COMPUTER AIDED DESIGN

Transcription

1 COMPUTER AIDED DESIGN (As per ANNA UNIVERSITY Revised Syllabus) (V th Semester Mechanical Engineering) Dr. S. Ramachandran, M.E.., Ph.D., Professor and Research Head,, Faculty of Mechanical Engineering Sathyabama University Jeppiaar Nagar, Chennai P. Vijayalakshmi Y.V.S. Karthick Bikash Kumar Thakur AIR WALK PUBLICATIONS (Near All India Radio) 80, Karneeshwarar Koil Street, Mylapore, Chennai Ph.: , aishram2006@gmail.com

2 First Edition: June, 2015 All Rights Reserved by the Publisher This book or part thereof should not be reproduced in any form without the written permission of the publisher. ISBN: Books will be door delivered after payment into AIR WALK PUBLICATIONS A/c No (IFSC: BKID ) Bank of India, Santhome branch, Mylapore, Chennai - 4 (or) S.Ramachandran, A/c.No (IFSC:IDIB000S201), Indian Bank, Sathyabama University Branch, Chennai Typeset by: aksharaa muthra aalayam, Chennai Ph.:

3 Contents C.3 SYLLABUS ME6501 COMPUTER AIDED DESIGN UNIT I Fundamentals of Computer Graphics 9 Product cycle- Design process- sequential and concurrent engineering- Computer aided design CAD system architecture- Computer graphics co-ordinate systems- 2D and 3D transformationshomogeneous coordinates - Line drawing -Clipping- viewing transformation UNIT II Geometric Modeling 9 Representation of curves- Hermite curve- Bezier curve- B-spline curves-rational curves-techniques for surface modeling surface patch- Coons and bicubic patches- Bezier and B-spline surfaces. Solid modeling techniques- CSG and B-rep UNIT III Visual Realism 9 Hidden Line-Surface-Solid removal algorithms shading colouring computer animation. UNIT IV Assembly of Parts 9 Assembly modelling interferences of positions and orientation tolerance analysis-massproperty calculations mechanism simulation and interference checking. UNIT V CAD Standards 9 Standards for computer graphics- Graphical Kernel System (GKS) - standards for exchangeimages- Open Graphics Library (OpenGL) - Data exchange standards - IGES, STEP, CALSetc. - communication standards. Total : 45 Periods

4 C.1 Computer Aided Design Contents I. COMPUTER AIDED DESIGN (CAD) 1.1 Introduction Product Cycle Typical product cycle Design Process Sequential Engineering Concurrent Engineering Characteristics of concurrent engineering Need for implementation of concurrent engineering Comparison of Sequential And Concurrent Engineering Computer Aided Design Why should we go for CAD? Factors considered for selecting CAD system Role of computer in CAD Benefits of CAD Engineering Applications of CAD Other applications of CAD Computer Graphics Coordinate Representation System Cartesian coordinate system World coordinate system Normalised coordinate system

5 Contents C Device coordinate system Two Dimensional Transformation Translation Scaling Rotation Three Dimensional Transformations Translation Scaling Rotation Homogeneous Coordinates Representation Homogeneous Transformation Matrices Translation Scaling Rotation Shear Application of homogeneous coordinate representation Line Drawing Digital differential analyzer algorithm Bresenham s line drawing algorithm Procedure for line generation when slope m Procedure for generating the line when slope m Clipping Viewing Transformation Possible Part : B Questions

6 C.3 Computer Aided Design 2. GEOMETRIC MODELING 2.1 Introduction Representation Of Curves Analytic curve Synthetic curves Hermite Curve Shape control of Hermite curve Limitations of Hermite curve Bezier Curve Disadvantages of Bezier curves Difference Between the Bezier curve and Hermite curve B - Spline Curves Advantages of B - spline curve Rational Curves Techniques Of Modelling Geometric Modelling Surface Modelling Surface Patch The Coons Patch The Bicubic Patch Bezier surface B-spline surface Solid Modelling Boundary representation method (b-rep) The Advantage of B-rep Disadvantages of B-rep

7 Contents C Constructive Solid Geometry (CSG) The Advantages of CSG or C-rep Disadvantages of CSG The Salient Features of Solid Modelling Package Includes Feature based design Sketched Feature Method Pick-and place Feature Modelling Tools VISUAL REALISM 3.1 Introduction Approaches or Techniques For Visual Realism Object Space and Image Space Methods Difference between Object space method and Image-space method Visibility Visibility techniques for improving efficiency of algorithms Bounding Box or Minimax Test Containment Test Backface Culling (or) Surface Test Silhouette Detection Edge Intersections Segment (scanline) comparisons Homogeneity Test Sorting Coherence

8 C.5 Computer Aided Design 3.5 Hidden Line Removal Priority Algorithm Area-oriented Algorithm Hidden Line Removal for curved surfaces Hidden Surface Removal Painter s Algorithm Depth-Buffer (Z-buffer) algorithm Warnock s Area Subdivision Algorithm Hidden Solid Removal Ray Tracing Shading Shading Methods Flat shading or constant intensity shading Smooth shading Gourand shading Phong shading Difference between Flat and Smooth shading Shading Enhancements Colouring Colour models RGB colour model CMY (or) CMYK colour model YIQ colour model HSV colour model HSL colour model User Interface For Shading And Colouring

9 Contents C Computer Animation Types of Animation Frame Buffer Animation Real-Time Playback Real Time Animation Computer Animation Techniques Keyframing Procedural Animation Motion capture Computer Animation hardware and software Animation problems ASSEMBLY OF PARTS 4.1 Introduction Assembly Modeling Heat Application Continuous welding General design points of continuous welding Localised welding Brazing and soldering Chemical Application Technique Mechanical Joints Automated Assembly Facilitating Assembly Processes Interference of Position And Orientation Tolerance Analysis

10 C.7 Computer Aided Design Definition and need for tolerances Tolerance Vs Cost Tolerance presentation Tolerance allocation Tolerance grade Geometric tolerances Geometric tolerancing examples When to use geometric tolerancing Mass Property Calculations Mechanism Simulation Interference Checking CAD STANDARDS 5.1 Introduction Standards For Computer Graphics Special Interest Group on graphics (Siggraph CORE) Other graphics standards Graphics Kernel System (GKS) Standards For Exchanging Images Bitmaps Open Graphics Library Data Exchange Standards Development of Data Exchange Format IGES IGES File structure Error handling

11 Contents C STEP Difference between IGES and STEP CALS Communication Standards Local Area Networks Wide Area Networks Fiber optic links

12 Chapter I COMPUTER AIDED DESIGN (CAD) Product cycle - Design process - Sequential and concurrent engineering - Computer aided design - CAD system architecture - Computer graphics - Coordinate systems - 2D and 3D transformations - homogeneous coordinates - Line drawing - Clipping - Viewing transformation. 1.1 INTRODUCTION The present century is known for rapid development in the fields of computer in both hardware and software. It has become the most important tool in all technological developments. The computers are becoming larger in memory and faster in computation speed. With the advancement of very large scale integration technology, computer hardware is gradually getting cheaper and now they are within the financial range of most of the industries/organizations. The entry of computers in design and manufacturing has led to the emergence of new areas known as Computer Aided Design (CAD) and Computer Aided Manufacturing (CAM). Traditionally design and manufacturing are two distinct and separate activities. However, the integration of CAD/CAM system is a boon for the design and manufacturing of engineering products. The term CIM (Computer Integrated Manufacturing) is associated with the application of computers to the manufacturing of products starting from the drawing office to the machine tools on production floor, and assembly shop to the quality control department, and stores department for shipping, and finally to the dealers for marketing.

13 1.2 Computer Aided Design PRODUCT CYCLE Design Process Customer Feed Back Product Concept Design Drafting And Documentation Manufacturing Process Order New Equipment and Tooling Process Planning Marketing Packing Quality Control Production Production Planning Fig:1.1 Product Cycle for Design and Manufacture Manufacturing process The process planning specifies the sequence of production operations required to make the product. New equipment and tools must sometimes be acquired to produce the product. The next stage is scheduling which provides a plan that commits the company to the manufacture of certain quantities of the product by certain dates. Once all of these plans are formulated, the product goes into production, followed by quality testing, and delivery to the customer.

14 Computer Aided Design Typical product cycle The impact of CAD/CAM involves in all the different activities of the product cycle, which is shown in Fig Computer - aided design and automated drafting are utilized in the conceptualization, design, and documentation of the product. Computers are used in production to monitor and control the manufacturing operation. In quality control, computers are used to perform inspections and performance tests on the product and its components. Design process Figure 1.1 shows the various steps involved in the product cycle. The product cycle is driven by customers and markets which demand the product. It is realistic to think of these as a large collection of diverse industrial and consumer markets rather than one monolithic market. Depending on the particular customer group there will be differences in the way the product cycle is activated. In some cases, the design functions are performed by the customer and the product is manufactured by a different firm. In other cases, design and manufacturing is accomplished by the same firm. But somehow, the product cycle begins with an idea of product or product concept. This concept is generated, refined, analyzed, improved and translated into a plan for the product through the design engineering process. The plan is documented by drafting a set of engineering drawings showing how the product is made and providing a set of specifications indicating how the product should perform.

15 1.4 Computer Aided Design With the engineering changes (i.e) drafting which typically follow the product throughout its life cycle this completes the design activities. Computer Aided design Computer Automated Drafting and Documentation Product Control Design Engineering Drafting Customers and Markets Order New Equipment and Tooling Process Planning Computer Aided Process Planning Quality Control Production Scheduling Computer Aided Quality Control Computer Controlled Robots, Machines, Etc Fig:1.2 Product Cycle with CAD / CAM Computerized Scheduling Material Requirement Planning, Shop Floor Control In the design and production operations of modern manufacturing techniques, the computer has become a pervasive, useful and indispensable tool. It is strategically important and competitively imperative that manufacturing firms and the people who are employed by them understand CAD/CAM.

16 Computer Aided Design DESIGN PROCESS Design process is an activity that facilitates the realization of new products and processes through which technology the human needs and aspirations are satisfied. Design process cannot be summarized in a formula. It can be the work of an individual or efforts of a group of people. Design process is not straight forward but it is an iterative process. It means that after processing every step of design process one should go to the previous steps. There are many ways of defining the steps in a traditional design process. In 1975 Deutschman has summarized the design process in the following nine steps. (i) Recognition of need (ii) Problem definition and specification (iii) Feasibility study (iv) Design synthesis (v) Analysis and preliminary design (vi) Detailed design (vii) Prototype building and testing (viii) Design for mass production (ix) Product release Later on, in 1983, Shirley has combined few of the design steps and redefined the design process in six steps. A typical block diagram for the classical or conventional design approach is shown in Fig 1.3.

17 1.6 Computer Aided Design Recognition of Needs Problem Definition Modification Synthesis Modification Analysis and Optimization Design Review Presentation Fig: 1.3 Conventional Design Process 1. Recognition of need The design process involves initially in identifying the need. The product begins with a need based on market survey and customer s demand. The data is collected via observation and/or a detailed survey. There may be: Adoption of existing design Modifications in the existing design Completely new design

18 Computer Aided Design Problem definition In the problem definition, the designer s task is defined and criterion for the performance of designed product is specified. The designer collects different information about the existing products of similar type, about the market potential, about the manufacturing constraints, about the legal requirements and standards and so on. The specifications, constraints and design criteria may be: Specifications: Such as power required, life of product, efficiency, reliability, cost, temperature range, etc. Constraints: (i.e) Maximum and minimum values of the specifications. Criteria: Used to decide the goodness of the design amongst the alternative design process, e.g, for shaft design, the strength and stiffness criteria should be specified, diameter of the shaft based on certain theory of failure. 3. Synthesis Synthesis is nothing but the conceptualization. Synthesis forms a design solution to satisfy the need. The end goal of synthesis is a conceptual design of the product. In this phase, sketches of different components and assembly are drawn. The feedback received from the marketing professionals also help to build up a strong concept of design. Synthesis requires a sound technical background, creativity and experiences of the designer. In synthesis, the design parameters are adjusted to get a perfect fit; if fit does not occur, the designer can

19 1.8 Computer Aided Design change the specifications or sometimes even modify the need specified in Recognition of need. 4. Analysis and optimization Analysis must be followed for every synthesis. Analysis is a highly iterative process and requires good mathematical knowledge. Analysis means critically examining an already existing or proposed design to judge the suitability for the task that is to be performed by the designer. Analysis determines whether the performance complies with the requirements or not. The analysis subprocess selects suitable material and its associative mechanical properties. Calculations are performed to determine the size or parameters using the physical laws such as laws of momentum, motion, energy conservation, etc. The different types of engineering analysis are stress-strain analysis, kinematic analysis, dynamic analysis, vibration analysis, thermal analysis, fluid-flow analysis, etc. Optimization means the best possible solution for the given objectives. All possible solutions are analyzed and optimum is selected. After every phase of design process, the designer may go to the previous steps and modify them. 5. Design review Design review is nothing but evaluation. Evaluation means measuring the design against the specifications set in the problem definition. It usually involves prototype building and testing of the product to ascertain operating performance or factors such as reliability. The result of evaluation phase may yield a satisfactory design or it may lead to further modifications in the design parameters. The changes into the prototype assembly are incorporated

20 Computer Aided Design 1.9 during continued testing of the product. This process is repeated until satisfactory performance of the component and assembly is achieved. 6. Presentation Presentation means drafting. The final stage in design process is the presentation and documentation of the design on paper. This forms an interface between the design and the manufacture. Production drawing shows various design parameters, machining parameters, tolerances etc. The design is presented using the drawings, parts list, materials, specifications, etc. The design is not complete if one cannot sell it. Therefore a great deal of effort should be applied in the presentation of the design. 1.4 SEQUENTIAL ENGINEERING The conventional product cycle is sequential. It contains quality control, product design, manufacturing process with every activity is carried out in a sequential manner. Design Planning Manufacturing Quality Marketing Fig:1.4 Sequential Engineering

21 1.10 Computer Aided Design In sequential engineering, each department is insulated i.e. each department functions separately. There is no interaction among the groups. This is time consuming as for example, if any flaw is encountered during the quality check stage, the product has to go through the whole cycle from the start. 1.5 CONCURRENT ENGINEERING Concurrent engineering is known as simultaneous engineering. Here, while the product is designed, the design and manufacturing processes are carried out simultaneously. This technique facilitates the design engineer to improve the efficiency of product design and process. This is effective interaction of process planning and product design. Concurrent engineering also influences the cycle cost of product. Concurrent engineering also unites people from different functional areas. The block diagram of concurrent engineering is shown in Fig.1.5. Inspection Manufacturing Marketing Servicibility Design Co-ordinator Sales Assembly Packaging Function Fig:1.5 Simultaneous (or) Concurrent Engineering

22 Computer Aided Design 1.11 In a traditional designing process, complete design descriptions are produced in the form of engineering drawings and diagrams and these are then issued by the design department of a company for analytical evaluation, and for the preparation of plans and instructions for manufacture. Inevitably, the manufacturing specialists and design analyst find aspects of the design that should be improved, and so the design is returned to the design department for modification and reissue of the drawings. In some cases reissue may occur many times - one large aerospace manufacturer is said to change each drawing an average of 4.5 times before final release - and thus the whole process is both time consuming and costly. Furthermore because the considerations of manufacturing and other specialists are taken into account after the design drawings have been produced, the design department tends to concentrate on functional aspects of the design at the expense of ease of manufacture, maintainability and so on. Concurrent engineering aims to overcome all of these limitations, by bringing together a design team with the appropriate combination of specialist expertise to consider early in the design process, all elements of the product life cycle from conception through manufacture and use in service to maintenance and disposal Characteristics of concurrent engineering Constant and un-interrupted evaluation of design process and development process. Fast and speedy information exchange achieved through internet, LAN etc. Rapid prototyping.

23 1.12 Computer Aided Design More attention and concern for satisfying customer needs. Focus on new technologies Need for implementation of concurrent engineering In order to effectively implement concurrent engineering, suitable training programs need to be organized. The power should be decentralized which allows effective participation of workers from all levels to work together and solve the problem. Concurrent engineering ensures that the problem between design and manufacturing, design and production, etc. are removed. In concurrent engineering there is simultaneous interaction between the groups, moreover all the procedures are split into simple tasks which are easier to complete. 1.6 COMPARISON OF SEQUENTIAL AND CONCURRENT ENGINEERING Sequential engineering is followed in conventional manufacturing. As mentioned earlier, this process flows in one direction and back-tracking at any stage is time consuming and has to be started from first step. Moreover, the activities of each department is localised and isolated. Thus interaction among the group is lacking. On the other hand, concurrent engineering facilitates an effective interaction between various departments, such as production planning, production development and

24 Computer Aided Design 1.13 manufacturing. Thus the spirit of team work is developed. Moreover specialists from different departments interact with each other and improve the efficiency of the production design. Concurrent engineering includes special methods such as DFMA (Design for manufacturing and assembly) and FMEA (Failure mode and effect analysis) for flaw finding and design optimizing. Another difference is, in the infant stages numerous changes will be encountered in the product cycle and these changes progressively come down for the rest (i.e.) remaining period for concurrent engineering. In case of sequential engineering changes may not be constant and predicted, but the magnitude of change differ at every stage. This comparison is depicted in the Fig Concurrent Engineering Number of Change Sequential Engineering Product Development Cycle Fig:1.6 Comparison Graph Detailed comparison is shown in Fig. 1.7.

25 1.14 Computer Aided Design Sequential Product Development Individuals Slow Changes Long Lead Time Lower Quality Start Requirements Definition Product Definition Concept Detail Embodiment Process Definition Production and Distribution Start Requirements Definition Errors, Changes and Corrections Concurrent Product Development Teams Fast Changes Short Lead Time Higher Quality Product Definition Concept Embodiment CE Life Cycle Time Detail Process Definition Production and Distribution Errors, Changes And Corrections Finish Time Saved Fig:1.7 Sequential Versus Concurrent Product development from Start to Finish Finish

26 1.7 COMPUTER AIDED DESIGN In the field of computer science and technology the advancements have resulted in the emergence of very powerful hardware and software tools that offer scope in the conventional design process, which results in the improvement of quality of the product. Thus, Computer aided design is the automation of design process. CAD is the use of computer to aid in the design process of an individual part, a subsystem or total system. CAD is the process of creation and development of a prototype on a computer to assist the engineer in the design process. CAD creates a three dimensional geometric model on the computer to examine the geometric and manufacturing requirements of an object Why should we go for CAD? There are four fundamental reasons for implementing the CAD system, which are as follows. (i) To increase the productivity of the designer. (ii) To improve the quality of the design. (iii) To improve communications. Computer Aided Design 1.15 (iv) To create a database for engineering. 1. To increase the productivity of the designer The product and its components, subassemblies and parts can be visualised quickly by the designer using CAD. Time for synthesis, analysis and documentation of the

27 1.16 Computer Aided Design design will be reduced. Even it reduces design time and cost. 2. To improve the quality of design Without any error, quick alterations can be made in the design with the help of CAD. 3. To improve communications Better documentation of the design, fewer drawing errors with greater legibility will be provided by CAD. 4. To create database for engineering The product geometries and dimensions, bill of materials, etc., will make a design database, which are essential input for manufacturing of the product Factors considered for selecting CAD system (a) Reliability (b) Compatibility with other systems (c) Cost factors (d) Memory size and storage requirement. (e) Type of peripherals required Role of computer in CAD (i) Computer improves accuracy of design. (ii) Various dimensions, and other design attributes can be conveniently manipulated by computers. (iii) Another role played by computers is creation of part libraries for standard components. Similarly multiple components can be included in these part libraries.

28 Computer Aided Design 1.17 (iv) Moreover the modification of the model is very simple which helps the designer to look in for further improvement. (v) Calculation of various geometric properties such as area, volume, and dimenioning can be accurately done. The application of computers to the designing process is shown in Fig Traditional Design Process Recognition Of Need Computer Aided Design Problem Definition Synthesis Geometric Modeling Analysis And Optimization Engineering Analysis Evaluation Design Review And Evaluation Presentation Automated Drafting Fig:1.8 Application of Computer to the Design Process

29 1.18 Computer Aided Design BENEFITS OF CAD Some important benefits of CAD are: CAD is faster, consistent and more accurate than the classical design process. The manipulation of various dimensions, attributes is easily possible under the CAD environment. Some CAD software is parametric and possesses parent-child relationship between the component and assembly. The efficiency, effectiveness and creativity of the designer improve drastically, leading to high quality engineering designs. The added advantage of CAD is excellent graphical representation and production drawing of product with exchange facility between different phases through e-drawing. Easy modification and improvement of product is possible in CAD environment taking care of further needs. In CAD, it is not required to repeat the design or drawing of any component with modified dimensions. It is possible to copy and modify the designs as per the new dimension within seconds, including geometric transformations, material replacements, if needed. Graphics simulation and animation makes it possible to study the real-time behaviour of CAD assembly. This is useful for inspecting tolerance and interface between the matching components of the model.

30 Use of standard components in part libraries makes very fast CAD modeling. For a specific task, various components, subassembly may be stored in part libraries for future use. 3D visualization of model from several orientations eliminates the need of making a prototype. Computer Aided Design 1.19 The documentation at various design phases is efficient, easier, flexible and economical. The coordination among the groups and sharing of design data and results is possible in CAD environment. Most CAD software can link the geometric model directly to its manufacturing counterpart, i.e, CAM to carry out production. 1.9 ENGINEERING APPLICATIONS OF CAD The CAD system is extensively used in mechanical engineering and manufacturing industries. CAD increases productivity of designer through the visualization of components/assemblies. The engineering applications of computer aided design (CAD) are shown in Table 1.1. Applications 1. Structural design of Aircraft Detail CAD analyzes the turbulent flow pattern in aerospace structures

31 1.20 Computer Aided Design Applications 2. Aircraft simulation 3. Real time simulation 4. Automobile industries 5. Architectural design 6. Pipe routing and plan layout design 7. Electronic industries Detail The complex situation during the flight can be simulated in flight simulator using the CAD software, which avoids lengthy delay, saves fuel cost and provides better than pilots. It is possible to study the real-time behaviour and inspection of critical parts subjected to repeated stresses due to mechanical loading. CAD provides various types of space curves for the aerodynamic design of automobile surfaces. CAD has tremendous scope in architectural design of bridges, buildings, structures, etc. It is possible to estimate the building materials requirements for a similar design with different design parameters. CAD Design optimizes the pipe layout and plant layout in chemical plants. CAD is applicable in the design of Integrated circuits and printed circuit board design used in electronic equipment/machines.

32 Computer Aided Design 1.21 Applications Detail 8. Dynamic analysis of mechanical systems CAD design is useful for estimating the dynamic forces, reactive forces of mechanical systems at various time intervals. 9. Kinematic analysis Similar to dynamic force analysis, CAD estimates the kinematic quantities such as displacement, velocity and acceleration of various links for different configurations of the mechanism. 10. Mesh data preparation for finite element analysis The input data for FEA of a structure consists of geometrical and mechanical properties, loading and boundary conditions, CAD system generates the best mesh data suitable to a particular problem. It is possible to represent data graphically to quickly guess the results. Table: Applications of CAD So far, CAD systems have been described in very general terms. More specifically, they can be thought of as comprising: Hardware: The computer and associated peripheral equipment. Software: The computer program(s) running on the hardware.

33 1.22 Computer Aided Design Data: The data structure created and manipulated by the software. CAD systems are no more than computer programs perhaps using specialized computing hardware. The software normally comprises a number of different elements or functions that process the data stored in the database in different ways. These are represented diagrammatically in Fig. 1.9 and include elements for: Database Data Working Data Model Definition Functions Component Models Drawings Standards Geometry Associated Data Manufacturing Manipulation Picture Generation Input Output User Library Data Utilities Data Base Management Applications Model definition: For example, to add geometric elements to a model of the form of a component. Fig:1.9 The Architecture of Computer Aided Design System Model manipulation: To move, copy, delete, edit or otherwise modify elements in the design model.

34 Picture generation: The generate images of the design model on a computer screen or on some hard-copy device. User interaction: To handle commands input by the user and to present output to the user about the operation of the system. Database management: For the management of files that makeup the database. Applications: These elements of the software don t modify the design model, but use it to generate information for valuation, analysis or manufacture. Utilities: Computer Aided Design 1.23 Parts of the software that do not directly affect the design model, but modify the operation of the system in some way. For example, To select the colour to be used for display, or the units to be used for construction of a part model. These features may be provided by multiple programs operating on a common database or by a single program encompassing all of the elements COMPUTER GRAPHICS Computer graphics is the language of engineers, which provides a powerful tool for communication

35 1.24 Computer Aided Design among the team members associated with design, manufacturing and sales of a product. Computer graphics involves the creation, storage, manipulation and integration of models and images of the object by means of a digital computer. The shaded and coloured two-dimensional, three-dimensional and higher-dimensional models are generated to bring the realism in different objects such as natural scene, animation, flight simulation, navigation, commerce, advertising, etc. In recent years, computer graphics become a very powerful tool for the development of high quality pictures rapidly, consistently and economically. Computer graphics is an important tool in computer aided manufacturing (CAM) where the graphical data of the object, converted into machine data, operates CNC machines for production. The synthesis of real or imaginary objects from their computer model is concerned by computer graphics. The image processing is the reverse of computer graphics, which performs the analysis of pictures. The computer graphics and image processing techniques together deal with the computer processing of pictures. Both use raster displays, combined in interactive image processing. Computer graphics is very popular in industries, business, education, medicine, fashion, entertainment, etc. It has made things easier to visualize.

36 Computer Aided Design COORDINATE REPRESENTATION SYSTEM Every CAD/CAM system follows certain type of coordinate representation system. While displaying an image, the mapping of coordinates of the object consisting of 2D and 3D primitives occurs onto the display device or workstation. This is obtained through the coordinates transformations, also referred to as viewing transformations Cartesian coordinate system Cartesian coordinate system is mostly followed by the graphics software design. If coordinates of an image is defined in other coordinate system (eg., cylindrical or spherical coordinate system), they must be converted into the cartesian coordinates before using in the graphics software. Local Co-ordinate System ( Cartesian, Polar,Spherical) World Co-ordinate System ( Cartesian ) ( x w, y w ) Normalized Co-ordinate System ( x n, y n) Device Co-ordinate System ( x d, y d) Polar Co-ordinate Display Cartesian Co-ordinate Floating Point Numbers Floating Point Numbers Normalised Numbers Integer Numbers x w 0 x n 1 0 x d x max übçj0% Fig:1.10 Viewing Transformation Sequence from Local Co-ordinates to Device Co-Ordinates.

37 1.26 Computer Aided Design Fig shows the viewing transformation sequence from local coordinates to the device coordinates. Broadly, three types of coordinate system are required to input display and store the geometry of graphics model during the modeling process World coordinate system World coordinate system (WCS) is the working or user coordinate system, which describes the image in cartesian coordinates. Firstly the shape of objects is created in the form of graphics of image, using separate coordinate reference frames, known as local coordinate system. The units are the user units, which can be anything like mm, m, km, foot, etc., Once all the objects in graphics images are described by their individual local coordinate systems, they can be placed in the graphics images with reference to one single reference frame, i.e, cartesian coordinates. The WCS may (i) (ii) (iii) (iv) have numerical values that depend on the type of problem. have positive values or negative values. have range from to in both x and y directions. be represented by floating point numbers. (e.g ; mantissa = and exponent = 3) Normalised coordinate system For modeling, each graphics output device may follow different coordinates. In some images, we might want to specify objects dimensions in fraction of a foot, while for some other applications it may be mm (or) km.

38 Computer Aided Design 1.27 It is, therefore, desirable to convert the world coordinates into the normalized coordinates. i.e, Normalized coordinate system (NCS), to make the coordinate system independent of several graphics output devices. Normalization may be done from (0,0) to (1,1) with origin at (0,0) in the lower left corner and co-ordinate (1,1) on the right top corner of the display devices. To accommodate the differences in scales and aspect ratios, the mapping of normalized coordinates into square area of the displays is required to maintain the proper proportions of various images Device coordinate system The device coordinate system is one in which the image of normalized coordinate system will be displaced in the output device like monitor (soft device), printer/plotter (hard device). A graphics device understands the device coordinate system in terms of pixels, cm, inch, etc. Depending upon the pixel density, the DCS would vary from one system to another. The features of device control system are follows: (i) (ii) (iii) (iv) The pixel density (eg: ) of the display device depends on the maximum size. Positive values have to be considered. Always fixed in size (i.e. size of display surface) irrespective of the problem. It should be always represented by an integer number.

39 1.28 Computer Aided Design TWO DIMENSIONAL TRANSFORMATION Basic transformation The transformations are used to reposition and resize two-dimensional objects on the displays (alternatively, in the database). The three basic transformations are as follows: (i) Translation (ii) Scaling (iii) Rotation Any point is represented by its coordinates x, y, z from the reference datum in the three dimensional system. For simplicity and to make it easy to understand, we can analyse the two dimensional system. Then we can discuss the 3D system Translation Translation is one of the important types of transformation. This is used to move the entity. After moving, all points of new entity are parallel to all points of the old entity. Moving the drawing or model across the screen is called translation. This is accomplished by adding to the coordinate of each corner point the distance through which the drawing is to be moved x x m y y n...(1.1) where, x, y coordinates of the translated point x, y coordinates of the original point m, n movements in the x and y direction

40 Computer Aided Design 1.29 In matrix notation this can be represented as x, y x, y T...(1.2) where, T m, n, the translation matrix Problem 1.1: Translate a two dimensional rectangle as shown in figure, by adding 4 units in x - coordinate and 3 units in y - coordinate. Given data T 4, 3 ( 1,5 ) ( 4,5 ) x 1 1; y 1 1 x 2 4; y 2 1 x 3 4; y 3 5 ( 1,1 ) ( 4,1 ) x 4 1; y 4 5 Fig. To find New translated rectangle Solution From equation (1.2) We know that x, y x, y T...(1) Expanding the equation (1) for 4 coordinate rectangle x 1 x 2 x 3 x r y 1 x 1 y 2 x 2 y 3 x 3 y 4 x 4 y 1 y 2 y [T] 3 y 4

41 1.30 Computer Aided Design Substitute the given data values in equation (2), then we have...(2) x 1 x 2 x 3 x 4 y 1 1 y 2 4 y 3 4 y [4 3] x 1 x 2 x 3 x 4 y 1 5 y 2 8 y 3 8 y (3) Result Equation (3) is the New translated rectangle y 9 8 ( 5,8) ( 8,8) ( 1,5 ) ( 4,5 ) New Translated Rectangle ( 1,1 ) ( 4,1 ) ( 5,4 ( 8,4 ) Orig inal Rectangle Fig: Original and New Translated Rectangle. x

42 Computer Aided Design 1.31 Problem 1.2: Translate a triangle ABC with coordinates A (1,1) B (3,5), C (1,3) about the origin by 3 units in x - direction and 3 units in y direction. Given data To find Solution T 3, 3 x 1 1; y 1 1 x 2 3; y 2 5 x 3 1; y 3 3 Translate the triangle From equation (1.2), we know that, x, y x, y T...(1) Expanding the equation (1) for the triangle x 1 x 2 x 3 y 1 x 1 y 2 x 2 y 3 x 3 y 1 y 2 [T] y 3...(2) Substitute the given data in equation (2) so, equation (2) becomes, x 1 x 2 x 3 x 1 x 2 x 3 y 1 1 y 2 3 y 3 1 y 1 4 y 2 6 y [3 3]...(3)

43 1.32 Computer Aided Design Equation (3) is the new translated triangles coordinates. Result y 8 ( 6,8 ) ( 3,5 ) ( 4,6 ) Translated Triangle 4 3 ( 1,3 ) ( 4,4 ) 2 1 ( 1,1 ) Original Triangle Fig: Original and New Translated Triangle x Problem 1.3: Consider the line defined by, L Translate the line 3 units in x - direction and 4 units in y direction. Given data T 3, 4

44 Computer Aided Design 1.33 x 1 1; y 1 2 x 2 3; y 2 4 To find Translate the line Solution From equation 1.2 we get, x, y x, y T...(1) Expand the equation (1) for 2 points. So equation (1) becomes, x 1 x 2 y 1 x 1 y 1 y 2 x y 2 2 [T]...(2) Substitute the given data in equation (2) x 1 x 2 y 1 1 y [3 4] x 1 x 2 y y (3) Equation (3) is the new translated line coordinates.

45 1.34 Computer Aided Design Result y ( 3,4 ) ( 4,6 ) ( 6,8 ) Translated Line ( 1,2 ) Original Line Fig: Original and New Translated Line x Scaling Scaling of an element is used to enlarge it or reduce its size. A drawing can be made bigger by increasing the distance between the points of the drawing. In general, this can be done by multiplying the coordinates of the drawing by an enlargement or reduction factor called scaling factor, and this operation is called scaling. The coordinates of an object is multiplied uniformly by the scaling factor.

46 Computer Aided Design 1.35 p y p S x y y x Fig: Scaling of a Rectangle s x From Fig it is shown that a rectangle is enlarged using scaling. The scaling matrix is given as follows: S S x 0 0 S y...(1.3) where, S Scaling matrix S x Scaling in x direction S y Scaling in y direction S x and S y need not be equal. A circle can be transformed into an ellipse by unequal scaling factors S x and S y. If the scaling factors are less than 1, it will reduce

47 1.36 Computer Aided Design the size of the object and the object is moved towards the origin. If it is greater than 1, then it will enlarge the size of the object and object is moved away from the origin. P [x, y] [S x x, S y y] The above equation can be represented in a matrix form as follows: (i.e) P S x 0 0 x S y y...(1.4) P [S] [P] While zooming or magnifying the object, uniform scaling ((i.e) S x S y ) is applied. Zooming or magnifying is only a display attribute and is used only to the display and not stored in actual geometric database. Problem 1.4: A line AB factor of 2. Show the transformation. is enlarged by a scaling Given data S x S y 2 x 1 1 ; y 1 2 x 2 3 ; y 2 4 To find To obtain the transformation using scaling.

48 Computer Aided Design 1.37 Solution The scaling matrix, S S x 0 0 S y Results Original line matrix AB Scaled line matrix is determined as follows. A B y 2 0 [S] [AB] [A B] [A B] 4 8 are new scaled line coordinates 8 (6,8) (2,4) (3,4) New scaled Line (1,2) Original Line x

49 1.38 Computer Aided Design Rotation Rotation is also an another important transformation. In this transformation, all the points of an object are rotated about the origin (or) about any base point by an angle. For a positive angle, the object is rotated in anticlockwise direction and viceversa. Consider a point P in xy plane. P is rotated in the anticlockwise direction to get new position P through an angle of as shown in Fig y P ( x,y ) x 1 P ( x, y ) y 1 r r y O x x Fig:1.12 Rotation of a Square The new position, The original position is, [x, y]

50 Computer Aided Design 1.39 x, y The rotation of the object by some angle will also move the object. In matrix notation, the procedure would be as follows: where, R cos sin x, y x, y R...(1.5) sin cos Rotation matrix...(1.6) Problem 1.5: Consider the line of coordinates (1,1) and (2,4) Rotate the line about the origin by 30. Determine the transformation of the line. Given data To find 30 Solution x 1 1; y 1 1 x 2 2; y 2 4 Transformation of the line We know that from equation (1.5) x y x, y R...(1) where, R cos sin sin cos...(2) Apply as 30 in equation (2)

51 1.40 Computer Aided Design R cos 30 sin 30 R sin 30 cos 30 Apply equation (3) in (1) x y 1 2 x y (3) The effect of applying the rotation matrix to the line is shown in result. Result y 6 (-0.268,4.464) 5 4 ( 2,4 ) Rotated Line 3 2 Original Line 1 (0.366,1.366) ( 1,1) -x x

52 1.14 THREE DIMENSIONAL TRANSFORMATIONS Transformations by matrix methods can be extended to three-dimensional space. The procedure used for two dimensional transformations can be extended to three dimensional by adding z axis. The transformation matrix will then be Translation For a three dimensional element, the translation point will be given as, T m, n, p...(1.7) where, m, n, p are the coordinates of translation point or increment. In matrix notation, it is given as, Computer Aided Design 1.41 x, y, z x, y, z T...(1.8) Scaling The scaling transformation is given by, S m n p...(1.9) where, m, n and p are the units needed, to be scaled. For equal values of m, n and p, the scaling is linear Rotation For each axis, the rotation in three dimensions varies.

53 1.42 Computer Aided Design For Z axis matrix, Rotation about the Z axis by angle is given by the cos R z sin 0 sin cos (1.10) For Y axis Rotation about y - axis by angle is given by matrix. cos R y 0 sin sin 0 cos...(1.11) For x axis Rotation about x - axis by angle is given by matrix, 1 R x cos sin 0 sin cos 1.15 HOMOGENEOUS COORDINATES REPRESENTATION...(1.12) The difficulty in image manipulation, incorporating all the five types of geometric transformations, can be removed if represented by a single matrix equation. This will be possible only if points are represented in homogeneous coordinates. The homogeneous coordinates are obtained by adding the third coordinate to a point. This facilitates the image manipulation with a single transformation matrix for all types of geometric transformations.

54 Computer Aided Design 1.43 In homogeneous coordinate system, mapping between the n-dimensional spaces with n 1 dimensional spaces occur, it points in n-dimensional coordinates are represented by the corresponding n 1-dimensional coordinates. This is obtained by introducing a scale factor along the cartesian coordinates. For 2D coordinates, instead of being represented by a pair x, y, each point is represented by triple coordinates x, y, h where h 0, is the scalar factor. The relationship between the cartesian coordinates and homogeneous coordinates of a point is given by, (i.e) x x h ; y y h x xh; y yh...(1.13) Generally, h 1 represents a homogeneous coordinate x, y, 1 for a point x, y in computer graphics. Two sets of homogeneous coordinates x, y, h and x, y, h represent the same point if and only if one is a multiple of the other. Different homogeneous coordinates can represent each point. For example: homogeneous coordinates of a point (3,2) may be expressed by the coordinate triple as (3,2,1) or (6,4,2) or (9,6,3). If we take all triples representing the same point, we get a line in 3D space. Each homogeneous point represents a line in 3D space. If these points are homogenized points from the

55 1.44 Computer Aided Design plane, defined by equation h 1, in x, y, h space is shown in Fig Graphically, the scale factor h may be interpreted as the cartesian image on a plane parallel to the xy plane and at unit distance away from the origin along z - direction. The points with h 0 are called points at infinity (not represented on the planes), and will not appear very often in the discussion. y y h h=1 Third Principal A xis Perpendicular to xy Plane x Fig:1.13.The (x,y,h) Homogeneous Coordinate Space With h=1 Plane x This type of visualization is not possible in 3D geometric transformations. In computer graphics, we shift the coordinates of the object model from x, y coordinate space to x, y, 1 coordinate space, keeping h 1. Thus 1 should be added while defining any coordinate in 2D. For example, a point in 3D 3, 4, 2 should be represented as (3, 4, 2, 1).