1. CSCE 441 Lecture 2: Scan Conversion of Lines
Jinxiang Chai

2. Displays – Cathode Ray Tube

3. Displays – Liquid Crystal Displays

4. Displays – Plasma

5. Displays – Pixels Pixel: the smallest element of picture
- Integer position (i,j)
- Color information (r,g,b) y x
(0,0)

6. Digital Scan Conversion
Convert graphics output primitives into a set of pixel values for storage in the frame buffer

7. OpenGL Geometric Primitives
All geometric primitives are specified by vertices
GL_LINE_STRIP
GL_LINE_LOOP
GL_POLYGON
GL_LINES
GL_POINTS
GL_TRIANGLE_STRIP
GL_QUAD_STRIP
GL_TRIANGLES
GL_QUADS
GL_TRIANGLE_FAN

8. OpenGL Geometric Primitives
All geometric primitives are specified by vertices
GL_LINE_STRIP
GL_LINE_LOOP
GL_POLYGON
GL_LINES
GL_POINTS
GL_TRIANGLE_STRIP
GL_QUAD_STRIP
GL_TRIANGLES
GL_QUADS
GL_TRIANGLE_FAN

9. Outline How to draw a line? - Digital Differential Analyzer (DDA)
- Midpoint algorithm
Required readings
- HB 3.1 to 3.8

10. Problem
Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line

11. Problem
Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line

12. Problem
Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line

13. Special Lines - Horizontal

14. Special Lines - Horizontal
Increment x by 1, keep y constant

15. Special Lines - Vertical

16. Special Lines - Vertical
Keep x constant, increment y by 1

17. Special Lines - Diagonal

18. Special Lines - Diagonal
Increment x by 1, increment y by 1

19. How about Arbitrary Lines?
How to draw an arbitrary line?

20. Arbitrary Lines Slope-intercept equation for a line: m: slope
b: y-intercept
Slope-intercept equation for a straight line. Here me is the slope of the line and b is the y intercept

21. Arbitrary Lines Slope-intercept equation for a line:
How can we compute the equation from (xL,yL), (xH,yH)?
m: slope
b: y-intercept
Slope-intercept equation for a straight line. Here me is the slope of the line and b is the y intercept

22. Arbitrary Lines Slope-intercept equation for a line:
How can we compute the equation from (xL,yL), (xH,yH)?
m: slope
b: y-intercept
Slope-intercept equation for a straight line. Here me is the slope of the line and b is the y intercept

23. Arbitrary Lines Assume Other lines by swapping x, y or negating
Take steps in x and determine where to fill y
Slope-intercept equation for a straight line. Here me is the slope of the line and b is the y intercept

24. Digital Differential Analyzer
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi+1 + b

25. Digital Differential Analyzer
Simple algorithm:
- sample unit intervals in one coordinate
- find the nearest pixel for each sampled point.

26. Digital Differential Analyzer (DDA)
Start from (xL, yL) and draw to (xH, yH)
where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ ) //sample unit intervals in x coordinate
DrawPixel ( xi, Round ( yi ) ) //find the nearest pixel for each sampled point.
xi+1 = xi + 1
yi+1 = m ( xi + 1 ) + b

27. DDA Start from (xL, yL) and draw to (xH, yH) where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi + m + b

28. DDA Start from (xL, yL) and draw to (xH, yH) where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi + b + m

29. DDA Start from (xL, yL) and draw to (xH, yH) where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = m xi + b + m

30. DDA Start from (xL, yL) and draw to (xH, yH) where xL< xH
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = yi + m

31. DDA - Example

32. DDA - Example

33. DDA - Example

34. DDA - Example

35. DDA - Example

36. DDA - Example

37. DDA - Example

38. DDA - Example

39. DDA - Example

40. DDA - Example

41. DDA - Example

42. DDA - Example

43. DDA - Example

44. DDA - Example

45. DDA - Example

46. DDA - Problems Floating point operations Rounding Can we do better?
( x0, y0 ) = ( xL, yL )
For ( i = 0; i <= xH-xL; i++ )
DrawPixel ( xi, Round ( yi ) )
xi+1 = xi + 1
yi+1 = yi + m

47. Outline How to draw a line? - Digital Differential Analyzer (DDA)
- Midpoint algorithm

48. How to Improve It?
Question: if we draw the current pixel,, where is the next pixel?

49. How to Improve It?
Question: if we draw the current pixel,, where is the next pixel?
Next column, E or NE direction

50. Midpoint Algorithm
Given a point just drawn, determine whether we move E or NE on next step

51. Midpoint Algorithm
Given a point just drawn, determine whether we move E or NE on next step
Is the line above or below ?
Below: move E
Above: move NE

52. Above/Below the Line?

53. Two Issues How to evaluate if the midpoint is above or below the line?
How to incrementally evaluate this?

54. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

55. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

56. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

57. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

58. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

59. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

60. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?

61. Midpoint Algorithm – Implicit Forms
How do we tell if a point is above or below the line?
Properties
f(x,y)=0 (x,y) on the line
f(x,y)<0 (x,y) below the line
f(x,y)>0 (x,y) above the line

62. Midpoint Algorithm – Implicit Forms

63. Midpoint Algorithm – Implicit Forms

64. Midpoint Algorithm – Implicit Forms

65. Midpoint Algorithm – Implicit Forms

66. Midpoint Algorithm – Implicit Forms

67. Midpoint Algorithm – Implicit Forms

68. Two Issues How to evaluate if the midpoint is above or below the line?
How to incrementally evaluate this?

69. Above/Below the Line?

70. Midpoint Algorithm What about starting value?
What is the middle point?

71. Midpoint Algorithm What about starting value?
What is the middle point?
(xL+1,yL+1/2)

72. Midpoint Algorithm What about starting value?
What is the middle point?
(xL+1,yL+1/2)

73. What about starting value?
Midpoint Algorithm
What about starting value?

74. What about starting value?
Midpoint Algorithm
What about starting value?
is on the line!

75. What about starting value?
Midpoint Algorithm
What about starting value?

76. What about starting value?
Midpoint Algorithm
What about starting value?
Multiplying coefficients by 2 doesn’t change sign of f

77. Midpoint Algorithm Need value of to determine E or NE
Build incremental algorithm
Assume we have value of
Find value of if E chosen
Find value of if NE chosen

78. Midpoint Algorithm Need value of to determine E or NE
Build incremental algorithm
Assume we have value of
Find value of if E chosen

79. Midpoint Algorithm Need value of to determine E or NE
Build incremental algorithm
Assume we have value of
Find value of if E chosen
Find value of if NE chosen
79/110 79

80. Midpoint Algorithm
If E was chosen, find

81. Midpoint Algorithm
If E was chosen, find

82. Midpoint Algorithm
If NE was chosen, find

83. Midpoint Algorithm
If NE was chosen, find

84. Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c

85. Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial value
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c

86. Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial value
draw(x,y)
while ( x < xH) // iterate until reaching the end point
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c

87. Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial value
draw(x,y)
while ( x < xH) // iterate until reaching the end point
if ( sum < 0 ) // below the line and choose NE
sum += 2d
y++
x++
sum += 2c

88. Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial value
draw(x,y)
while ( x < xH) // iterate until reaching the end point
if ( sum < 0 ) // below the line and choose NE
sum += 2d
y++
x++
sum += 2c
- Update the current pixel
- Update the function value of midpoint
88/110 88

89. Midpoint Algorithm – Summary
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d //initial value
draw(x,y)
while ( x < xH) // iterate until reaching the end point
if ( sum < 0 ) // below the line and choose NE
sum += 2d
y++
x++
sum += 2c
- Draw the pixel based on the function value of midpoint
89/110 89

90. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
Dx =6 dy = 2
d =6 c = -2

91. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

92. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

93. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

94. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

95. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

96. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

97. Midpoint Algorithm – Example
x=xL y=yL
d=xH - xL c=yL - yH
sum=2c+d
draw(x,y)
while ( x < xH)
if ( sum < 0 )
sum += 2d
y++
x++
sum += 2c
d =6 c = -2

98. Midpoint Algorithm Only integer operations
Exactly the same as the more commonly found Bresenham’s line drawing algorithm
Extends to other types of shapes (circles)

99. Midpoint Algorithm: Circle

100. Midpoint Algorithm: Circle

101. Midpoint Algorithm: Circle

102. Midpoint Algorithm: Circle
Two issues:
What are two possible solutions?
What is the evaluation function?

103. Midpoint Algorithm: Circle

104. Midpoint Algorithm: Circle
(xk+1,yk)
(xk+1,yk-1)
Two issues:
What are two possible solutions?
What is the evaluation function?

105. Midpoint Algorithm: Circle
(xk+1,yk)
Midpoint: (xk+1,yk-1/2)
(xk+1,yk-1)
Two issues:
What are two possible solutions?
What is the evaluation function?

106. Midpoint Algorithm: Circle
(xk+1,yk)
Midpoint: (xk+1,yk-1/2)
(xk+1,yk-1)
Two issues:
What are two possible solutions?
What is the evaluation function?

107. Circle, Ellipse, etc Need implicit forms - circle: - ellipse:
Equations for incremental calculations
Initial positions
More details in section of the textbook

108. Next Lecture
How to draw a polygon?
- section