Kх = Kу = Kz = K, and consequently: 3K2 = 2. Solving the equation given above, we obtain K = 0,82. It means that in a rectangular isometric all the sizes of a represented subject change at 0,82 relatively its true value (Fig. 64). As Kх = Kу = Kz, then cos α = cos β = cos γ, and consequently α = β = γ. The equality of corners says that lines xPzP, xPyP, yPzP are equal among themselves and consequently also corners ∠xP0PzP, ∠xP0PyP, ∠yP0PzP are equal among themselves and each of them is equal 120°. The position of axonometric axes is given in Fig. 65. In practice, in constructing an isometric projection do not apply a distortion indicator equal to 0,82, and replace with the reduce indicators equal to one (K = 1). Rectangular dimetric projection. The position of axonometric axes is given in Fig. 66.

Fig. 65 Fig. 66

48 In rectangular dimetric the distortions on axes 0x and 0z are identical, i. e. Kх = Kz = K. The third indicator Kу can have infinite set of values, however it is accepted equal to 0,5 from axes 0x (0z). We have: 2K2 + K2 / 4 = 2. Hence:

Kх = Kz = 0,94; Kу = 0,47. In practice according to GOST 2.317-69 distortion factors on axes x and z accept equal to 1 and on an axis y – 0,5. The calculated corner between a horizontal line and an axis 0x is equal to 7°10′ (a parity 1 : 8), and between a horizontal line and an axis 0y – 41°25′ (a ratio 7 : 8). Oblique-angled axonometric projections. Frontal oblique-angled isometric projection. The position of axonometric axes is represented in Fig. 67. The angle of slope 0y axis to a horizontal line is usually accepted equal to 45°. The standard supposes drawing 0y axis at an angle 30° or 60° to a horizontal line.

Fig. 67

Horizontal oblique-angled isometric projection. The position of axonometric axes is given in Fig. 68. The angle of slope of axis 0y to a horizontal line is usually equal to 30°. It is supposed to build an axis 0y at an angle 45° or 60°, keeping a corner 90° between axes 0x and 0y. A horizontal oblique-angled isometric projection is built without distortion on axes 0x, 0y, 0z. The frontal dimetric projection. The position of axonometric axes is given in Fig. 69. The angle of slope of axis 0y to a horizontal line should be equal to 45°. The standard supposes possibility of carrying out of an axis 0y at an angle 30° or 60°. The distortion factor on an axis 0y is equal to 0,5 and on axes 0x and 0z is equal to 1.

Fig. 68

Fig. 69

The ways of constructing an oval in the rectangular isometric

During the construction of axonometric projections of any object it is necessary to build axonometric projections of circles. In most cases planes of circles are disposed parallel to some plane of projections (π1, π2 or π3). Let's consider the variants of construction of a circle in isometric axonometric projections (Fig. 70). To have more evident idea about the arrangement and size of axes of ellipses, in which circles are projected, the last are entered in cube sides. Points of a contact of ellipses are in the middle of edges of a cube. Ex- cept these four points it is possible to specify four more points belonging to the ends of the biggest and smaller diameters of an ellipse (big and small axes). In rectangular isometric projections of a direction of the big axes of ellipses are perpendicular to free axonometric axes – in a horizontal plane to axis 0z, in a frontal plane to axis 0y, in a profile plane to axis 0x, and small axes of ellipses coincide in a direction with free axonometric axes. In practice, considering certain complexities in ellipse construction, instead of this draw an oval that slightly influences the accuracy of the image.

Fig. 70

In Fig. 71 and 72 two ways of constructing an oval are presented.

Fig. 71 Fig. 72

Way of constructing an oval on intermediate points. Considering a rectangular isometric projection it is established (Fig. 71) that the big axis of an oval equals 1,22d (d – diameter of a circle), and a smaller axis – 0,71d. As in a horizontal plane the big axis is perpendicular to axes 0z and is located horizontally, the smaller axis coincides in a direction with axis 0z. We mark off a distance equal to 1,22d, and on axes 0z – a distance equal to 0,71d on a horizontal line. Thus, we find four points of the future oval. Other four points are located on axes 0x and 0y. The distance between them

51 is equal to diameter d of a circle. Connecting the 8 points of a smooth curve, we obtain a required oval. Way of constructing of an oval by means of circle arches. We build two circles with diameters equal to 1,22d (the size of the big axis) and 0,71d (the size of a smaller axis) (Fig. 72). We build axes 0x, 0y and 0z through the center of these circles. From the intersection point of a circle of diameter 1,22d (the big circle) with a vertical axis as from the center we build an arch with the radius, equal to the distance from this point to distant intersection point of a circle in radius 0,71d (a small circle) with a vertical axis, before their crossing with axes 0x and 0y (points 1, 2, 3 and 4). Then from the intersection points of a small circle with a horizontal axis as from the centers in the radius equal to the difference of radii of the big and small circles, we build arches before their crossing in points 1, 2, 3, 4 with axes 0x and 0y, i. e. before interface to earlier built arches, and the required oval is obtained.

52 Lecture 7. HELICES. THREAD

Helices

Helical motion of a point in the simplest case is the result of its uniform translational motion along with the simultaneous uniform rotation around it. If this movement makes any line that is formed helical surface. The most spread types of helices are cylindrical and bevel helical lines, although the helix can be built on any surface rotation. The coil line – a line described by a point undergoing a uniform motion along the cylinder, which rotates with a constant angular velocity about the axis of the cylinder. Fig. 73 illustrates the projection of the coil and its scan line (right). The frontal projection of a cylindrical helix is a sine wave, the horizontal – the circle. Helices are right-handed and left-handed. If the rise of the helix is anti- clockwise spiral is left, clockwise – right. Fig. 73 shows the right helix.

Fig. 73

We are following the helix elements – round, pitch and angle of eleva- tion. Revolution – is a part of the helix described by a point in one revolution of the generator along the axis of the cylinder. Step (P) – the distance between the starting point and end point turns, measured along the cylinder. Lifting angle (α) helix – is the angle defined by the expression α = arctg Р / πd, where P – step helix; d – diameter of the circumference of the cylinder.

53 A helical scan line is straight. The angle between the helix and form a cylinder of the helix remains constant.

Thread

If the surface of a right circular cylinder on one side touches an arbitrary flat shape so that its plane passes through the cylinder axis, the helical movement resulting shape without changing its position relative to the axis of the cylindrical surface of the obtained spiral protrusion. The cylinder with screw flange is called a cylindrical screw and the screw lug – threaded screw. The shape, forming a spiral ridge is called the thread profile. Depend- ing on the form of thread-forming screw projection bolts can be triangular, trapezoidal, rectangular, square and other threads. When all the manifold fittings may be classified into one of two types: 1) connection to the direct bolting of the parts, without the use of connecting parts; 2) connection is done using special fasteners, such as bolts, screws, studs, fittings, etc. The threads are classified according to several criteria: – depending on the shape of the profile (metric, trapezoidal, etc.); – depending on the surface shape (cylindrical, conical); – depending on the location on the surface of the thread (male, female); – for operational purposes – by fixing (metric, inch), O-Fixing (tube, tapered), running (trapezoidal, persistent), and others; – depending on the direction of the helix (right and left); – the number of calls (single-start, multi-start). Straight thread – thread formed on the cylindrical surface. Tapered thread – thread formed on the conical surface. Outer thread – thread formed on an outer cylindrical or conical surface. Inner thread – thread formed on an inner cylindrical or conical surface. In the threaded joint female thread is covering the surface. The right-hand thread – thread by loop formed by rotating clockwise and moving along the axis in the direction of the observer. In the drawing, right-hand thread is not explained. Left hand thread – thread formed loop rotating counterclockwise and moving along the axis in the direction of the observer. Metric thread. Metric thread is attaching thread and executed in accordance with the requirements of existing standards – ГОСТ 8724–81, ГОСТ 9150–81, ГОСТ 24705–81, ГОСТ 16093–81, etc.

54 Metric thread profile is an equilateral triangle with an apex angle of 60°. Metric threads are made with large and small step. A major step in the drawings is not specified. For example: M10, where M – the symbol notation metric thread, 10 – external (internal) diameter of the thread. Thread with fine pitch designated by the letter M indicating the nominal diameter and pitch, for example – M10 × 0,75. For left-hand threads capital letters LH, for example – M10LH or M36 × 1,5 LH are placed after the symbol. Multiple threads shall be designated by the letter M, nominal diameter, numerical value, speed and the numerical value in pa- rentheses pitch (P), for example – M36 × 3 (P1, 5) and M36 × 3 (P1, 5) LH. Fig. 74 Fig. 74 show the examples of drawing and designations of metric threads (male and female). Border carved image of the rod is carried out through the outer diameter of the thread solid base line, the inner diameter of the thread is drawn with a continuous thin line. On a left outer thread diameter of a circle drawn with a continuous main line, the internal diameter of an arc of 270° continuous thin line. Note, however, that the arc should not begin or end on the center lines. The chamfer on the form on the left is drawn. When drawing the inner diameter of the internal thread spend a solid base line and the outer – a continuous thin. Hatching carried out before the main line (inner diameter). Tapered thread (ГОСТ 6111–52). Tapered threads are widely used in industry. Profile conical inch thread – an equilateral triangle with an apex angle of 60°. With tapered threads get tight joints that do not require the use of sealing means. For such compounds, characterized by a more uniform load distribution, as well as reducing the time for the assembly and disassembly of the connection. Tapered thread is performed on the conical surfaces (taper 1:16). Conical inch thread denoted by the letter K with the addition of the thread diameter (in inches) and the number of ГОСТ (K1/4" ГОСТ 6111–52).

55 Examples of drawing and designations outside (top) and internal conical inch threads are shown in Fig. 75. Trapezoidal thread (ГОСТ 9484–81). Profile – isosceles trapezoid with the angle between the sides of 30°. Used as a running thread. Ex- amples of notation: Тr40×6; Тr40×8 (Р4), where Tr – symbol designations trapezoidal thread, 40 – outer diameter of the thread, 6 and P4 – pitch, 8 – move the thread. Schematic of trapezoidal

thread is not different from a Fig. 75 metric thread. Cylindrical pipe (GOST 6357–73) and a conical tube (ГОСТ 6211–69) of the thread. Profile cylindrical and tapered pipe threads – an isosceles triangle with an apex angle of 55°. Parallel thread is available in two accuracy classes – A and B, which must accompany the designation. The symbol of a cylindrical pipe thread is the letter G, R for conical external threads and Rc for internal threads. The examples refer to the internal pipe threads are shown in fig. 76, a tubular inner conical threading in Fig. 77. From the notation shows that the NPT is measured in inches.

Fig. 76 Fig. 77

Thread resistant (ГОСТ 10177–62). It is used in cases where it is necessary to transmit the force in one direction, such as in a vice, jacks, press. Thread profile – not isosceles trapezoid with a slope of 3° of the working and not working – 30°. The symbol to denote the buttress thread is the letter S.

56 The examples of designations trapezoidal resistant and special threads are shown in Fig. 78.

Fig. 78

Note, however, that the diameters, chamfers, steps, grooves threads defined according to the relevant standards and can not be taken arbitrarily.

57 Lecture 8. INTERSECTION OF SURFACES

Intersection of polyhedrons

The intersection line of two polyhedrons, named a transition line, represents some spatial broken line, which can break up into two and more separate parts. These parts can be flat polygons. Top points of intersection lines of polyhedrons are intersection points of edges of the first polyhedron with sides of the second polyhedron and edges of the second polyhedron with sides of the first polyhedron. The sides or links of intersection lines are lines, on which sides both polyhedrons are crossed.

Fig. 79

The construction of top points of intersection lines of two polyhedrons consists of the repeated decision of a task on intersection of a straight line with a plane, and construction of the parts of these lines – to the repeated decision of tasks on crossing two planes. Usually we prefer to find tops of lines.

58 It is thus obvious that only those pairs of tops can be connected with straight lines which lie in the side of the first polyhedron and at the same time in the side of the second polyhedron. If considered pair of tops at least in one polyhedron belongs to different sides, such tops do not incorporate. The order of connection tops of a intersection line in most cases is eas- ily defined, if after construction the question of visibility of edges of both polyhedrons is found out. The intersection of two prisms. In Fig. 79 two direct prisms – vertical pentahedral and horizontal tetrahedral are presented. We define the lines of their crossing. Horizontal and profile projections of the intersection line are obvious and coincide with a horizontal projection of a pentagon (the basis of a vertical prism) and with a profile projection of a quadrangle (the basis of a horizontal prism). A frontal projection of the intersection line is built on points of intersection of edges of first prism with sides of second prism. There are points 1′, 2′, 3′, 4′, 5′ on a horizontal projection and points 6′′′, 7′′′, 8′′′ and 9′′′ on a profile projection. On horizontal and vertical join lines it is found frontal projections of the specified points – 1", 2", 3", 4", 5", 6", 7", 8" and 9". Connecting consistently projections of points 1" and 6", 6" and 2", 2" and 3", 3" and 7", 7" and 8", 8" and 4", 4" and 5", 5" and 9", 9" and 1", we obtain a required broken line. The intersection of the prism and the pyramid. Intersecting line of a tetrahedral prism with a tetrahedral pyramid (Fig. 80) we build with aid border points of a broken line. For example, projections of points 1 and 3 of required lines is found by following way. Frontal projections 1" and 3" are obvious (crossing of edges of a pyramid and prism sides), then we find profile projections 1′′′ and 3′′′, using horizontal join lines. On two certain projections of points we found their horizontal projections 1′ and 3′. In the same sequence we find projections of points of 2 and 4 located on crossing edges of a prism and pyramid. We find projections of points 5, 6, 7 and 8 by means of the joint lines built from a frontal projection on horizontal and profile projections. Connecting consistently on a horizontal projection of points 1′, 2′, 3′ and 4 ′ (they are visible), and 5′, 6′, 7′ 8′ too (they are invisible) straight lines, we obtain projections of required intersection lines. On a profile projection of a intersection line 2′′′, 1′′′, 4′′′, 6′′′, 5′′′, 8′′′ are visible, and lines 2′′′, 3′′′, 4′′′ and 6′′′, 7′′′, 8′′′ are invisible.

Fig. 80

Intersection of surfaces of rotation

The line of crossing two surfaces of rotation generally represents a spatial curve line, which can break up into two and more parts. These parts can be flat curve lines. Usually the intersection line of surfaces of rotation is built on its separate points. The general way of construction of these points is the way of auxiliary surfaces-mediators. Crossing the given surfaces some auxiliary surface and defining lines of its crossing with considered surfaces, we find the points belonging to the required intersection line. Most often planes or spheres are applied as surfaces-mediators for constructing an intersection line of two rotation surfaces. The way of auxiliary planes is applied when both surfaces are able to cross on simple lines (on a circle). The way of auxiliary spheres is expedient for using in constructing an intersection line of such surfaces, which have the general plane of symme-

60 try located in parallel any plane of projections. Thus, each of surfaces should contain family of circles in which its auxiliary spheres can cross, the general for both surfaces. In certain cases the crossing lines of surfaces of the second order break up into flat curves of the second order. If the kind of these curves is known, it is possible to avoid a very difficult construction of an intersection line on points, and to build the construction of these curves on their basic elements. The intersection of cylindrical surface. In constructing the projections of intersection lines of rotation surfaces in the beginning there are so-called obvious (basic) points (without additional graphic constructions). Then the characteristic points located on extreme of forming rotation surfaces, the lines of transition separating a visible part from an invisible are defined. All other points of an intersection line are called intermediate. Generally in construction of intersection lines of cylindrical surfaces the auxiliary mutually parallel intersecting planes or spherical surfaces more often are used. In the case under consideration horizontal planes of level is chosen as auxiliary planes which are crossed with both surfaces on simple lines – to straight lines and circles, and circles are located in the planes parallel to the planes of projections. In constructing the projections of the intersection line of two cylinders, the axes of which are perpendicular (Fig. 81), we use a way of displaying without axes. In the beginning, as it was marked above, we find the projections of obvious (basic) points 1 and 7 and the characteristic point 4. The intermediate points is found, using auxiliary mutually parallel planes of the level, crossing both surfaces on the forming. So, for example, the intersecting plane of level PV2 cuts the horizontal cylinder on forming in points 2 and 21, and the vertical cylinder – on a circle. Using the profile projections of specified points 2′′′ and 21′′′ and as join lines and Monzh constant, we find the horizontal projections of the specified points. Then frontal points projections 2" and 21" are found in the known ways. Similarly there are horizontal and frontal projections of points 3, 5 and 6. Connecting the found projections of points 1′′, 2′′ … 7′′, we obtain a visible part of a frontal projection of the intersection line, and a point 1′′, 21′′, 31′′ … 7" – its invisible part. The intersection of cylinder and a cone. An example of the intersection line construction of a right circular truncated cone having a vertical axis, with a cylinder located horizontally, is shown in Fig. 82.

Fig. 81

At the next step we define the projections of obvious points 1 and 7. For the definition of intermediate points parallel intersecting planes it is used auxiliary mutually – horizontal planes of level which cross a cone on a circle, and the cylinder – on forming lines. Required points are on crossing forming lines with circles. To define, for example, the horizontal projections of points 2′ and 21′ from the centre of a horizontal projection of the basis of a cone, we draw a horizontal projection of a circle in diameter d2, on which the auxiliary intersecting plane crosses a cone.

From the profile projections of points 2′′′ and 21′′′ it is built on a horizontal projection of the join line which intersects a circle in diameter d2 in points 2′ and 21′. Similar image we find projections of other intermediate points. Having horizontal and profile projections of points, on join lines are their frontal projections found. Connecting consistently the founded points with smooth lines, we obtain visible and invisible parts of the intersection line. The construction of an intersection line of surfaces by means of concentric auxiliary spheres. Instead of auxiliary intersecting planes under certain conditions, it is convenient to apply auxiliary intersecting surfaces to the construction of a line of crossing surfaces.

62 In comparison with the method of auxiliary intersecting planes, the method of auxiliary concentric spheres has an advantage, for example, the frontal projection of a line of intersection is under construction without application of two other projections of crossed surfaces at any arrangement of surfaces. Auxiliary spherical surfaces can be applied to the construction of an intersection line of surfaces of rotation under the following conditions: – intersected surfaces should be only rotation surfaces; – axes of surfaces should be crossed, the point of the intersection of axes is the centre of auxiliary spheres; – axes of surfaces of rotation should be parallel to any plane of projections.

Fig. 82 In Fig. 83 the construction of an intersection line of surfaces of two cylinders which axes are crossed at an acute angle is shown. Auxiliary spherical surfaces is built from point 0" crossing axes of cylinders. Points 1" and 5" are obvious.

Fig. 83 Let's construct, for example, a frontal projection of point 2" – an intermediate point of a line of crossing. For this purpose from point 0" we draw a spherical surface in radius R2. The circle in radius R2 crosses the horizontal cylinder in points c" and d", and the second cylinder – in points a" and b". In crossing of the lines a"b" and c"d" we find a required point 2" of intersection line. We find the position of other intermediate points having similar construction. The limits of radii of spherical surfaces are found as follows (Fig. 83). The largest circle of a spherical surface should be crossed with outline forming lines I–I and II–II, and the smallest should be a tangent to one of the crossed surfaces and should crossed with the forming lines of other surfaces. In Fig. 84 the example of construction of a intersection line of the right truncated cone and the horizontal direct cylinder is shown. The principle of finding the intermediate points of an intersection line is the same as in above considered example (Fig. 83). It is necessary to take only into consideration that the smallest radius of the intersecting sphere is defined by the size of a perpendicular, which is drawn from point 0" to the cone outlines. If the surfaces of rotation (for example, two cones) are circumscribed about a sphere, they are crossing a sphere on two circles. Cir- cles are crossed in two points. Planes, in which these circles lie, are crossed on a straight line the connecting points of intersection of con-

64 tact lines of cones with a sphere. The circles are projected on a frontal plane of projections in the form of straight lines.

Fig. 84 In Fig. 85 two circular cylinders with the axes crossed in point 0" that are circumscribed about a sphere with the center as in point 0". The frontal projection of a sphere will be a circle, a tangent forming outlines of cylinders. The lines of crossing these cylinders – the ellipses, the frontal projections of which are represented in the form of straight lines a"b" and c"d". If two circular cones (Fig. 86) with the axes crossed in point 0" are circumscribed about a sphere with the centre in the same point 0", the frontal projection of a sphere will be a circle, a tangent to forming outlines of cones. The lines of crossing these cones represent the ellipses the frontal projections of which are represented as straight lines.

Fig. 85 Fig. 86

65 LECTURE 9. AUTOMATION OF PERFORMANCE OF DESIGN WORKS

The function of the automated designing systems

Before making any object (subject), the designer should represent evi- dently it, i.e. to prepare the engineering specifications. Technical creativity is closely related both to science and manufacture. The designer is obliged to know and use the data of basic physical, mathematical and other scientif- ic disciplines, should take into account the opportunities of modern manufacture. Besides technical creativity is connected to art as the designer is obliged to provide a modern design to the product. Design activity in general can be divided conditionally in: – designing as planning – creative predetermination of the future technical device or a technological method at which calculations, by sketches or experimentally make preliminary study; result – a substantia- tion for the subsequent design the device or development of a method; – designing as constructor’s work – development of the graphic representations reflecting a plan of the technical device; result – reception of drawings of a new product or new technological process. Despite of a significant amount of the routine operations making all process of designing, its formalization is difficult and rather labour- consuming. Only with the advent of microprocessor techniques this process became an objective reality, that has led to a wide circulation of systems of Automated Design or Computer-Aided Design systems (CAD). CAD cover all spectrum of the problems connected to design activity (graphic, analyti- cal, economic, ergonomic, aesthetic, etc.). It is obvious, that any enterprise is interested in reduction of terms from the idea before starting new products in manufacture. Using CAD appreciably reduces the duration of this stage. The advan- tages of CAD: – faster performance of drawings. The designer using CAD can carry out drawings on average three times faster than working with a pencil; – the increase of accuracy of performance of drawings. Accuracy of the drawing constructed manually is defined by the designer’s abilities qua- litatively to carry out graphic representations. The drawing constructed with the help of computer means, transfers the image of an object more precisely. For more detailed viewing elements of the drawing there is the

66 software, allowing to increase any part of the given drawing. Except for it, CAD provides the designer with many special means inaccessible at manual plotting; – the opportunity of repeated copying and editing of the drawing. The construction of the image of the drawing or its part can be kept for the fur- ther work. Usually it is useful when the structure of the drawing includes the components having the identical form. The kept drawing can be used for the subsequent designing; – the acceleration of calculations and the analysis at designing. Now there are a large number of software products which allow to carry out practically all design calculations; – the high level of designing. Powerful means of computer modeling (for example, a method of final elements) allow to project non-standard geometrical models which can be modified and optimized quickly that allows to lower the general expenses to such degree which has been unattain- able because of big waste of time earlier; – the reduction of expenses for improvement. The means of imitation and analysis, included in CAD, allow to reduce sharply waste of time and means for researches and improvement of prototypes which are expensive stages of the process of designing; – the integration of designing with other kinds of activity. The inte- grated computer network with high-quality means of the communications provides CAD closer interaction with other engineering divisions. To the designer and operator the system offers some set of software which should facilitate its work. Finally the decision remains for the designer, and the machine provides humanly with an opportunity of a choice. Proceeding from the afore- said, it is obvious, that the designer should know well the rules of registra- tion of the drawing documentation (standards of ESKD), to own the software necessary for work, and to know about the structure and opportunities of the automated workplace.

Characteristics of modern systems of the automated designing

The development and perfection of the automated systems of designing was carried out in the following directions: – increase the quantity of program functions; – transition from two-dimensional plotting to three-dimensional; – transition to solid-state modeling;

67 – work on the unification which have provided an opportunity of assembly of unit on the drawing from standard elements. In the process of perfection of systems of designing occurs division depending on a level of complexity and cost on systems of a high, average and low level (basically on the basis of personal computers). The leading place in the system of engineering designing and its software is occupied by firm AutoCAD Mechanical Desktop. Among Russian- speaking versions the greatest distribution has the system of an average level COMPAS (firm ASKON, Saint Petersburg) which does not concede to similar foreign systems on key parameters. Strictly speaking, the systems of a low level can be related to CAD conditionally. They are the graphic editors intended for automation of engineering- graphic works, together with a computer and the monitor represent «electronic Kuhlman drafting unit», that is the a good tool for performance of the design documentation. These systems are called two-dimensional. Now more and more application is found in the systems, allowing to build three-dimensional models. Designing occurs at a level of solid-state models to attraction of powerful design-technology libraries, with the use of the modern mathematical device for carrying out necessary calculations. Besides it systems allow to simulate moving with the help of means of animation of working bodies of a product (for example, manipulators of the robot). They trace a trajectory of movement of the tool by the development and the control of the technological process of manufacturing of designed products. The restriction in use of three-dimensional systems is their high cost. The process of three-dimensional modeling is very labour-consuming, as the development of model occupies many man-hours. However, if to consider this process within the framework of the whole production cycle it considerably raises the efficiency of designing and manufacture in many branches. Three-dimensional systems are successfully applied to the creation of complex drawings at designing accommoda- tion of the factory equipment, pipelines, building constructions, etc., where traditionally prototyping was used for these purposes.

The basic directions of automation of engineering-graphic works

The process of designing can be divided into a number of stages or kinds of activity, and the order of their description has no value as in practice there is a transition from one kind of activity to another without obvious priorities.

68 Basically it is possible to allocate the following kinds of activity: – creation – an opportunity to carry out projective drawings of new products which do not exist yet; – editing – an opportunity to make change in developed drawings of a product in the process of their occurrence; – calculations – at a level of typical calculations of machine’s details; – a choice – decision-making, on what way to direct development of the project to the detriment of other variants on the basis of specifications (for example, drawings of prototypes of products, calculations, etc.); – search – work with archives (search of already existing decisions, acquaintance with a history of modification of a product), and the circle of a choice and search, as a rule, is limited to prototypes of concrete branch. The listed kinds of activity can be automated thanks to the modern software. The drawing of any complexity is under construction on the basis of simple graphic elements (graphic primitives): points, pieces, circles and curves. The method of construction of each separate drawing in most cases depends on required accuracy. Graphic primitives (a point, a piece, a circle, etc.), commands of their editing (deleting, transferring, copying, etc.), commands of installation of primitive properties (the task of thickness, type and color of graphic objects) correspond to tools of manual plotting in the automated environment. To choice a sheet of the necessary format and scale of the drawing in system there are corresponding commands of adjustment of the drawing. For drawing the size it is necessary for a designer to set only a place of its arrangement on the drawing. Dimensional and remote lines, and also arrows and inscriptions are carried out automatically, and in last versions of the systems of designing there is a mode of full automation drawing of the sizes. The corresponding commands of the system allow to increase the image of the drawing on the screen or to reduce it if necessary (similarly to viewing of the image through lens) and also to move borders of a part of the drawing seen on the screen without changing the scale of the image. The system gives the designer an opportunity to unite graphic objects in the uniform block which is stored under the certain name and is, if necessary, inserted into any drawing that relieves the designer of drawing of the same frequently repeating elements of the drawing. The designer can also create the images of separate elements of the drawing or separate details of assembly on various layers. It allows to su- pervise the compatibility of details at configuration. Switching on or switching off layers, it is possible to enter or remove the details from the general configuration, creating thus convenience in selecting various va-

69 riants of a design of a product. The layers are useful for using even in simple drawings, placing the preparation of the drawing, contour, the sizes, inscriptions, axial lines for the subsequent opportunity of a fast choice of group of objects and their editing on each separate layer. Besides the creation of two-dimensional drawings, CAD allows to model three-dimensional objects and to give a photographic reality to three-dimensional drawings. The development of machine schedules has allowed to create the spe- cialized systems of the automated manufacturing of drawings and other engineering specifications with the help of special plotters, providing an out- put of the information to a drawing paper. The desktop plotter is given in Fig 87.

Fig. 87 In Fig. 88 a plotter of A1 format is given. It produces high quality drawings at maximum plotting speed of 600 mm/sec with a maximum acceleration of 3G. Precision micro-stepped motors are incorporated to achieve an astonishing resolution factor as high as 1.5625 micron/step.

Fig. 88

70 Practice shows, that existing CAD with the advanced graphic system allow to create complexes which are expedient for using by students trained to master of the bases of descriptive geometry and projective plotting, fur- ther – in performance of the course and degree projects. Thus, essentially new opportunities are created it is possible to accompany automatic synchronous construction of the second, third projections with the construction of one projection. It is possible to construct quickly a plenty of images t changing of the sizes of elementary crossed surfaces and to investigate laws revealed at it. The using of the method of auxiliary planes can be shown on the example of building up of lines of mutual crossing of anyone surfaces. Thus, various kinds of the curves which are turning out in sections will be shown. The demonstration of ki- nematic ways of the surfaces formation both on orthogonal projections and on an axonometric is possible. In any computer graphic system there is an editor of drawings and other specifications and technical documentation. It allows printing the document on the display and provides the performance of commands for creation, changing, transferring, viewing the contents of the document on the graph plotter or the printer. The editor of drawings contains an extensive set of commands for editing which allow to move, copy, repeat some times, interchange the position, mirror to reflect, to erase partially or completely, turn, and also to stretch or compress vertically and horizontally any objects or groups of objects. Students master all this during training. The majority of programs of the automated performance of the documentation allows to create the conditions for information interchange with other objects which take place on significant distance that is widely used in a national economy. To use successfully the considered systems of designing and modern technical means in the future work young experts should have basic knowledge both in the field of machine-building plotting and in the field of the automated designing.

71 REFERENCES

1. S. K. Bogolyubov. Exercises in machine drawing. Translated from Russiian by Leonid Levant. – М.: Mir publishers, 1985. 2. S. K. Bogolyubov, A. V. Voinov. Engineering drawing. М.: Mir publishers, 1986.

72 CONTENTS

INTRODUCTION ...... 3

Lecture 1. THE BASIC PRINCIPLES OF THE ORTHOGONAL PROJECTION ...... 5 The Subject and the Method of Descriptive Geometry ...... 5 Brief history of the development of descriptive geometry ...... 5 Legend ...... 6 The basic properties of projection ...... 7 The basic properties of parallel projection ...... 8 Comprehensive drawing of the point ...... 10 A complex drawing consisting of the three orthogonal projections ...... 12 The system of rectangular coordinates ...... 13 Methods of setting of the plane on the drawing ...... 14 The location of direct and of the plane comparatively to the projections of planes ...... 14

Lekture 2. DETERMINING THE ACTUAL SIZE OF A LINE SEGMENT ...... 20 Traces a straight line ...... 20 Traces of the plane ...... 21 Direct and to the point of the plane ...... 21 Determining the actual size of a line segment ...... 22

Lecture 3. SURFACE ...... 26 Surface. Methods of specifying the surface ...... 26 Sided surfaces and polyhedra ...... 27 Surfaces of revolution ...... 28 Dots and lines on the surface ...... 29

73 Lecture 4. THE INTERSECTION OF POLYHEDRA PLANES STRAIGHT LINE ...... 32 The intersection of the plane of the polyhedra of private provision 32 The intersection of the prism and pyramid straight lines ...... 36

Lecture 5. CROSSING OF SURFACES OF ROTATION BY PLANE AND STRAIGHT LINE ...... 38 Crossing of surfaces of rotation. Planes of private position ...... 38 Intersection cylindrical, conic and spherical surfaces by a straight line ...... 44

Lecture 6. AXONOMETRIC PROJECTIONS ...... 46 Axonometric projections ...... 46 The ways of construction of an oval in the rectangular isometric 50

Lecture 7. HELICES. THREAD ...... 53 Helices ...... 53 Thread ...... 54

Lecture 8. INTERSECTION OF SURFACES ...... 58 Intersection of polyhedrons ...... 58 Intersection of surfaces of rotation ...... 60

Lecture 9. AUTOMATION OF PERFORMANCE OF DESIGN WORK ...... 66 The function of the automated designing systems ...... 66 Characteristics of modern systems of the automated designing 67 The basic directions of automation of engineering-graphic works 68

REFERENCES ...... 72

Textbook

Georgiy Kasperov Aleksandr Kaltygin Vitaly Gil

ENGINEERING AND COMPUTER GRAPHICS

Texts of lectures

Computer-aided makeup A. S. Aristova Proofreading A. S. Aristova

Publisher and printing performance: EI «Belarusian State Technological University». Cвидетельство о государственной регистрации издателя, изготовителя, распространителя печатных изданий № 1/227 от 20.03.2014. Sverdlova Str. 13а, 220006 Мinsk.

Учебное издание

Касперов Георгий Иванович Калтыгин Александр Львович Гиль Виталий Иванович

ИНЖЕНЕРНАЯ И МАШИННАЯ ГРАФИКА

Тексты лекций

Компьютерная верстка А. С. Аристова Корректор А. С. Аристова

Издатель: УО «Белорусский государственный технологический университет». Cвидетельство о государственной регистрации издателя, изготовителя, распространителя печатных изданий № 1/227 от 20.03.2014. Ул. Свердлова, 13а, 220006, г. Минск.