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CS U540 Computer Graphics

CS U540 Computer Graphics. Prof. Harriet Fell Spring 2009 Lecture 6 – January 15, 2009. (x, y). Drawing Circles - 1. x = Rcos(  ) y = Rsin(  ) for  = 0 to 360 do x = Rcos(  ) y = Rsin(  ) draw(x, y). R. (0, 0). . (x, y). Drawing Circles 2. x 2 + y 2 = R 2

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CS U540 Computer Graphics

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  1. CS U540Computer Graphics Prof. Harriet Fell Spring 2009 Lecture 6 – January 15, 2009

  2. (x, y) Drawing Circles - 1 x = Rcos() y = Rsin() for = 0 to 360 do x = Rcos() y = Rsin() draw(x, y) R (0, 0) 

  3. (x, y) Drawing Circles 2 x2 + y2 = R2 forx = -R to R do y = sqrt(R2 - x2) draw(x, y) draw(x, -y) R (0, 0) 

  4. (-x, y) (x, y) (-y, x) (y, x) (-y, -x) (y, -x) (-x, -y) (x, -y) Circular Symmetry (0, 0)

  5. Midpoint Circle Algorithm IN THE TOP OCTANT: If (x, y) was the last pixel plotted, either (x + 1, y) or (x + 1, y - 1) will be the next pixel. Making a Decision Function: d(x, y) = x2 + y2 – R2 d(x, y) < 0 (x, y) is inside the circle. If d(x, y) = 0 (x, y) is on the circle. d(x, y) > 0 (x, y) is outside the circle.

  6. Decision Function Evaluate d at the midpoint of the two possible pixels. d(x + 1, y - ½) = (x + 1)2 + (y - ½)2 – R2 d(x + 1, y - ½)< 0 midpoint inside circle choose y If d(x + 1, y - ½)= 0 midpoint on circle choose y d(x + 1, y - ½)> 0 midpoint outside circle choose y - 1

  7. Computing D(x,y) Incrementally D(x, y) = d(x + 1, y - ½) = (x + 1)2 + (y - ½)2 – R2 D(x + 1, y) – D(x, y)= (x+2)2 + (y - ½)2 – R2 – ((x + 1)2 + (y - ½)2 – R2) =2(x + 1)+ 1  integer D(x + 1, y - 1) – D(x, y)= (x+2)2 + (y – 3/2)2 – R2 – ((x + 1)2 + (y - ½)2 – R2) =2(x+1) + 1 – 2(y – 1)  integer D(0, R) = 5/4 – R but we can round to 1 – R without changing any decisions.  integer You can also compute the differences incrementally.

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