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 © EI «Belarusian State Technological University», 2014 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 depart- ments. Textbooks and methodical literature are developed and pub- lished 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, geome- trical 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 lan- guage he or she speaks. The role of descriptive geometry is important in the process of study- ing natural sciences, when studied or analyzed properties are accompanied by accessible to the human perception visual geometric models, which al- low 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 ob- jects 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 draw- ing 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 de- scriptive 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 oth- ers – made great contributions into descriptive geometry teaching devel- opment 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 corres- ponding 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 inter- section 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 dimen- sions of the original, so as it doesn’t remain parallel direct and rela- tions segments. Therefore, in practice the method of parallel projec- tion (in particular, the orthogonal projection) is often used. Parallel projection assumes a given plane projections π1 and the di- rection 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 projec- tion 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.