1st chapter.doc

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Chapter 1 Introduction

Chapter One

1.1 Computer Graphics and Computer Aided Design (CG and CAD)

Graphics are used in many different areas such as industry, education,

physical field, economic for drawing histograms and engineering fields

architecture, mechanical, electrical and mechanical.[1].

A mechanical engineer converts ideas into initial drawing. With the

implementation of computers into academic and industrial institutions,

computer graphics introduced using the computer as a drawing tool.

Generative graphic, image processing and (cognitive) graphics are the maintaonomy of the !G.

Generative graphic involves generation, representation and

manipulation of the graphic o"#ects in a suita"le manner, the related non$

graphic information resides in computer file. %ince !G provide the user with

computer interaction together, the !G is em"edded in to large system such

as !A& system' hence !G has "ecome the primary tool of !A& system [].

t was not until the 1*+ that the !A& system started to appear on the

mar-et. arious types of !A& system currently eist, and they reflect

different stages in its development [/].

0he specification of the geometry of a part or product when held on a

!A& system is -nown as model and the techniue that is used to represent

the model is -nown as geometric modeling. 0here are three types of

conventional geometric modeling techniues used widely in !A& system

namely wire frame, surface modeling and solid modeling.

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Chapter One

1.2 Geometric modeling

Geometric modeling deals with the mathematical representation of

curves, surfaces, and solids necessary in the definition of comple physical

or engineering o"#ects. 0he associated field of computational geometry is

concerned with the development, analysis, and computer implementation of

algorithms encountered in geometric modeling. 0he o"#ect are concerned

within engineering range form the simple mechanical parts (machine

elements) to comple sculptured o"#ect such as ships, automo"iles,

airplanes, tur"ine and propeller "lades, etc.Geometric modeling attempts to provide a complete, flei"le, and

unam"iguous representation of the o"#ect, so that the shape oh the o"#ect can

"e2

$ easily visuali3ed (rendered)

$ easily modified (manipulated)

$ increased in compleity

$ converted to a model that can "e analy3ed computationally

$ manufactured and tested

!omputer graphic is an important tool in this process as visuali3ation

and visual inspection oh the o"#ect are fundamental parts of the design

iteration. !omputer graphics and geometric modeling have evolved

into closely lin-ed field within the last / years, especially after the

introduction of high$resolution graphics wor-station, which are now

pervasive in the engineering environment [4].


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Chapter One

1.3 Geometric Modeling Forms

&ifferent forms of geometric modeling can "e distinguished "ased on

eactly what is "eing represented, the amount and type of information

directly availa"le without derivation, and what other information can and

cannot "e derived[4].

1.3.1 WireFrame Modelers

n this modeler only the edges of a part geometry are represented

through lines or wires. 0his model is stored in computer as a set of points inorder form the vertices of the part. %ome !A& systems handle only $& wire

frame model where as the others may handle .5$& or over /& modeler.

Wire frame modelers are adeuate for many drafting applications [5].

6igure (1$1) 0he wireframe model of a computer mouse

1.3.2 !ur"ace Modelers

0he information and specification of the polygonal faces enclosed the

edges are employed in this modeling techniue. %hading and hidden surface

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Chapter One

removal is also treated in this -ind of modeling which leads to increase in

the visi"ility of the o"#ect. 0herefore surface modeling is more comple than

wire$frame modeling [5,7], surface modeling using !A& system are widely

used for parts "eing machined where new surfaces are "eing created there

are different types of surfaces[+].

6igure (1$) the parametric surface model of a computer mouse

#$pe o" sur"aces

A%lane !ur"ace

0his is the simplest surface, reuires / non$coincidental points to

define an infinite plane.

0he plane surface can "e used to generate cross sectional views "y

intersecting a surface or solid model with it.

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Chapter One

&'uled (lo"ted) !ur"ace

0his is a linear surface. t interpolates linearly "etween two "oundary

curves that define the surface. 8oundary curves can "e any wire frame

entity. 0he surface is ideal to represent surface that do not have any twists or

-in-s.

C!ur"ace o" 'eolution

0his is aisymmetric surface that can model aisymmetric o"#ects. t

is generated "y rotating a planer wire frame entity in space a"out the ais of

symmetry of a given angle.

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Chapter One

D#aulated !ur"ace

0his is surface generated "y translating a planar curve a given

distance along a specified direction. 0he plane of the curve is perpendicular

to the ais of the generated cylinder.

*&ilinear !ur"ace

0his /$& surface is generated "y interpolation of 4 endpoints. 8i$

linear surfaces are very useful in finite element analysis. A mechanical

structure is dispraised into elements, which are generated "y interpolation 4

node points to form a $& solid element.

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Chapter One

FCoons %atch

!ons patch or surfaces are generated "y the interpolation of 4 edge

curve as shown.

G&!pline !ur"ace

0his is a synthetic surface and dose not passes through all data points.0he surface is capa"le of giving very smooth contour, and can "e reshaped

with local controls.

9athematical derivation of the 8$%pline surface is "eyond the scope of this

course. :nly limited mathematical consideration will give here.

!omputer generated surface play a very important part in manufacturing of

engineering products. A surface generated "y a !A& program provides a

very accurate and smooth surface, which can "e generated "y ;! machine

without any room for misinterpretation. 0herefore, in manufacturing

computer generated surface are preferred. %ince surfaces are mathematical

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Chapter One

models, we can uic-ly find the centroid, surface area, etc. another

advantage of !A& surfaces is that they can "e easily modified.

+&e,ier !ur"ace


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Chapter One

1.3.3 !olid Modelers

0his is the most powerful tool for representing three > dimensional

o"#ects.

0he o"#ects represented "y the solid model techniue not only have

edges and surface "ut they also include volumetric and mass information.

%ome solid models are represented in computer data"ase of there edges and

vertices of the part and this is called 8$rep. %olis models can also "e created

from solid primitives such as "oes, "loc-s, cylinders, cones and spheres.

0he final part geometric is then created "y performing 8oolean operations

(#oin, intersection and su"traction) on the primitives. 0his form of geometry

is -nown as !%G or !$rep [7, ?]Geometric modeling capa"ilities are an integral part of the !A&

system. nterfacing !A& system with other systems such as !A9 is the

driving force "ehind the development of much of geometric modeling.

@erhaps one of the most fertile applications of geometric modeling is !A9.

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Chapter One

6or eample, geometric modeling ma-es possi"le process planning and

machine tool path$verification systems completely automatic [*]. 0hese

capa"ilities are very important in design and manufacturing of sculptured

surfaces. 0hese surfaces have wide applications in the aircraft and

automo"ile industries [1].

@arametric free form curves and surfaces form as essential part of the

!A& system. %chemes for defining these entities employ a wide range of

mathematical sophistication [11]. epresentation of /$& free$form surfaces

on a computer is one of the most difficult tas-s to "e handles "y the designengineers. nterpolation techniue is used to produce surfaces in $& and

/$& from sampled points data [1].

6igure (1$/) the solid model of a computer mouse

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Chapter One

1.3.3.1 Constructie !olid Geometr$C!G

!onstructive %olid Geometry (!%G) is one of the most popular

representation schemes for solid modeling "ecause it is well understood, and

easy to interface with the user and to chec- for validity.

A !%G model assumes that physical o"#ects can "e created "y

com"ining "asic elementary shapes through specific rules. 0hese "asic

shapes form what are commonly -nown as primitives, which are themselves

valid "ounded !%G models represented "y r$sets. A wide variety of

primitives are availa"le in solid modeling systems, "ut the most commonlyused are "loc-s cylinders cones and spheres as shown in 6igure 1.4, the solid

primitives in the !%G representation are defined mathematically as the

com"ination of un"ounded geometric entities separating the B5space into

infinite portions. 0hese entities are called half>spaces. 0he most commonly

used half$spaces are planar cylindrical spherical and conical and relate to the

natural uadric surfaces.

!%G primitives are represented "y the intersection of a set of half$

spaces. 0he primitive "loc- is formed "y the regulari3ed intersection of si

planar spaces. Bach half$space is epressed "y one limit of three ineualities

forming the primitive. A solid modeler supporting these primitives must "e

a"le to calculate the intersections of the given half$spaces. 9ore details on

the creation of uadric primitives and calculation of uadric intersections.

Cuadric surfaces are commonly used in !%G "ecause they represent

the most commonly used surfaces in mechanical design produced "y the

standard operations of milling turning rolling and so forth. 6or eample,

planar surfaces are o"tained through rolling and milling cylindrical surfaces

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Chapter One

through turning and spherical surfaces through cutting done with a "all$end

cutting tool.

6igure

(1$4)

6rom the userDs point of view and regardless of how the primitive is created

internally "y the system only its location geometric data and orientation data

are needed. 0he location data for each primitive encompasses the

esta"lishment of a local coordinate system and the position of the origin. 0he

geometric and orientation data are usually input "y the user. All primitives

have a default si3e guaranteed "y most modeling system.

0he 8oolean operations union difference intersection descri"ed in this

%ection are used to com"ine the r$sets formed "y the solid primitives 6igure

(1$5) shows an eample of this process.

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Chapter One

2C-/ &OC0!olid1 3C- !OD2

6igure (1$5) 8oolean operations

1.3.3.2 &inar$ #ree

0he !%G is also referred to as method to store a solid model in the

data"ase . 0he resulting solid can "e easily represented "y what is called a

"inary tree . n a "inary tree , the terminal "ranches (leaves) are the various

primitives that are lin-ed together to ma-e the final solid o"#ect (the root).

0he "inary tree is an effective way to represent the steps reuired to

construct the solid model . !omplicated solid models can "e modeled "y

considering the different com"inations of 8oolean operations reuired in the

"inary tree . 0he provides a convenient and intuitive way of modeling that

imitates the manufacturing process . A "inary tree is an effective way to plan

your modeling strategy "efore you start creating anything .

1/

a " c d e

6ig.(1$7) 0he technical illustration pipeline "ased on !%Gprimitives. (a) @rimitive types to "e selected' (") 0he

o"#ect composed from primitives' (c) 0he o"#ect after

hidden surface removal' (d) %patial layout of the o"#ectto "e rendered' (e) 0he resultant illustration


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Chapter One

1.3.3.3 C!G &oolean operations

CSG shapes can be combined by one of Following Boolean

operations to dene a more complex

CSG shape:

=nion. The resulting shape consists of all regions either

in the rst in the second or in both input Shapes.

ntersection. The resulting shape is the region Common to

both input shapes.

%u"traction. The resulting shape is the region of the rst

shape reduced by the region of the second.

6igure (1$+). 9odeling a dice using !%G. A cu"e and a sphere are

intersected' from the result the dots of the dice are su"tracted.

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Chapter One

1. Fundamentals o" Computer Graphics 4sing MA#A&

9A0EA8 [1/] is a powerful environment for linear alge"ra with graphical

presentation [14], and is availa"le on a wide range of computer platforms.

=nli-e a general$purpose language, 9A0EA8 development goes much

faster and code is dramatically shorter. n some regards, it is a higher

language than most common programming languages li-e ! or 6:0A;.

9A0EA8 is therefore a great computation environment for learning the

fundamentals of computer graphics. 9any 9A0EA8 files have "een

developed in the pastfew years "y the author and his students to help effectively presenting -ey

concepts and visuali3ing these mathematical epressions.

9ost tet"oo-s [15] covering these graphics su"#ects are primarily written

for computer science ma#ors. Algorithms to implement these concepts are

efficient "ut difficult to "e programmed in the conventional programming

languages that engineering students are familiar with. 9any engineering

students feel the comple mathematical epressions and programs hinder the

learning of these concepts.

ntroduction of computer graphics addresses , among other topics,

parametric curves and surfaces, including 8$spine and 8e3ier curves. 0hese

su"#ects applied to the design of airfoils, auto "odies and ship hulls, as well

as to commercial advertising and movie ma-ing. Without good

understanding of these graphics fundamentals, !A& users can not

effectively use associated tools.

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Chapter One

Graphics 4sing MA#A&

1. 2D Graphic

(1). plot

$& line plot.

!$nta5

plot(F)

plot(1,F1,...)

Description6

plot(F) plots the columns of F versus their inde if F is a real num"er. f F

is comple, plot(F) is euivalent to plot(real(F),image(F)). n all other uses

of plot, the imaginary component is ignored. @lot(1, F1,...) plots all lines

defined "y n versus Fn pairs. f only n or Fn is a matri, the vector is

plotted versus the rows or columns of the matri, depending on whether the

vectorHs row or column dimension matches the matri. f n is a scalar and

Fn is a vector, disconnected line o"#ects are created and plotted as discrete

points vertically at n.

Bample2

graphic.m

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x!linspace"# $%pi #'(

plot"x sin"x' )c) x cos"x' )g)'( *+c++g+ means the color of

the line.

xlabel"),nput -alue)'( * the name of axis

ylabel")Function -alue)'( * the name of / axis

title")Two Trigonometric Functions)'( * the title of the graphic

legend")y ! sin"x'))y ! cos"x')'( * annotation

grid on( * open the grid

'esults6 Figure 1

(2). !uplot

!reate aes in tiled positions.

!$nta5

su"plot(m,n,p)

su"plot(mnp)

Description6

%u"plot divides the current figure into rectangular panes that are num"ered

rowwise. Bach pane contains an aes o"#ect. %u"seuent plots are output to

the current pane.

%u"plot(m,n,p) or su"plot(mnp) "rea-s the figure window into an m$"y$n

matri of small aes, selects the pth aes o"#ect for the current plot, and

returns the aes handle. 0he aes are counted along the top row of

the figure window, then the second row, etc.

*5maple6

graphic.m

subplot"$$&'( plot"x sin"x''(

1+


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Chapter One

subplot"$$$'( plot"x cos"x''(

subplot"$$0'( plot"x sinh"x''(

subplot"$$1'( plot"x cosh"x''(

6igure (1$?)2 result of graphic1.m 6igure(1$*)2 result of graphic.m

2. 3D Graphic

(1) plot3

/$& line plot.

!$nta5

plot/(1,F1,I1,...)

Description

0he plot/ function displays a three$dimensional plot of a set of data points.

plot/(1,F1,I1,...), where 1, F1, I1 are vectors or matrices, plots one or

more lines in three$dimensional space through the points whose coordinates

are the elements of 1, F1, and I1.

*5maple6

graphic/.m

x!20:#.3:0(

y!20:#.3:0(

1?


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Chapter One

45-6!meshgrid"xy'(

7!25.819-.8125.8$2-.8$2$%5%-(

mesh"7'(

xlabel")x)'(

ylabel")y)'(

label"))'(

esult2 6igure (1$*)

6igure(1$1)2 result of graphic/.m

(2). mesh sur" sur"c sur"l

9esh2 !reate mesh plot

%urf2 !reate /$& shaded surface plot

%urfl2 %urface plot with colormap$"ased lighting

%urfc2 !reate /$& shaded surface plot with contour plot

*5ample6graphic4.m

x! 2&.3:#.$:&.3(y!2&:#.$:&(

4/6!meshgrid"xy'(

p!s;rt"12.8$


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Chapter One

subplot"$$&'(mesh"p'(legend")>?S@)'(

subplot"$$$'(surf"p'(legend")S5AF)'(

subplot"$$0'(surfc"p'(legend")S5AFC)'(

subplot"$$1'(sur"p'(legend")S5AF)'(

'esults 6 Figure(117)

6igure (1$11)2 result of graphic4.m

1st chapter.doc

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