1.11 Bresenham's Algorithm

7/30/2019 1.11 Bresenham's Algorithm

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Advantage & Disadvantage of

DDA algo


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Adva

It is a faster method for calculating pixel positions than

the direct use.

Disadv

The accumulation of round off error in successiveadditions of the floating point increment, can cause the

calculated pixel positions to drift away from the true line

path for long line segments. Rounding operations and floating point procedure in

lineDDA are still time consuming.


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Bresenhams Algorithm

TheoryFirst consider scan conversion process for lines

with +ve slopes


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Starting from the left end point(x0,y0) of a

given line, we step to each successive

column (x position) and plot the pixel

whose scan line y value is closest to the

line path.


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Bresenham's line algorithm

d1

d2

x x+1

y

y = m(x+1) + b

y = mx + b


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Assume the given endpoints are (xa,ya)

and (xb,yb), that xa < xb.

Define x = xb - xa; y = yb - ya;

As in the DDA algorithm we will iteratethrough the relevant columns and set one

pixel in each column.


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Assume that in the k-th iteration, the pixel

(xk,yk) is set.

Then in the next iteration xk+1,,, ,We will

select either (xk+1,yk) or (xk+1,yk+1)


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The y coordinate on the mathematical line at pixel

position xk+1 is calculated as

Y = m(Xk+1) + b

d1=y yk

= m(xk+1)+b - yk

d2=(yk+1)-y

=yk+1 m(xk+1)-b


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But xk+1=xk+1

Pk+1 PK

= 2 yxk + 2 y 2 x(yk+1) + c

-[ 2 yxk - 2 xyk +c

2 y - 2 x(yk+1 yk)


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Bresenham's line algorithm

(slope 1)

input line endpoints, (x0,y0) and (xn, yn)

calculate x = xn - x0 and y = yn - y0 calculate parameter p0 = 2 y - x

set pixel at position (x0,y0) repeat the following steps until (xn, yn) is reached:

if pi < 0

set the next pixel at position (xi +1, yi )

calculate new pi+1 = pi + 2 y if pi 0

set the next pixel at position (xi +1, yi + 1 )

calculate new pi+1 = pi + 2(y - x)


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Compute using Bresenhams

End points (20,10) and (30,18) Slope= 0.8

x=10 y=8

Initial decision parameter has the value

P0=2 y - x

=6

Successive decision parameters

increments r

2 y = 16 , 2 y 2 x = - 4


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We plot initial point (20 , 10) anddetermine successive pixel position along

the line path from the decision parameters

as


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K Pk (xk+1,yk+1) K Pk (xk+1,yk+1)

0 6 (21,11) 6 2 (27,16)

1 2 (22,12) 7 -2 (28,16)2 -2 (23,12) 8 14 (29,17)

3 14 (24,13) 9 10 (30,18)

4 10 (25,14)

5 6 (26,15)


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DDA versus Bresenhams

Algorithm DDA works with floating point arithmetic Rounding to integers necessary

Bresenhams algorithm uses integer

arithmetic

Constants need to be computed only once

Bresenhams algorithm generally faster

than DDA


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Bresenhams Algorithm Code

void lineBres(int xa, int ya, int xb, int yb)

{

int dx = abs(xb xa); int dy =abs( yb ya);

Int p=2*dy dx;

int twoDy = 2*dy;

int twoDyDx=2*(dy dx);

Int x, y,xend;

If(xa>xb)

{

x = xb;

y = yb;

xEnd=xa;

}


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else{

x = xa;

y = ya;

xEnd=xb;}

setPixel(x,y);

While (x < xEnd){

x++;

If (p

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1.11 Bresenham's Algorithm