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BRESENHAM’S ALGORITHM

BRESENHAM’S LINE ALGORITHM

UNIT - II

MET71 COMPUTER AIDED DESIGN

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

straight line. In this algorithm either pixels along X or Y directions are incremented by one unit

depending upon the slope of the line. The increment in the other direction is determined by

examining the error or distance between actual line location and the nearest grid locations.

The principle of Bresenham’s algorithm can be explained with the aid of Fig a. If the slope

of the line (in the first octant) is more than 1/2, the pixel point in the Y direction is shifted by one.

Thus lines L1 and L2 passes through pixel (0, 0). For line L2 slope is greater than 1/2; hence the

pixel point is (1, 1) whereas for L1 the slope is less than 1/2 and hence (1, 0) is the pixel point.

Bresenham’s algorithm selects optimum raster locations with minimum computation. To

accomplish this, the algorithm always increments by one unit in either X or Y depending upon the

slope of the line. The increment in the other variable either zero or one is determined by

examining the distance (error) between the actual line location and the nearest grid location. Only

the sign of this error needs be examined.

Consider the line of slope m = 0.4 and passing through (0, 0) in Fig (a). The error team e

is initialized to –1/2. The next raster point can be determined by adding the slope (m) to the error

term.

i.e. e = e + m

e = – 0.5 + 0.4 = – 0.1

Since e is negative, the line will pass below the middle of the pixel. Hence the pixel is at

the same horizontal level i.e., (0, 1). For the next location X is incremented to 2. Error e = -0.1 +

0.4 = 0.3. Since e is positive the line passes above the middle of the raster line. Hence the

location of pixel is (2, 1). Before examining the next pixel location the error term has to be re-

initialized as its value is positive. Re-initialization is done by subtracting one from the current e

value. Hence e = 0.3 - 1 = -0.7. Adding the slope 0.4 we get e = -0.3. Table 3.2 shows the

computed values and the location of pixels. A plot of the pixel location is shown in Fig. (b).

Table: Calculation of Pixel Position

M.SRINIVASSAN, ASST. PROF 1 RGCET


Fig. a Location of Pixels Using Fig. b Pixels for Line of

Bresenham Algorithm Slope, m = 0.4

The speed of the Bresenham’s algorithm can be increased by using integer arithmetic and

eliminating division to determine slope.The pseudo code and a C-program for implementing

Bresenham’s algorithm are given below:

Pseudo code for Bresenham’s line-drawing algorithm

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 dy

inc is the larger of dx, dy

plotx is x1

ploty is y1 (the beginning of line)

x starts at 0

y starts at 0

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.

Program in Turbo-C to draw a line

# 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 (100, 100, 50, 50) ;

}

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

{

{

{

else

}

int dx, dy, inc, ix, iy, x, y, plot, plotx, ploty, i ;

int gd, gm ;

gd = DETECT ;

initgraph (&gd, &gm, “ “) ;

dx = x1 – x2 ;

dy = y1 – y2 ;

ix = abs (dx) ;

iy = abs (dy) ;

inc = max (ix, iy) ;

x = y = 0 ;

plot x = x1 ;

plot y = y1 ;

for (i = 0 ; i <inc ; i ++)

x + = ix ;

y + iy ;

plot = 0

if (x > inc)

plot = 1 ;

x – = inc ; if

(dx < 0) plot

x – = 1 ;

plotx + = 1 ;

if (y > inc)


{

else

}

else

}

plot = 1 ;

y – = inc ;

if (dy)

ploty – = 1 ;

ploty + = 1 ;

if (plot)

putpixel (plotx, ploty, 1)


getch ( ) ;

closegraph ( ) ;

BRESENHAM’S CIRCLE ALGORITHM

An efficient algorithm for generating a circle has been developed by J. Bresenham.

Values of a circle centered at the origin are computed in the sector X = 0 to X = R /2 where R is

the radius of the circle. The symmetry of the circle is used to obtain the pixels corresponding to

other sectors.

Bresenham’s circle algorithm can be explained as follows:

Consider an origin-centred circle. The algorithm begins at X = 0 and Y = R. In the first

quadrant of the circle, Y is a monotonically decreasing function of X. Referring to Fig., (Xi, Yi) is a

point on the circle. For clockwise generation of the circle there are only three possible selections

of the next pixel, which represents the circle. These positions are also shown in Fig. 3.4. The

algorithm is designed to choose the pixel which minimizes the square of the distance between

one of these pixels and the true circle, i.e., the minimum of

H = [ (Xi + 1) 2 + (Yi)2 – R 2]

V = [ (Xi ) 2 + ( Yi – 1) 2 – R 2 ]

D = [ ( Xi + 1) 2 + ( Yi – 1 ) 2 – R 2 ]

Fig. First Quadrant Pixel Position



A flow chart to obtain the pixel values for representing a circle is given in Fig. It is

sufficient to obtain the pixel values for 1/8th of a circle, the remaining obtained by

symmetry.

A program which implements Bresenham’s algorithm follows.

include <stdio.h>

include <graphics.h>

Void draw circle (int xc, int yc, int y) ;

Void symmetry (int x, int y, int xc, int yc) ;

double ratio ;

Void main (void)

{

draw circle (300, 150, 50) ;

}

int d, x ;

int gd, gm ;

gd = DETECT ;

initgraph (&gd, &gm, “ “) ;

d = 3 - 2 * y ;

ratio = 1.5

x = 0 ;

while (x < y)

{

symmetry (x, y, xc, yc) ;

if (d < 0)

d + = 4 * x + 6 ;

else

{

d + = 4 * (X – Y) + 10 ;

y - :

}

x++ ;

if (x = = y)

symmetry (x, y, xc, yc) ;

}

getch ( ) ;

closegraph ( ) ;

}

void symmetry (int x, int y, int xc, int yc)

{


int x start, x end, x out ;

int y start, y end, y out ;

x start = x * ratio

x end = (x + 1) * ratio ;

y start = y * ratio ;

y end = (y + 1) * ratio


for (x out = x start ; x out < x end ; ++ x out)

{

put pixel (x out + xc, y + yc, 1 ) ;

put pixel (x out + xc, –y + yc, 1) ;

put pixel (-x out + xc, –y + yc, 1) ;

put pixel (-x out + xc, y + yc, 1) ;

}

(y out = y start ; y out < y end ; ++y out)

{

put pixel (y out + xc, x + yc, 1 ) ;

put pixel (y out + xc, –x + yc, 1) ;

put pixel (-y out + xc, –x + yc, 2) ;

put pixel (-y out + xc, x + yc, 1) ;

}



Flow Chart to Determine Pixel Values

EXAMPLE OF BRESENHAM’S ALGORITHM

Consider a circle with centre at the origin and radius equal to 10 units. Because of

symmetry only the first octant is considered here.

X = 0

Y = 10

i = 2 (1–10) = –18

LIMIT = 0

PLOT (0, 10)

Y1 > LIMIT CONTINUE

i < 0 GOTO 2


2 = 2 (–18) + 2 (10) –1 = –17 < 0 GOTO 10

10 X = 0 + 1 = 1

i = –17 + 2 + 1 = –14

GOTO 1

1 PLOT (1, 10)

Yi > LIMIT CONTINUE

i < 0

= 2 (–14) + 2 (10) –1

= –9 < 0 GOTO 10

X = 1 + 1

i = –14 + 2 (2) + 1

= –9

GOTO 1

PLOT (2, 10)


The procedure is to be continued till the required point is reached. The results are given in Table

and are plotted in Fig. shown below.

Pixel Values for Circle

Plot of First Octant of a Circle


TRANSFORMATION IN GRAPHICS


Geometric transformations provide a means by which an image can be enlarged in size,

or reduced, rotated, or moved. These changes are brought about by changing the co-ordinates of

the picture to a new set of values depending upon the requirements.

CO-ORDINATE SYSTEMS USED IN GRAPHICS AND WINDOWING

Transformations can be carried out either in 2-dimensions or in 3-dimensions. The theory

of two-dimensional transformations is discussed first in this chapter. This is then extended to

three dimensions. When a design package is initiated, the display will have a set of co-ordinate

values. These are called default co-ordinates. A user co-ordinate system is one in which the

designer can specify his own co-ordinates for a specific design application. These screen

independent coordinates can have large or small numeric range, or even negative values, so that

the model can be represented in a natural way. It may, however, happen that the picture is too

crowded with several features to be viewed clearly on the display screen. Therefore, the designer

may want to view only a portion of the image, enclosed in a rectangular region called a window.

Different parts of the drawing can thus be selected for viewing by placing the windows. Portions

inside the window can be enlarged, reduced or edited depending upon the requirements. Figure

shows the use of windowing to enlarge the picture.

VIEW PORT

It may be sometimes desirable to display different portions or views of the drawing in

different regions of the screen. A portion of the screen where the contents of the window are

displayed is called a view port. Let the screen size be X = 0 to 200 and Y = 0 to 130. A view port

can be defined by the co-ordinates say X1 = 65, X2 = 130, Y1 = 50 and Y2 = 100. If we use the

same window as in Fig., the definition of this view port will display the image in the right hand top

quarter of the screen (Fig.) choosing different view ports multiple views can be placed on the

screen. Fig. shows four views of a component displayed using view port commands.



CLIPPING

Clipping is the process of determining the visible portions of a drawing lying within a

window. In clipping each graphic element of the display is examined to determine whether or not

it is completely inside the window, completely outside the window or crosses a window boundary.

Portions outside the boundary are not drawn. If the element of a drawing crosses the boundary

the point of inter-section is determined and only portions which lie inside are drawn. Readers are

advised to refer to books on computer graphics for typical clipping algorithms like Cohen-

Sutherland clipping algorithm. Fig. Shows an example of clipping.

HIDDEN SURFACE REMOVAL

One of the difficult problems in computer graphics is the removal of hidden surfaces from

the images of solid objects. In Fig. (a) An opaque cube is shown in wire frame representation.

Edges 15, 48, 37, 14, 12, 23, 58 and 87 are visible whereas edges 56, 67 and 26 are not visible.

Correspondingly, surfaces 1265, 2673 and 5678 are not visible since the object is opaque. The

actual representation of the cube must be as shown in Fig. (b).



There are a number of algorithms available for removal of hidden lines and hidden surfaces.

Table gives a list of algorithms for hidden line removal and hidden surface removal.

Table Algorithms for Hidden Line and Hidden Surface

There are two popular approaches to hidden surface removal. These are scan line based

systems and Z-buffer based systems. Other important approaches are area subdivision and

depth list schemes.

2-D & 3-D TRANSFORMATION

2-D TRANSFORMATIONS

In computer graphics, drawings are created by a series of primitives which are

represented by the co-ordinates of their end points. Certain changes in these drawings can be

made by performing some mathematical operations on these co-ordinates. The basic

transformations are scaling, translation and rotation.

ROTATION

Another useful transformation is the rotation of a drawing about a pivot point. Consider

Fig. Point P1 (40, 20) can be seen being rotated about the origin through an angle, =45°, in

the anti-clockwise direction to position P2. The co-ordinates of P2 can be obtained by multiplying

the co-ordinates of P1 by the matrix:



SCALING

Changing the dimensions of window and view port, it is possible to alter the size of

drawings. This technique is not satisfactory in all cases. 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 co-ordinates of the drawing by an enlargement or reduction factor called scaling

factor and the operation is called scaling. Referring to Fig., P1 (30, 20) represents a point in the

XY plane. In matrix form, P1 can be represented as:P1 = [30, 20]

If we multiply this by a matrix



An example of scaling in the case of a triangle is shown in Fig. Fig. (a) Shows the original picture

before scaling. Fig. (b) Shows the triangle after the co-ordinates are multiplied by the scaling

matrix.

TRANSLATION

Moving drawing or model across the screen is called translation. This is accomplished by adding

to the co-ordinates of each corner point the distance through which the drawing is to be moved

(translated). Fig. shows a rectangle (Fig.(a)) being moved to a new position (Fig.(b)) by adding 40

units to X co-ordinate values and 30 units to Y coordinate values. In general, in order to translate

drawing by (TX , TY ) every point X, Y will be replaced by a point X1 , Y1 where

X1 = X + TX

Y = Y + TY M.SRINIVASSAN, ASST. PROF 13 RGCET


REFLECTION

Shading is an important element in 3-D computer graphics, as it gives the necessary

realism to the representation of the object. Fig. shows what happens when light is incident on a

surface. Light gets partly reflected, partly scattered, partly absorbed and partly transmitted. The

relative magnitudes of these are influenced by many factors like the opaqueness of the solid,

surface texture etc. The intensity and wave length of light reflected from a surface depends on

the incident angle, the surface roughness, incident wave length and the electrical properties of

the surface. In computer graphics designer can model reflected light and transmitted light.

Reflected light could be categorized into two types:

Diffuse reflection: Diffuse light is scattered in all directions and is responsible for the color of the

object. The light is reflected from a surface due to molecular interaction between incident light

and the surface material. A yellow object for example, absorbs white light and reflects yellow

component of the light. This property is attributed to diffuse reflection. When light from a point

source is incident on a solid object, the diffuse reflection depends upon the angle of inclination of

the surface with that of the incident beam. More important source of illumination of the objects is

ambient light, which is the result of multiple reflections from the walls and other objects in the

vicinity and is incident on a surface in all directions.



Specular reflection: A perfectly matt surface scatters light in all directions. Most of the surfaces

that we deal with, however, have different levels of glossiness. The specular deflection deals with

the reflection of the surface due to glossiness. Consider Fig. which shows the reflection of light

on a surface. If the surface is perfectly glossy the reflected light is in the direction of R. If the

surface becomes more and more matt, the reflection intensity varies as in a profile shown as the

shaded area of the figure.

A technique to model reflection from an object based on specular reflection has been proposed

by Phong. This model assumes that:

• Light sources are point sources.

• All geometry except the surface normal is ignored.

• Diffuse and specular components are modeled as local components

• The model to simulate the specular term is empirical.

• The color of specular reflection is that of the light source

• The ambient lighting is constant.

SHEARING

A shearing transformation produces distortion of an object or an entire image. There are two

types of shears: X-shear and Y-shear. A Y-shear transforms the point (X, Y) to the point (X1, Y1)

by a factor Sh1, where

X1 = X

Y1 = Sh1. X + Y

Fig. shows Y shear applied to a drawing.



HOMOGENEOUS TRANSFORMATIONS

Each of the above transformations with the exception of translation can be represented as

a row vector X, Y and a 2 X 2 matrix. However, all the four transformations discussed above can

be represented as a product of a 1 X 3 row vector and an appropriate 3 X 3 matrix. The

conversion of a two-dimensional co-ordinate pair (X, Y) into a 3-dimensional vector can be

achieved by representing the point as [X Y 1]. After multiplying this vectorby a 3 X 3 matrix,

another homogeneous row vector is obtained [X1 Y1 1]. The first two terms in this vector are the

co-ordinate pair which is the transform of (X, Y). This three dimensional representation of a two

dimensional plane is called homogeneous coordinates and the transformation using the

homogeneous co-ordinates is called homogeneous transformation. The matrix representations of

the four basic transformations are given below.


Translation: COMBINATION TRANSFORMATIONS


Sequences of transformations can be combined into a single transformation using the

concatenation process. For example, consider the rotation of a line about an arbitrary point. Line M.SRINIVASSAN, ASST. PROF 17 RGCET


AB is to be rotated through 45° in anticlockwise direction about point A (Fig (a)). Fig. (b) Shows

an inverse translation of AB to A1B1. A1B1 is then rotated through 45° to A2B2. The line A2B2 is

then translated to A3B3

Since matrix operations are not commutative, care must be taken to preserve the order in which

they are performed while combining the matrices.

3-D TRANSFORMATIONS

It is often necessary to display objects in 3-D on the graphics screen. The transformation

matrices developed for 2-dimensions can be extended to 3-D.



PROJECTIONS

In drawing practice, a 3-dimensional object is represented on a plane paper. Similarly in

computer graphics a 3-dimensional object is viewed on a 2-dimensional display. A projection is a

transformation that performs this conversion. Three types of projections are commonly used in

engineering practice: parallel, perspective and isometric.

PARALLEL (ORTHOGONAL) PROJECTION

This is the simplest of the projection methods. Fig. shows the projection of a cube on to a

projection plane. The projectors, which are lines passing through the corners of the object are all

parallel to each other. It is only necessary to project the end points of a line in 3-D and then join

these projected points. This speeds up the transformation process. However a major

disadvantage of parallel projection is lack of depth information.


PERSPECTIVE PROJECTION


The perspective projection enhances the realism of displayed image by providing the

viewer with a sense of depth. Portions of the object farther away from the viewer are drawn

smaller than those in the foreground. This is more realistic as it is the way we see an object. In

perspective projection the projections connect the eye with every point of the object and therefore

all projections converge to the eye.

As the display screen is a two-dimensional space, we cannot display three-dimensional

objects but only their projections. Computationally, projection transformations are in general quite

expensive. Since the generation of a perspective view of a given object may require the

projection transformation of a considerable number of points, the projection applied is usually

restricted to the central projection and sometimes to even simpler parallel or orthographic

projection in order to keep the execution time for the generation of a perspective view within

reasonable limits.

Figure explains the central projection as it is usually applied in computer graphics. The

problem is to determine the projection of an object point, located somewhere in a three-

dimensional space, onto a plane in that space, called the image plane. This projection is called

the image point of the corresponding object point. In a central projection, the center of projection,

also called the viewpoint, is located on one of the axes of the three dimensional orthogonal co-

ordinate systems. In Figure the viewpoint is arbitrarily located on the Z-axis. This fact can also be

expressed by saying that the optical axis is aligned with the Z-axis of the co-ordinate system. The

image plane is perpendicular to the optical axis; i.e., in figure it is parallel to the xy-plane of the

co-ordinate system. This fact accounts for the simplicity of a central projection. Let the co-

ordinates in the two-dimensional co-ordinate system of the image plane, which we may call the

image co-ordinate system, be denoted by X and Y. Let the distance of the image plane to the

origin of the spatial co-ordinate system be denoted by Z and the distance of the viewpoint to the

origin of the co-ordinate system by Z.



This projection is called orthographic. The orthographic projection is a special form of the

parallel projection by which parallel lines of the three-dimensional object are transformed into

parallel lines of its image.

ISOMETRIC PROJECTION

In isometric projection the three orthogonal edges of an object are inclined equally to the

projection plane. Because of the relative ease of projection and the ability to give 3-D perception,

isometric projection is widely used in computer aided design. In computer aided design the co-

ordinates of the drawing are available in their natural co-ordinate system. These are transformed

suitably to enable the viewer different views or rotate the object in such away that all the faces of

the object are made visible continuously.

There are several uses for this technique in product design. Hence good design packages

incorporate several viewing transformation techniques. The viewing parameters depend upon the

system graphics standard followed in developing the graphics package. The algorithms for these

viewing transformations are available in literature.

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