Dr. Scott Schaefer Scan Conversion of Lines

2/78 Displays – Cathode Ray Tube

3/78 Displays – Liquid Crystal Displays

4/78 Displays – Plasma

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

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

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

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

9/78 Special Lines - Horizontal

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

11/78 Special Lines - Vertical

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

13/78 Special Lines - Diagonal

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

15/78 How about Arbitrary Lines?

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

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

18/78 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

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

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

21/78 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

22/78 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

23/78 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

24/78 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

25/78 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

26/78 Simple Algorithm - Example

27/78 Simple Algorithm - Example

28/78 Simple Algorithm - Example

29/78 Simple Algorithm - Example

30/78 Simple Algorithm - Example

31/78 Simple Algorithm - Example

32/78 Simple Algorithm - Example

33/78 Simple Algorithm - Example

34/78 Simple Algorithm - Example

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

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

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

38/78 Midpoint Algorithm

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

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

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

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

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

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

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

46/78 Midpoint Algorithm – Implicit Forms

47/78 Midpoint Algorithm – Implicit Forms

48/78 Midpoint Algorithm – Implicit Forms

49/78 Midpoint Algorithm – Implicit Forms

50/78 Midpoint Algorithm – Implicit Forms

51/78 Midpoint Algorithm – Implicit Forms

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

53/78 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

54/78 Midpoint Algorithm If E was chosen, find

55/78 Midpoint Algorithm If E was chosen, find

56/78 Midpoint Algorithm If NE was chosen, find

57/78 Midpoint Algorithm If NE was chosen, find

58/78 Midpoint Algorithm What about starting value?

59/78 Midpoint Algorithm What about starting value?

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

61/78 Midpoint Algorithm What about starting value?

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

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)

64/78 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)

65/78 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)

66/78 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)

67/78 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)

68/78 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)

69/78 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)

70/78 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)

71/78 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)

72/78 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)

73/78 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)

74/78 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)

75/78 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)

76/78 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)

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

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