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Scan Conversion of Lines

Scan Conversion of Lines. Dr. Scott Schaefer. Displays – Cathode Ray Tube. Displays – Liquid Crystal Displays. Displays – Light-Emitting Diode. Displays – Pixels. Pixel: the smallest element of picture - Integer position (i,j) - Color information (r,g,b). y. x. (0,0). Problem.

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Scan Conversion of Lines

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  1. Scan Conversion of Lines Dr. Scott Schaefer

  2. Displays – Cathode Ray Tube

  3. Displays – Liquid Crystal Displays

  4. Displays – Light-Emitting Diode

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

  6. 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. 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. 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. Special Lines - Horizontal

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

  11. Special Lines - Vertical

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

  13. Special Lines - Diagonal

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

  15. How about Arbitrary Lines?

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

  17. Arbitrary Lines Slope-intercept equation for a line: How can we compute the equation from (xL,yL), (xH,yH)? m: slope b: y-intercept

  18. Arbitrary Lines Slope-intercept equation for a line: How can we compute the equation from (xL,yL), (xH,yH)? m: slope b: y-intercept

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

  20. Simple Algorithm • 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 = mxi+1 + b

  21. Simple Algorithm • 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

  22. Simple Algorithm • 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 = mxi + m + b

  23. Simple Algorithm • 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 = mxi + b + m

  24. Simple Algorithm • 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 = mxi + b + m

  25. Simple Algorithm • 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

  26. Simple Algorithm - Example

  27. Simple Algorithm - Example

  28. Simple Algorithm - Example

  29. Simple Algorithm - Example

  30. Simple Algorithm - Example

  31. Simple Algorithm - Example

  32. Simple Algorithm - Example

  33. Simple Algorithm - Example

  34. Simple Algorithm - Example

  35. Simple Algorithm - 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

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

  37. 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. Midpoint Algorithm

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

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

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

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

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

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

  45. 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. Midpoint Algorithm – Implicit Forms

  47. Midpoint Algorithm – Implicit Forms

  48. Midpoint Algorithm – Implicit Forms

  49. Midpoint Algorithm – Implicit Forms

  50. Midpoint Algorithm – Implicit Forms

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