Lecture02 Bresenham Line Algo

7/30/2019 Lecture02 Bresenham Line Algo

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Line Drawing Algorithm

Prepared By:

Prof. Bhavini R. Bhatt


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Algorithms for Drawing line

Introduction

- Pixel addressing

- Primitives and attributes

Line drawing algorithms

- DDA

- Bresenham


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Pixel addressing in raster graphics and how

does computer draw line?

Pixel

address

Pixel

x x+1 x+2 x+3 x+4

y

y+1

y+2

y+3

Theoretical length

Actual length

Screen made of pixels

High-level language specifies line System must color pixels


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Line drawing algorithms

imagessymbols &

y = 2x + 5x0 = 100

y0 = 50

d = 100

thickness = 4

descriptions


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Line raster representation


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DDA ( Digital Differential Algorithm )

m < 1


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DDA ( Digital Differential Algorithm )

m > 1


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Line Drawing Algorithms

Line drawn as pixels Graphics system

Projects the endpoints to their pixel locations inthe frame buffer (screen coordinates as integers)

Finds a path of pixels between the two

Loads the color

Plots the line on the monitor from frame buffer(video controller)

Rounding causes all lines except horizontal orvertical to be displayed as jigsaw appearance (lowresolution)

Improvement: high resolution, or adjust pixelintensities on the line path.


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Line equation

Slope-intercept form

y = m . x +

bslope m

y-intercept b

Slope

y-intercept

x

y

xx

yym

end

end

0

0

00mxyb


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xmy m

yx

For lines with slope |m|


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DDA (Digital Differential Analyzer)

Scan conversion line algorithm based on calculating

eitherx ory

Line sampled at regular intervals of x, then

corresponding y is calculated and rounded

From left to right

)1=(m

1

+x=x1.0,>mif

)1=(m+y=y1.0,mif

yk1+k

xk1+k

)-1=(

m

1-x=x1.0,>mif

)-1=(m-y=y1.0,mif

yk1+k

xk1+k


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Line Drawing Algorithmsvoid lineDDA (int x0, int y0, int xEnd, int yEnd)

{int dx = xEnd - x0, dy = yEnd - y0, steps, k;

float xIncrement, yIncrement, x = x0, y = y0;

if (fabs (dx) > fabs (dy))

steps = fabs (dx);

elsesteps = fabs (dy);

xIncrement = float (dx) / float (steps);

yIncrement = float (dy) / float (steps);

setPixel (round (x), round (y));

for (k = 0; k < steps; k++) {x += xIncrement;

y += yIncrement;

setPixel (round (x), round (y));

}

}


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DDA Algorithm (continued)

but floating point operations and rounding operations

are expensive

Y_inc

X_inc


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DDA Algorithm

Start with starting and ending coordinates of the line:(x0, y0) and (x1, y1)

Color first pixel (round to nearest integer)

Suppose x1-x0 > y1-y0 (gentle slope) There will be x1-x0 steps (# pixels to be colored)

Set x=x0, y=y0

At each step,

increment x by (x1-x0)/numsteps, and increment y by (y1-y0)/numsteps

For each step, round off x and y to nearest integer,and color pixel


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DDA Pseudo-code

// assume that slope is gentle

DDA(float x0, float x1, float y0, float y1) {

float x, y;

float xinc, yinc;

int numsteps;

numsteps = Round(x1) Round(x0);

xinc = (x1 x0) / numsteps;

yinc = (y1 y0) / numsteps;

x = x0;

y = y0;

ColorPixel(Round(x),Round(y));

for (int i=0; i


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DDA Example

Suppose we want todraw a line starting atpixel (2,3) and endingat pixel (12,8).

What are the valuesof the variables x andy at each timestep?

What are the pixelscolored, according tothe DDA algorithm?

numsteps = 12 2 = 10

xinc = 10/10 = 1.0

yinc = 5/10 = 0.5

t x y R(x) R(y)

0 2 3 2 3

1 3 3.5 3 42 4 4 4 4

3 5 4.5 5 5

4 6 5 6 5

5 7 5.5 7 6

6 8 6 8 6

7 9 6.5 9 7

8 10 7 10 7

9 11 7.5 11 8

10 12 8 12 8


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DDA Algorithm

Advantage Does not calculate coordinates based on the complete

equation (uses offset method)

Disadvantage Round-off errors are accumulated, thus line diverges more

and more from straight line

Round-off operations take time

Perform integer arithmetic by storing float asintegers in numerator and denominator andperforming integer arithmetic.


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Bresenhams Line Drawing

Algorithm

Efficient line drawing algorithm using onlyincremental integer calculations

Can be adapted to draw circles and other

curves Principle

Vertical axes show scan line positions

Horizontal axes show show pixel columns

At each step, determine the best next pixel basedon the sign of an integer parameter whose value is

proportional to the difference between the vertical

separations of the two pixel positions from the

actual line.


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Bresenhams Algorithm

Uses only integer calculations

Uses distance between ideal y-coordinate and the upper

and lower pixel (assuming gentle slope)

dupper

dlower


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General idea how Bresenham

works first consider the scan-conversion process for lines with positive

slope less than 1.

Pixel positions along a line path are then determined by sampling at

unit x intervals.

Start ing from the left endpoin t(xo, yo) of a given line, we step toeach successive column (x p os it ion) andplot the pixel whose scan-

line y value is closest to the line path.

Figure 3-7 demonstrates the Kth step in this process.

Assuming we have determined that the pixel at (xk, yk) is to be

displayed, we next need to decide which pixel to plot in

column xk+1,

Our ch oices are the pixels at

posit io ns (Xk+l, Yk) and (Xk+l, Yk+l).


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Bresenhams Line Drawing

Algorithm

Section of the screen grid showing a pixel in column xkon scanline yk that is to be plotted along the path of a line segment withslope

O


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Bresenhams Line Drawing Algorithm

Distances between pixel positions and the line y coordinate atsampling position xk+ I.


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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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Bresenham's line algorithm (slope 1)

1. Input the two line endpoints and store the leftendpoint in (x0, y0).

2. Set the color for the frame-buffer position (x0, y0)i.e. plot the first point.

3. Calculate the constantx, y, 2y and 2y 2x,and set the starting value for the decision parameter

as p0 = 2y x

4. At each xk along the line, from k=0, perform thefollowing test:if pk


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Number of Operations in

Bresenhams AlgorithmQ: In each step, how many floating point operations are there?

A: 0

Q: In each step, how many integer operations are there?A: 3 or 4


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Bresenhams Algorithm Example

Suppose we want to drawa line starting at pixel(2,3) and ending at pixel(12,8).

What are the values ofp0, dx and dy?

What are the values ofthe variable p at eachtimestep?

What are the pixelscolored, according toBresenhams algorithm?

dx = 12 2 = 10

dy = 8 3 = 5p0 = 2dy dx = 15

t p P(x) P(y)

0 0 2 3

1 -10 3 42 0 4 4

3 -10 5 5

4 0 6 5

5 -10 7 6

6 0 8 6

7 -10 9 7

8 0 10 7

9 -10 11 8

10 0 12 8

2dy = 10

2dy 2dx = -10


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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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Lecture02 Bresenham Line Algo