The lines of this object appear continuous However, they are made of pixels 2April 13, 2015

The lines of this objectappear continuous

However, they aremade of pixels

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Some useful definitions

Rasterization: Process of determining which pixels provide the best approximation to a desired line on the screen.

Scan Conversion: Combination of rasterization and generating the picture in scan line order.

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• Straight lines must appear as straight lines.

• They must start and end accurately• Lines should have constant brightness along their length•Lines should drawn rapidly

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For horizontal, vertical and 45º lines, the choice of raster elements is obvious. This lines exhibit constant brightness along the length:

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For any other orientation the choice is more difficult:

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Rasterization of straight lines.

Rasterization yields uneven brightness: Horizontal and vertical lines appear brighter than the 45º lines.

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The equation of a straight line is given by: y=m.x+b

Algorithm 1: Direct Scan Conversion

1. Start at the pixel for the left-hand endpoint x12. Step along the pixels horizontally until you reach the right-hand endof the line, xr3. For each pixel compute the corresponding y value4. Round this value to the nearest integer to select the nearest pixel

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x = x1;while (x <= xr){y = m*x + b;y = Round (y);PlotPixel (x, y); /* Set the pixel at (x,y) on */x = x + 1;}

The algorithm performs a floating-point multiplication for every step in x.This method therefore requires an enormous number of floating-pointmultiplications, and is therefore more expensive

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Help for programming

Algorithm 2 : Digital Differential Analyzer (DDA)

The differential equation of a straight line is given by:

OR

The solution of the finite difference approximation is:

You need only compute m once, as the start of the scan-conversion

The DDA algorithm runs rather slowly because it requires real arithmetic (floating-point operations).

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DDA algorithm for lines with -1 < m < 1 ( No. of Colum more than number of rows)

x = x1;y = y1;while (x <= xr){y = y + m;y = Round (y);PlotPixel (x, y);x = x + 1;}

yx x2 – x1 > y2 – y1

Slope (m) = xy

ifThen step on Column and find x

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Reverse the rolesof x and y usinga unit step in y,and 1/m for x.

Gaps occurwhen m > 1

Switching the roles of x and y when m>1

yx Step on the rows and find y

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Help to write General program

dx = x2 – x1 ; dy = y2 – y1;If dx > dy length = dx ;else length = dy ;Xinc = dx / length ; Yinc = dy / length ;x = x + 0.5 ; y = y + 0.5 ;for ( i=1 ; i <= length ; i++){ putpixel (x, y, c) ; x = x + Xinc ; y = y + Yinc ;}

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292726252423222120 28 30

step x y

1 20 10

2 21 10.8

3 22 11.6

4 23 12.4

5 24 13.2

6 25 14

7 26 14.8

8 27 15.6

9 28 16.4

10 29 17.2

11 30 18

Dx = 30 – 20 = 10 Dy = 18 – 10 = 8 Larger length = 10 = Dx

Yinc = 8 / 10 = 0.8 Xinc = 10 / 10 = 1

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Implementation DDA algorithm by hand

Algorithm 3 : Bresenham’s algorithm (1965)

This algorithm uses only integer arithmetic, and runs significantly faster.

Key idea: (error) is the distance between the actual line and the nearest grid locations.

Initialize error:e=-1/2Error is given by:e=e+mReinitialize error:when e>0

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To draw line with Bresenham’s algorithm follow the following steps

1. In each step change one of the axis in the value 12. The other axis change or not change depended on the

value of (error (e))

3. Determine the largest length ( Dx or Dy )

4. Loop will be from 1 to largest length

5. Find the initiate value of(e) as following

e = - 0. 5x

y

6. Start to draw line with the point ( x1, y1 )

7. In each step plot-pixel and determine the new values of ( x, y)

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8. In each step check sign of (e)

9. If the sign of (e) positive then (y) will be incremented by (1), and subtract one from (e). { in case of Dx is the largest length}, otherwise {if Dy is the largest length then x will be incremented}

10. If (e) is negative or zero, then the value of (y) will remained as it without change.

11. In each step the value of (x) will increase by (1).

12. Also in each step (e) will increase with x

y

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However, this algorithm does not lead to integer arithmetic. Scaling by: 2* dx

void Bresenham (int x1, int y1, int xr, int yr){int x,y; /* coordinates of pixel being drawn */int dy, dx;int ne; /* integer scaled error term */x = x1; y = y1; /* start at left endpoint */ie = 2 * dy - dx; /* initialize the error term */while (x <= xr){ /* pixel-drawing loop */PlotPixel (x,y); /* draw the pixel */if (ie > 0) {y = y + 1;ne = ne - 2 * dx; /* can replaces e = e - 1 */}x = x + 1;ne = ne + 2 * dy; /* can replaces e = e + m */}}

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Help for programming

Go through the steps of the Bresenham line drawing algorithm (by hand) for a line going from (21,12) to (29,16)

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step x y

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In each step (across the X-axis) the height increased with (M) (slop)

If H was positive we change the row value and subtract one from height

If the slop was gentle

H > 0.5

H = H +M

H = H + M - 1

Rewrite H > 0.5 H- 0.5 > 0

Multiply by 2 get 2H – 1 > 0

H = x

y

Multiply by Dx get 2 . Dx . H – Dx > 0

Suppose G = 2 . Dx . H – Dx

G > 022April 18, 2023

H will be

OR

Now how G change from Colum to another

If Hnew = Hold + M

If Hnew = Hold + M - 1

Key is to use G > 0 instead of use H > 0.5

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If the end points was integer numbers the initial value for H equal M

Then the initial value for G equal

For each Colum check G if it was positive then move to next row and add

To G

Other wise keep the same row and add

To G

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Home work