1. Dr. Scott Schaefer Scan Conversion of Lines

2. 2/78 Displays – Cathode Ray Tube

3. 3/78 Displays – Liquid Crystal Displays

4. 4/78 Displays – Plasma

5. 5/78 Displays – Pixels Pixel: the smallest element of picture - Integer position (i,j) - Color information (r,g,b) x y (0,0)

6. 6/78 Problem Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line

7. 7/78 Problem Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line

8. 8/78 Problem Given two points (P, Q) on the screen (with integer coordinates) determine which pixels should be drawn to display a unit width line

9. 9/78 Special Lines - Horizontal

10. 10/78 Special Lines - Horizontal Increment x by 1, keep y constant

11. 11/78 Special Lines - Vertical

12. 12/78 Special Lines - Vertical Keep x constant, increment y by 1

13. 13/78 Special Lines - Diagonal

14. 14/78 Special Lines - Diagonal Increment x by 1, increment y by 1

15. 15/78 How about Arbitrary Lines?

16. 16/78 Slope-intercept equation for a line: Arbitrary Lines m: slope b: y-intercept

17. 17/78 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

18. 18/78 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

19. 19/78 Arbitrary Lines Assume Other lines by swapping x, y or negating Take steps in x and determine where to fill y

20. 20/78 Simple Algorithm 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

21. 21/78 Simple Algorithm 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

22. 22/78 Simple Algorithm 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

23. 23/78 Simple Algorithm 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

24. 24/78 Simple Algorithm 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

25. 25/78 Simple Algorithm 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

26. 26/78 Simple Algorithm - Example

27. 27/78 Simple Algorithm - Example

28. 28/78 Simple Algorithm - Example

29. 29/78 Simple Algorithm - Example

30. 30/78 Simple Algorithm - Example

31. 31/78 Simple Algorithm - Example

32. 32/78 Simple Algorithm - Example

33. 33/78 Simple Algorithm - Example

34. 34/78 Simple Algorithm - Example

35. 35/78 Simple Algorithm - 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

36. 36/78 Midpoint Algorithm Given a point just drawn, determine whether we move E or NE on next step

37. 37/78 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

38. 38/78 Midpoint Algorithm

39. 39/78 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

40. 40/78 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

41. 41/78 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

42. 42/78 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

43. 43/78 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

44. 44/78 Midpoint Algorithm – Implicit Forms How do we tell if a point is above or below the line?

45. 45/78 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

46. 46/78 Midpoint Algorithm – Implicit Forms

47. 47/78 Midpoint Algorithm – Implicit Forms

48. 48/78 Midpoint Algorithm – Implicit Forms

49. 49/78 Midpoint Algorithm – Implicit Forms

50. 50/78 Midpoint Algorithm – Implicit Forms

51. 51/78 Midpoint Algorithm – Implicit Forms

52. 52/78 Need value of to decide E or NE Build incremental algorithm Assume we have value of  Find value of if E chosen  Find value of if NE chosen Midpoint Algorithm

53. 53/78 Midpoint Algorithm Need value of to decide E or NE Build incremental algorithm Assume we have value of  Find value of if E chosen  Find value of if NE chosen

54. 54/78 Midpoint Algorithm If E was chosen, find

55. 55/78 Midpoint Algorithm If E was chosen, find

56. 56/78 Midpoint Algorithm If NE was chosen, find

57. 57/78 Midpoint Algorithm If NE was chosen, find

58. 58/78 Midpoint Algorithm What about starting value?

59. 59/78 Midpoint Algorithm What about starting value?

60. 60/78 Midpoint Algorithm What about starting value? is on the line!

61. 61/78 Midpoint Algorithm What about starting value?

62. 62/78 Midpoint Algorithm What about starting value? Multiplying coefficients by 2 doesn’t change sign of f

63. 63/78 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)

64. 64/78 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)

65. 65/78 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)

66. 66/78 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)

67. 67/78 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)

68. 68/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

69. 69/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

70. 70/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

71. 71/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

72. 72/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

73. 73/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

74. 74/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

75. 75/78 Midpoint Algorithm – Example x=x L y=y L d=6 c=-2 sum=2c+d draw(x,y) while ( x < x H ) if ( sum < 0 ) sum += 2d y++ x++ sum += 2c draw(x,y)

76. 76/78 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)

77. 77/78 Circle, ellipse, etc… Need implicit forms - circle: - ellipse: Equations for incremental calculations Initial positions More details in section 6.4-6.5 of the textbook

78. 78/78 OpenGL: Drawing Lines glBegin (GL_LINES); glVertex2f (p1.x, p1.y); glVertex2f (p2.x, p2.y); glVertex2f (p3.x, p3.y); glVertex2f (p4.x, p4.y); glVertex2f (p5.x, p5.y); glVertex2f (p6.x, p6.y); glEnd(); P1 P2P3 P4 P5P6