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BRESENHAM’S LINE ALGORITHM

Bresenham’s algorithm enables the selection of optimum raster locations to represent a

straight line

Fig. a Location of Pixels Using Fig.  Pixels fo! Line of

B!esen"a# Algo!it"# Slo\$e% # & '.(

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Pse)*o co*e fo! B!esen"a#’s line+*!a,ing algo!it"#

Given a line from x1, y1 to x2, y2...

dx is the difference between the x components of end points

dy is the difference between the y components of end points

ix is the absolute value of dx

iy is the absolute value of dyinc is the larger of dx, dy

plotx is x1

ploty is y1 (the beginning of line

x starts at !

y starts at !

plot a pixel at plotx, ploty

increment x using ix increment y using iy

plot is false

if x is greater than inc

plot is true

decrement x using inc

increment plotx if dx is positive

decrement plotx id dx is negative

if y is greater than inc

plot is true

decrement y using inc

increment ploty if dy is positive

decrement ploty if dy is negative

if plot is true, plot a pixel at plotx, ploty increment i.

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" include #stdio. h\$

" include #graphics. h\$

" include #stdlb. h\$

void draw line (int x1, int y1, int x2, int y2,void main (void

%

draw line (1!!, 1!!, &!, &! '

void draw line (int x1, int y1, int x2 m int y2

%

int dx, dy, inc, ix, iy, x, y, plot, plotx, ploty, i 'int gd, gm '

gd ) *++- '

initgraph (gd, gm, / / '

dx ) x1 0 x2 '

dy ) y1 0 y2 '

ix ) abs (dx '

iy ) abs (dy '

inc ) max (ix, iy '

x ) y ) ! '

plot x ) x1 '

plot y ) y1 '

for (i ) ! ' i #inc ' i 

%

x  ) ix '

y  iy '

plot ) !

if (x \$ inc

%  plot ) 1 '

x 0 ) inc '

if (dx # !

plot x 0 ) 1 '

else

plotx  ) 1 '

 if (y \$ inc

%

plot ) 1 '

y 0 ) inc '

if (dy

ploty 0 ) 1 '

else  ploty  ) 1 '

if (plot

putpixel (plotx, ploty, 1

else

getch (  ' closegraph (  '

P!og!a# in T)!o+- to *!a, a line

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BRESENHAM’S -IR-LE ALGORITHM

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GEOMETRI- MOELLING

wire frame, surface and solid modeling

-LASSIFI-ATION OF GEOMETRI- MOELING

-omputer representation of the geometry of a component using software is called

a geometric model. Geometric modeling is done in three principal ways. hey are

/. 0i!e f!a#e #o*eling

1. S)!face #o*eling

2. Soli* #o*eling

hese modeling methods have distinct features and applications.

UNIT + 2

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0IRE FRAME MOELING

3n wire frame modeling the ob4ect is represented by its edges

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2 5 * 6odels 75* 8ire 9rame 6odels

1. +nds (vertices of lines are represented

by their 3 an* 4 coordinates

2. -urved edges are represented by

circles, ellipses, splines etc.

A**itional 5ie,s an* sectional 5ie,s

are necessary to represent a complex

ob4ect with clarity.

7. 75* image reconstruction is te*io)s.

:. ;ses only one gloal coo!*inate

s6ste#

1. +nds of lines are represented by their

3% 4 an* 7 coordinates.

2. -urved surfaces are represented by

suitably spaced generators. Hi**en

line o! "i**en s)!face eli#ination is

a must to interpret complex

components correctly.

7. 25* views as well as various pictorial

views can be generated easil6.

:. 6ay re

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SURFA-E MOELING

3n this approach, a component is represented by its surfaces which in turn are

represented by their vertices and edges.

For example, eight surfaces are put together to create a box, as

shown in

>urface modeling has been very popular in

aerospace product design and automotive

design.

=part from standard surface types

available for surface modeling

(ox% \$6!a#i*% ,e*ge% *o#e%

s\$"e!e% cone% to!)s% *is" an*

#es" techni

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SOLI MOELING

he representation of solid models uses the fundamental idea that a physical

ob4ect divides the 75* +uclidean space into two regions, one exterior and one interior,

separated by the boundary of the solid. >olid models are

? bounded

? @omogeneously three dimensional

? 9inite

here are six common !e\$!esentations in solid modeling.

i. S\$atial En)#e!ation 3n this simplest form of 7* volumetric raster model, a

section of 7* space is described by a #at!ix  of evenly spaced cubic volume

elements called 5oxels.

ii. -ell eco#\$osition his is a hierarchical adaptation of spatial enumeration.

7* space is sub5divided into cells. -ells could be of different siAes. hese simple

cells are glued together to describe a solid ob4ect.

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iii. Bo)n*a!6 Re\$!esentation he solid is represented by its boundary which

consists of a set of faces, a set of e*ges  and a set of 5e!tices  as well as their

topological relations.

iv. S,ee\$ Met"o*s 3n this techni

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-onst!)cti5e Soli* Geo#et!6 9-SG:

• 3n a ->G model, physical ob4ects

are created by co#ining asic

ele#enta!6 s"a\$es  Cnown as

primitives liCe blocCs, cylinders,

cones, pyramids and spheres.

• he Boolean operations liCe )nion

(∪, *iffe!ence  (0 and

inte!section E are used to carry out

this tasC. 9or example, let us

assume that we are using two

primitives, a blocC and a cylinder

which are located in space as

shown in 9ig.

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Bo)n*a!6 Re\$!esentation

Boundary representation is built on

the concept that a physical ob4ect is

enclosed by a set of faces which

themselves are closed and orient able

surfaces. 9ig. >hows a B5rep model of

an ob4ect. 3n this model, face is bounded

by edges and each edge is bounded by

vertices. he entities which constitute a

B5rep model are

Geometric entities opological

entities

Point

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0o!=ings of -SG

-SG E l

14/24

-SG Exa#\$le

+ =

REMOVE   INTERSEC

15/24

SALIENT FEATURES OF SOLI MOELING

FEATURE+BASE ESIGN

• he most fundamental aspect in creating a solid model is the concept of feature5based

design. • 3n typical 25* -=* applications, a designer draws a part by adding basic geometric

elements such as lines% a!cs% ci!cles an* s\$lines.

• 3n solid modeling a 75* design is created by starting a ase feat)!e and then a**ing other

feat)!es, one at a time, until the accurate and complete representation of the part’s

geometry is achieved.

• = feature is a basic building blocC that describes the design, liCe a =e6,a6 on a s"aft.

+ach feature indicates how to add material (liCe a !i or remove a portion of material

(liCe a c)t or a "ole.

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SURFA-E MOELING

• =ll physical ob4ects are 75dimensional.

• 3n a number of cases, it is sufficient to describe the boundary of a solid ob4ect in order

to specify its shape without ambiguity. his fact is illustrated in 9ig..

he boundary is a collection of faces forming a closed surface

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= surface can be created in several ways

i. -reating a plane surface by the linear s,ee\$ of

a line or series of lines.

ii. Re5ol5ing  a st!aig"t line about an axis.

-ylindrical, conical surfaces etc. can be generated

by this techni

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6odeling of curves and surfaces is essential to describe

ob4ects that are encountered in several areas of mechanical

engineering design. -urves and surfaces are the basic building

blocCs in the following designs

i. Bo*6 \$anels of passenger cars

ii. Ai!c!aft  bulC heads and other fuselage structures, slats,

flaps, wings etc.

iii. Ma!ine structures

iv. -onsumer products liCe \$lastic containe!s, telephones etc.

v. +ngineering products liCe mixed flo, i#\$elle!s, fo)n*!6 \$atte

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