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

2. 2/110 Displays – Cathode Ray Tube

3. 3/110 Displays – Liquid Crystal Displays

4. 4/110 Displays – Plasma

5. 5/110 Displays – Pixels Pixel: the smallest element of picture - Integer position (i,j) - Color information (r,g,b) x y (0,0)

6. 6/110 Digital Scan Conversion Convert graphics output primitives into a set of pixel values for storage in the frame buffer

7. 7/110 OpenGL Geometric Primitives All geometric primitives are specified by vertices GL_QUAD_STRIP GL_POLYGON GL_TRIANGLE_STRIP GL_TRIANGLE_FAN GL_POINTS GL_LINES GL_LINE_LOOP GL_LINE_STRIP GL_TRIANGLES GL_QUADS

8. 8/110 OpenGL Geometric Primitives All geometric primitives are specified by vertices GL_QUAD_STRIP GL_POLYGON GL_TRIANGLE_STRIP GL_TRIANGLE_FAN GL_POINTS GL_LINES GL_LINE_LOOP GL_LINE_STRIP GL_TRIANGLES GL_QUADS

9. 9/110 Outline How to draw a line? - Digital Differential Analyzer (DDA) - Midpoint algorithm Required readings - HB 6.1 to 6.8

10. 10/110 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. 11/110 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. 12/110 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. Line-drawing using “Paint”

14. 14/110 Special Lines - Horizontal

15. 15/110 Special Lines - Horizontal Increment x by 1, keep y constant

16. 16/110 Special Lines - Vertical

17. 17/110 Special Lines - Vertical Keep x constant, increment y by 1

18. 18/110 Special Lines - Diagonal

19. 19/110 Special Lines - Diagonal Increment x by 1, increment y by 1

20. 20/110 How about Arbitrary Lines? How to draw an arbitrary line?

21. 21/110 Slope-intercept equation for a line: Arbitrary Lines m: slope b: y-intercept

22. 22/110 Slope-intercept equation for a line: How can we compute the equation from (x L,y L ), (x H,y H )? Arbitrary Lines m: slope b: y-intercept

23. 23/110 Slope-intercept equation for a line: How can we compute the equation from (x L,y L ), (x H,y H )? Arbitrary Lines m: slope b: y-intercept

24. 24/110 Arbitrary Lines Assume

25. 25/110 Arbitrary Lines Assume Other lines by swapping x, y or negating - e.g., reverse the role of x and y if m>1

26. 26/110 Arbitrary Lines Assume Other lines by swapping x, y or negating - e.g., reverse the role of x and y if m>1 - e.g., negate y if

27. 27/110 Arbitrary Lines Assume Other lines by swapping x, y or negating Take steps in x and determine where to fill y

28. 28/110 Digital Differential Analyzer Start from (x L, y L ) and draw to (x H, y H ) where x L < x H ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) DrawPixel ( x i, Round ( y i ) ) x i+1 = x i + 1 y i+1 = m x i+1 + b

29. 29/110 Digital Differential Analyzer Simple algorithm: - sample unit intervals in one coordinate - find the nearest pixel for each sampled point.

30. 30/110 Digital Differential Analyzer (DDA) Start from (x L, y L ) and draw to (x H, y H ) where x L < x H ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) // sample unit intervals in x coordinate DrawPixel ( x i, Round ( y i ) ) // find the nearest pixel for each sampled point. x i+1 = x i + 1 y i+1 = m ( x i + 1 ) + b

31. 31/110 DDA Start from (x L, y L ) and draw to (x H, y H ) where x L < x H ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) DrawPixel ( x i, Round ( y i ) ) x i+1 = x i + 1 y i+1 = m x i + m + b

32. 32/110 DDA Start from (x L, y L ) and draw to (x H, y H ) where x L < x H ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) DrawPixel ( x i, Round ( y i ) ) x i+1 = x i + 1 y i+1 = m x i + b + m

33. 33/110 DDA Start from (x L, y L ) and draw to (x H, y H ) where x L < x H ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) DrawPixel ( x i, Round ( y i ) ) x i+1 = x i + 1 y i+1 = m x i + b + m

34. 34/110 DDA Start from (x L, y L ) and draw to (x H, y H ) where x L < x H ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) DrawPixel ( x i, Round ( y i ) ) x i+1 = x i + 1 y i+1 = y i + m

35. 35/110 DDA - Example

36. 36/110 DDA - Example

37. 37/110 DDA - Example

38. 38/110 DDA - Example

39. 39/110 DDA - Example

40. 40/110 DDA - Example

41. 41/110 DDA - Example

42. 42/110 DDA - Example

43. 43/110 DDA - Example

44. 44/110 DDA - Example

45. 45/110 DDA - Example

46. 46/110 DDA - Example

47. 47/110 DDA - Example

48. 48/110 DDA - Example

49. 49/110 DDA - Example

50. 50/110 DDA - Problems Floating point operations Rounding Can we do better? ( x 0, y 0 ) = ( x L, y L ) For ( i = 0; i <= x H -x L ; i++ ) DrawPixel ( x i, Round ( y i ) ) x i+1 = x i + 1 y i+1 = y i + m

51. 51/110 Outline How to draw a line? - Digital Differential Analyzer (DDA) - Midpoint algorithm

52. 52/110 How to Improve It? Question: if we draw the current pixel, where is the next pixel? (x k,y k )(x k+1,y k+1 )

53. 53/110 How to Improve It? Question: if we draw the current pixel, where is the next pixel? (x k,y k )(x k+1,y k+1 )

54. 54/110 How to Improve It? Question: if we draw the current pixel, where is the next pixel? (x k,y k )(x k+1,y k+1 )

55. 55/110 How to Improve It? Question: if we draw the current pixel, where is the next pixel? Next column: E or NE direction; why? x k+1 = x k +1 y k+1 = y k or y k +1

56. 56/110 Midpoint Algorithm Given a point just drawn, determine whether we move E or NE on next step x k+1 = x k +1 y k+1 = y k or y k +1

57. 57/110 Midpoint Algorithm Given a point just drawn, determine whether we move E or NE on next step Is the line above or below midpoint ? Below: move E Above: move NE

58. 58/110 Above/Below the Line?

59. 59/110 Two Issues How to evaluate if the midpoint is above or below the line? How to incrementally evaluate this? How to avoid floating point operations?

60. 60/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

61. 61/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line? Properties y=mx+b (x,y) on the line y<mx+b (x,y) below the line y>mx+b (x,y) above the line

62. 62/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line? Properties y=mx+b (x,y) on the line y<mx+b (x,y) below the line y>mx+b (x,y) above the line But this still requires floating point operations!

63. 63/110 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

64. 64/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

65. 65/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

66. 66/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

67. 67/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

68. 68/110 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

69. 69/110 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

70. 70/110 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 Note that e is an integer because

71. 71/110 Midpoint Algorithm – Implicit Forms

72. 72/110 Midpoint Algorithm – Implicit Forms

73. 73/110 Midpoint Algorithm – Implicit Forms

74. 74/110 Midpoint Algorithm – Implicit Forms

75. 75/110 Midpoint Algorithm – Implicit Forms

76. 76/110 Midpoint Algorithm – Implicit Forms

77. 77/110 Two Issues How to evaluate if the midpoint is above or below the line? How to incrementally evaluate this?

78. 78/110 Above/Below the Line?

79. 79/110 Midpoint Algorithm What about starting value? What is the middle point? (x L,y L )

80. 80/110 Midpoint Algorithm What about starting value? What is the middle point? (x L +1,y L +1/2) (x L,y L )

81. 81/110 Midpoint Algorithm What about starting value? What is the middle point? (x L +1,y L +1/2) (x L,y L )

82. 82/110 Midpoint Algorithm What about starting value?

83. 83/110 Midpoint Algorithm What about starting value? is on the line!

84. 84/110 Midpoint Algorithm What about starting value?

85. 85/110 Midpoint Algorithm What about starting value? Multiplying coefficients by 2 doesn’t change sign of f

86. 86/110 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

87. 87/110 Midpoint Algorithm Need value of to determine E or NE Build incremental algorithm Assume we have value of  Find value of if E chosen

88. 88/110 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

89. 89/110 Midpoint Algorithm If E was chosen, find

90. 90/110 Midpoint Algorithm If E was chosen, find

91. 91/110 Midpoint Algorithm If NE was chosen, find

92. 92/110 Midpoint Algorithm If NE was chosen, find

93. 93/110 Midpoint Algorithm – Summary x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

94. 94/110 Midpoint Algorithm – Summary x=x L y=y L d=x H - x L c=y L - y H sum=2c+d //initial value draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

95. 95/110 Midpoint Algorithm – Summary x=x L y=y L d=x H - x L c=y L - y H sum=2c+d //initial value draw(x,y) while ( x < x H ) // iterate until reaching the end point if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

96. 96/110 Midpoint Algorithm – Summary x=x L y=y L d=x H - x L c=y L - y H sum=2c+d //initial value draw(x,y) while ( x < x H ) // iterate until reaching the end point if ( sum < 0 ) // below the line and choose NE sum += 2d y++ x++ sum += 2c draw(x,y)

97. 97/110 Midpoint Algorithm – Summary x=x L y=y L d=x H - x L c=y L - y H sum=2c+d //initial value draw(x,y) while ( x < x H ) // iterate until reaching the end point if ( sum < 0 ) // below the line and choose NE sum += 2d y++ x++ sum += 2c draw(x,y) - Update the current pixel - Update the function value of midpoint

98. 98/110 Midpoint Algorithm – Summary x=x L y=y L d=x H - x L c=y L - y H sum=2c+d //initial value draw(x,y) while ( x < x H ) // iterate until reaching the end point if ( sum < 0 ) // below the line and choose NE sum += 2d y++ x++ sum += 2c draw(x,y) - Draw the pixel based on the function value of midpoint

99. 99/110 x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) Midpoint Algorithm – Example d =6 c = -2

100. 100/110 Midpoint Algorithm – Example x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) d =6 c = -2

101. 101/110 Midpoint Algorithm – Example x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) d =6 c = -2

102. 102/110 x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) Midpoint Algorithm – Example d =6 c = -2

103. 103/110 Midpoint Algorithm – Example x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) d =6 c = -2

104. 104/110 Midpoint Algorithm – Example x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) d =6 c = -2

105. 105/110 x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) Midpoint Algorithm – Example d =6 c = -2

106. 106/110 x=x L y=y L d=x H - x L c=y L - y H sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y) Midpoint Algorithm – Example d =6 c = -2

107. 107/110 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)

108. 108/110 Midpoint Algorithm: Circle

109. 109/110 Midpoint Algorithm: Circle

110. 110/110 Midpoint Algorithm: Circle

111. 111/110 Midpoint Algorithm: Circle Two issues: - What are two possible solutions? - What is the evaluation function?

112. 112/110 Midpoint Algorithm: Circle

113. 113/110 Midpoint Algorithm: Circle (x k +1,y k ) (x k +1,y k -1) Two issues: - What are two possible solutions? - What is the evaluation function?

114. 114/110 Midpoint Algorithm: Circle (x k +1,y k ) Midpoint: (x k +1,y k -1/2) (x k +1,y k -1) Two issues: - What are two possible solutions? - What is the evaluation function?

115. 115/110 Midpoint Algorithm: Circle (x k +1,y k ) Midpoint: (x k +1,y k -1/2) (x k +1,y k -1) Two issues: - What are two possible solutions? - What is the evaluation function?

116. 116/110 Circle, Ellipse, etc Need implicit forms - circle: - ellipse: Equations for incremental calculations Initial positions More details in section 6.4-6.6 of the textbook

117. 117/110 Next Lecture How to draw a polygon? - section 6.10- 6.13