1. Dr. S.M. Malaek Assistant: M. Younesi
Output Primitives
Dr. S.M. Malaek
Assistant: M. Younesi

2. Output Primitives
Output Primitives: Basic geometric structures (points, straight line segment, circles and other conic sections, quadric surfaces, spline curve and surfaces, polygon color areas, and character strings)
These picture components are often defined in a continuous space.

3. Output Primitives
In order to draw the primitive objects, one has to first scan convert the object.
Scan convert: Refers to he operation of finding out the location of pixels to the intensified and then setting the values of corresponding bits, in the graphic memory, to the desired intensity code.

4. Output Primitives
Each pixel on the display surface has a finite size depending on the screen resolution and hence a pixel cannot represent a single mathematical point.

5. Scan Converting A Point

6. Scan Converting A Point
A mathematical point (x,y) needs to be scan converted to a pixel at location (x´, y´).

7. Scan Converting A Point
x´=Round(x) and y´=Round(y)
All points that satisfy:
are mapped to pixel (x´,y´)

8. Scan Converting A Point
x´=Round(x+0.5) and y´=Round(y+o.5)
All points that satisfy:
are mapped to pixel (x´,y´)

9. Scan Converting A Line

10. Scan Converting A Line
The Cartesian slope- intercept equation for a straight line is:

11. Scan Converting A Line
These equation form the basic for determining deflection voltage in analog devices.
|m|<1
|m|>1

12. Scan Converting A Line
On raster system, lines are plotted with pixels, and step size (horizontal & vertical direction) are constrained by pixel separation.

13. Scan Converting A Line
We must sample a line at discrete positions and determine the nearest pixel to the line at each sampled position. X2

14. Digital Differential Analyzer
(DDA Algorithm)

15. DDA Algorithm Algorithm is an incremental scan conversion method.
Based on calculating either or
If |m|<1,

16. DDA Algorithm
If |m|>1, If

17. Bresenham’s Line Algorithm

18. Bresenham’s Line Algorithm
A highly efficient incremental method for scan converting lines.
Using only incremental integer calculation.
By testing the sign of an integer parameter, whose value is proportional to the difference between the separation of two pixel positions from the actual line path.

19. Bresenham’s Line Algorithm
A highly efficient incremental method for scan converting lines.
Using only incremental integer calculation.
By testing the sign of an integer parameter, whose value is proportional to the difference between the separation of two pixel positions from the actual line path.

20. Bresenham’s Line Algorithm

21. Bresenham’s Line Algorithm

22. Bresenham’s Line Algorithmy yk
yk+1
xk+1 } d2 d1 y yk
yk+1
xk+1 } d2 d1 y yk
yk+1
xk+1 } d2 d1

23. Bresenham’s Line Algorithm
d1 = y - yk = m (xk + 1) + b - yk
d2 = (yk + 1) - y = yk m (xk + 1) –b
d1 - d2 = 2 m (xk + 1) - 2 yk + 2b -1

24. Bresenham’s Line Algorithm
Pk = ΔX ( d1 - d2) =
2 ΔY . xk - 2 ΔX . yk + c
c = constante = 2 ΔY + ΔX (2b -1)

25. Pk+1 = Pk + 2ΔY or = Pk + 2(ΔY – ΔX( Let's get rid of multiplications
Pk+1 = 2 ΔY . xk ΔX .y + c
Pk+1 - Pk = 2 ΔY (xk+1 - xk) -2 ΔX (yk+1 - yk) (get rid of the constance)
Pk+1 = Pk + 2 ΔY - 2 ΔX (yk+1 - yk)
Pk+1 = Pk + 2ΔY  
   or  
   = Pk + 2(ΔY – ΔX(
with (yk+1 - yk) = 0 or 1 depending on Pk sign

26. Bresenham’s Line Algorithm
P0 =ΔX (d1-d2)
=ΔX[2m(x0+1)-2y0+2b-1]
=ΔX[2(mx0+b-y0)+2m-1]
P0=2ΔY -ΔX
m=Δy/x

27. Bresenham’s Line Algorithm
Example: Digitize the line with endpoint (20,10) and (30,18)
ΔX= ΔY= 8
P0=2ΔY –Δx=6
2ΔY =16
2(ΔY – ΔX)=-4
P0=2ΔY –ΔX
Pk+1 = Pk + 2ΔY  
   or  
   = Pk + 2(ΔY – ΔX(

28. Circle Generation Algorithms

29. Circle Generation Algorithms
The equation of a circle:
We could solve for y in terms of x

30. Circle Generation Algorithms
The spacing between plotted pixel positions is not uniform.

31. Circle Generation Algorithms
Computation can be reduced by considering the symmetry of circles
8 -Way symmetry

32. Midpoint Circle Algorithm

33. Circle Generation Algorithms
As in the line algorithm, we sample at unit intervals and determine the closet pixel position to the circle path at each step.

34. Circle Generation Algorithms
Points are generated from 90º to 45º, moves will be made only in the +x and –y direction.
positive x direction over this octant and use a decision parameter

35. Circle Generation Algorithms
We define a circle function:

36. Circle Generation Algorithms
Midpoint
Consider the coordinates of the point halfway between pixel T and pixel S
Midpoint

37. Circle Generation Algorithms
We use it to define a decision parameter

38. Circle Generation Algorithms
If is negative, the midpoint is inside the pixel, and we choose pixel T.
If we choose pixel S.

39. Circle Generation Algorithms
Parameter for the next step is:
since

40. Circle Generation Algorithms
If T is chosen ( ) we have:
If pixel S is chosen ( ) we have

41. Circle Generation Algorithms
In terms of

42. Circle Generation Algorithms
Initial value for the decision parameter using the original function of (0,r(
When r is an integer we can simply set

43. Circle Generation Algorithms

44. Circle Generation Algorithms
int x = 0; int y = radius; int p = 1 - radius; circlePoints(xCenter, yCenter, x, y, pix); while (x < y) { x++; if (p < 0) { p += 2*x+1; } else { y--; p += 2*(x-y+1); } circlePoints(xCenter, yCenter, x, y, pix); }

45. Circle Generation Algorithms
Example: A circle radius r=10
x=0 to x=y
P0=1-r = -9

46. Exercises?

47. Scan Converting Arcs And Sectors?

48. Character Generation

49. Character Generation
Letters, numbers, and other character can be displayed in a variety of size and styles.

50. Character Generation
Typeface: The overall design style for a set of characters is call typeface: Zar, nazanin, Titr.
Font: Referred to a set of cast metal character forms in a particular size and forma: 10 point Zar.

51. Character Generation
Two different representation are used for storing computer fonts:
Bitmap font (or bitmapped font)
Outline font

52. Bitmap font
Bitmap font (or bitmapped font): A simple method for representing the character shapes in a particular typeface is to use rectangular grid pattern.

53. Bitmap font
The character grid only need to be mapped to a frame buffer position.
Bitmap fonts required more space, because each variation (size and format) must be stored in a font cash.
Italic
Bold

54. Outline Font
Graphic primitives such as lines and arcs are used to define the outline of each character.
Require less storage since variation does not require a distinct font cash.

55. Outline Font
We can produce boldface, italic, or different size by manipulating the curve definition for the character outlines.
It does take more time to process the outline fonts, because they must be scan converted into frame buffer.
Outline font