Commonly a fixed area of the system memory is reserved for the frame buffer, Video controller can direct access to the frame-buffer Frame-buffer locations, and the corresponding screen positions, are referenced in Cartesian coordinates. Some system employ lower-left corner as origin But most common system employ upper-left corner as origin. Scan lines are labeled from ymax, at the top of the screen to 0 at the bottom. Along each scan line, screen pixel positions are labeled from 0 to xmax

1. Scan Conversion
Dept of Computer Science & Engineering
Islamic University, Kushtia

2. Graphics Primitives
 The basic building blocks for pictures are referred to as output
primitives.
 They include:
 character strings and
 geometric entities,
 Points,
 Straight lines,
 Curved Lines,
 Circle, Ellipse
 Filled areas (polygons, circles, etc.), and
 Shapes defined with arrays of color points.
 Routines for generating output primitives provide the basic
tools for constructing pictures.

3. Coordinates

4. Pixel addressing in raster graphics
Pixel
address
Pixel
x x+1 x+2 x+3 x+4
y
y+1
y+2
y+3
Theoretical length
Actual length

5. POINTS AND LINES
 Point plotting is accomplished by converting a single
coordinate position furnished by an application program
into appropriate operations for the output device in use.
 For example, CRT monitor, the electron beam is
turned on to illuminate the screen phosphor at the
selected location

6. POINTS AND LINES
 Line drawing is accomplished by calculating
intermediate positions along the line path
between two specified endpoint positions.
 Digital devices display a straight line segment by
plotting discrete points between the two
endpoints.
Discrete coordinate positions along the line
path are calculated from the equation of the
line.

7. Theoretical Line

8. Line on Display Devices

9. Line drawing algorithms
 Geometric
 Display devices
 DDA
 Bresenham

10. Line drawing algorithms
 The Cartesian slope-intercept
equation for a straight line is
 with m representing the slope of
the line and b as they intercept

11. Line drawing algorithms
 Problems in Geometric Equations:
 To plot a line need
 Multiplication, Division or more complex task
 Computer needs lot of task
 This degrade overall performance
 Interactive graphics need faster response
 Use DDA, Bresenham’s algorithm
 Interactive Graphics: where user dynamically controls
the presentation of graphics model on a computer
display

12. Requirements
1. Must compute integer coordinates of pixels which lie on or
near a line or circle.
2. Pixel level algorithms are invoked hundreds or thousands of
times when an image is created or modified.
3. Lines must create visually satisfactory images.
 Lines should appear straight
 Lines should terminate accurately
 Lines should have constant density
 Line density should be independent of line length and
angle.
4. Line algorithm should always be defined.

13. DDA (Digital Differential Analyzer)
m < 1

14. DDA (Digital Differential Analyzer)
m > 1
m > 1

15. DDA Algorithms
 DDA is a scan-conversion line algorithm based on
calculating either y or x
 If the slope is less than or equal to 1, we sample at unit x
intervals ( x = 1) and compute each successive y value
as
where k=1 for first point and increases by 1until the final
endpoint is reached
 Calculated y values must be rounded to the nearest
integer

16. DDA Algorithms
 For lines with a positive slope greater than 1, reverse the
roles of x and y.
 we sample at unit y intervals (y = 1) and calculate
each succeeding x value as
equation (a) and (b) are based on the assumption that
lines are to be processed from the left endpoint to the
right endpoint

17. DDA Algorithms
 If this processing is reversed, that is from right to left
 The DDA algorithm is a faster method for calculating
pixel positions than the direct use
 It eliminates the multiplication by making use of raster
characteristics

18. DDA at a glance
 input line endpoints, (x1,y1) and (x2, y2)
 set pixel at position (x1,y1)
 calculate slope m
 Case |m|≤1: repeat the following steps until (x2, y2) is
reached:
yi+1 = yi + y/ x
xi+1 = xi + 1
set pixel at position (xi+1,Round(yi+1))
 Case |m|>1: repeat the following steps until (x2, y2) is
reached:
xi+1 = xi + x/ y
yi+1 = yi + 1
set pixel at position (Round(xi+1), yi+1)

19. Bresenham’s Line Algorithm
 Bresenham’s scan converts lines using only incremental
integer calculations that can be adapted to display
circles and other curves.
|m|<1 |m|>1

20. Bresenham’s Line Algorithm
|m|<1

21. Bresenham’s Line Algorithm (m<1)
 Starting from the left endpoint (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.
 Let we have determined the pixel at (xk, yk) to be
displayed,
 we next need to decide which pixel to plot in column
xk+1,
 Our choices are the pixels at positions (xk+1, yk) and
(xk+1, yk+1)

22. Bresenham’s Line Algorithm
d1
d2
xk xk+1=x+1
yk
y = m(x+1) + b
y = mx + b
yk+1=y+1

23. Bresenham’s Line Algorithm

24. Bresenham’s Line Algorithm
 They coordinate on the mathematical line at pixel column
position xk+1 is calculated as
 The difference between these two separations is

25. Bresenham’s Line Algorithm
where pk = x(d1 – d2) is decision parameter for the kth
step in the line algorithm
The sign of p, is the same as the sign of (d1 – d2) since
x > 0
and c is constant and c = 2y + x(2b – 1),

26. Bresenham’s Line Algorithm
 If (d1 < d2) then pk is negative
 We plot lower pixel (xk+1, yk)
 If (d1  d2) then pk is positive
 We plot lower pixel (xk+1, yk+1)
 For next decision parameter pk+1
 Increment x as xk+1 = xk+1

27. Bresenham’s Line Algorithm
 The term (yk+1–yk) of eq-(f) either 0 or 1 depending on
sign of pk

28. Bresenham’s Line Algorithm
 If pk < 0 true line close to lower pixel and yk +1 = yk
pk+1 = pk + 2y -- g(a)
 If pk  0 true line close to lower pixel and yk +1 = yk+1
pk+1 = pk + 2y – 2x -- g(b)
 First decision parameter p0 from eq-(d) as considering the line
start from (0, 0) as
p0 = 2y.0 – 2x.0 + c
p0 = c = 2y + x(2b – 1)
but y = mx + b or b = 0
p0 = 2y - x

29. Bresenham’s Line Algorithm (m<1)

30. Bresenham’s Line Algorithm (m<1)

31. Bresenham’s Line Algorithm (m<1)

32. Bresenham’s Line Algorithm