Raster conversion algorithms for line and circle

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Raster conversion algorithms for line and circle

• Introduction
• - Pixel addressing
• - Primitives and attributes
• Line drawing algorithms
• - DDA
• - Bresenham
• Circle generating algorithms
• - Direct method
• - Bresenham algorithm

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Pixel addressing in raster graphics
3
Raster conversion algorithms requirements

• visual accuracy
• spatial accuracy
• speed

4
Line drawing algorithms
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Line raster representation
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DDA ( Digital Differential Algorithm )
m lt 1
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DDA ( Digital Differential Algorithm )
m gt 1
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DDA ( Digital Differential Algorithm )
m gt 1
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Digital Differential Algorithm

• input line endpoints, (x0,y0) and (xn, yn)
• set pixel at position (x0,y0)
• calculate slope m
• Case m1 repeat the following steps until (xn,
  yn) is reached
• yi1 yi ?y/ ?x
• xi1 xi 1
• set pixel at position
  (xi1,Round(yi1))
• Case mgt1 repeat the following steps until (xn,
  yn) is reached
• xi1 xi ?x/ ?y
• yi1 yi 1
• set pixel at position
  (Round(xi1), yi1)

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Bresenham's line algorithm
y mx b
y mx b
y
x
x1
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Bresenham's line algorithm (slope 1)

• input line endpoints, (x0,y0) and (xn, yn)
• calculate ?x xn - x0 and ?y yn - y0
• calculate parameter p0 2 ?y - ?x
• set pixel at position (x0,y0)
• repeat the following steps until (xn, yn) is
  reached
• if pi lt 0
• set the next pixel at position (xi 1, yi )
• calculate new pi1 pi 2 ?y
• if pi 0
• set the next pixel at position (xi 1, yi 1 )
• calculate new pi1 pi 2(?y - ?x)

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DDA versus Bresenhams Algorithm

• DDA works with floating point arithmetic
• Rounding to integers necessary
• Bresenhams algorithm uses integer arithmetic
• Constants need to be computed only once
• Bresenhams algorithm generally faster than DDA

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Circle generating algorithms

• Direct
• Polar coordinate based
• Bresenhams

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Direct circle algorithm

• Cartesian coordinates
• Circle equation
• ( x - xc )2 ( y - yc )2 r2
• Step along x axis from xc - r to xc r and
  calculate
• y yc ? r2 - ( x - xc )2

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Polar coordinates

• Polar coordinate equation
• x xc r cos?
• y yc r sin?
• step through values of ? from 0 to 2p

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Optimisation and speed-up

• Symmetry of a circle can be used
• Calculations of point coordinates only for a
  first one-eighth of a circle

(x,y)
(-x,y)
(y,x)
(-y,x)
(-y,-x)
(y,-x)
(x,-y)
(-x,-y)
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Bresenhams circle algorithm

• 1. Input radius r
• 2. Plot a point at (0, r)
• 3. Calculate the initial value of the decision
  parameter as p0 5/4 r 1 r

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• 4. At each position xk, starting at k 0,
  perform the following test
• if pk lt 0
• plot point at (xk 1, yk)
• compute new pk1 pk 2xk1 1
• else
• plot point at (xk 1, yk 1)
• compute new pk1 pk 2xk1 1 2yk1
• where xk1 xk 1 and yk1 yk - 1

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• 5. Determine symmetry points in the other seven
  octants and plot points
• 6. Repeat steps 4 and 5 until x ? y