OUTPUT%20PRIMITIVES%20/%20DISPLAY%20TECHNIQUES

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OUTPUT PRIMITIVES / DISPLAY TECHNIQUES

• LINE DRAWING
• TO DRAW LINE FROM (X1 Y1) TO (X2 Y2)

(x2 y2)
(x1y1)
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RASTER SCREEN
(x2 y2)
(x1 y1)
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DIGITAL DIFFERENTIAL ANALYZER (DDA)

• y m x b
• m (y2 - y1) / (x2 - x1) ?y / ?x dely / delx
• xk1 xk ? delxyk1 yk ? dely
• ? ?????

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TWO ALTERNATIVES

• ? 1 / max (x2-x1), (y2 - y1) (BOOK)
• ? 1 / 2 K WHERE K IS SUCH THAT2K-1 lt (max
  (x2-x1), (y2 - y1)) ? 2K

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SIMPLE DDA

• delx x2 - x1 dely y2 - y1 steps
  abs(dely)
• if abs (delx) gt abs (dely) then steps
  abs(delx)
• xinc delx / steps yinc dely / steps
• x x1 y y1
• drawpixel (round(x), round(y))
• for k 1 to steps do begin
• x x xinc y y yinc
• drawpixel (round(x), round(y))
• end
• end

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BRESENHAMS ALGO

• WHICH PIXEL NEXT ??

? 2
(xk yk)
? 1
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BRESENHAMS ALGO (contd)

• Deviations from the specified line path
• d1 y - yk m (xk 1) b - yk
• d2 yk 1 -y yk 1 - m (xk 1) - b
• d1 - d2 2 m (xk 1) - 2yk 2b -1
• m (?y / ?x)
• pk ?x (d1 - d2) 2?y . xk - 2?x . yk c

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Pk error at beginning of next pixel
If (at end of iteration k) true-line is below
centre of pixel (pk lt 0), draw pixel 1 else draw
pixel 2 (in iteration k1)
2
1
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Brezenhams algo (contd)

• pk1 - pk 2 ?y (xk1 -xk) - 2?x (yk1 -yk) 2
  ?y - 2?x (yk1 -yk)
• For pixel 1 2 ?y
• For pixel 2 2 ?y - 2?x
• Hence if pk lt 0, choose pixel 1 in iteration k1,
  pk1 pk 2 ?y
• else choose pixel 2 in iteration k1, pk1 pk
  2 ?y - 2 ?x
• Initial p0 2 ?y - ?x

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Brezenhams algo (contd)
pk m - 1
pk m
pk p0 m - 0.5
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Circle Generation

• (x - a)2 (y - b)2 r2
• OR x a r cos ? y b r sin ?
• (Fixed angular step size with lines between them)
• (Step Size 1/r)
• OR m dy / dx - (x - a) / (y - b)
• USE DDA with m as above

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MIDPOINT CIRCLE ALGO
Next midpoint
?? 1
yk
Yk - 1
?? 2
xk
xk 1
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MIDPOINT CIRCLE

• f (x, y) x2 y2 - r2
• pk (xk 1)2 (yk - 0.5)2 - r2 (error at next
  midpoint)
• If pk lt 0 then Pixel 1 Else Pixel 2 (is closer to
  circle in iteration k1)
• pk1 (xk 1) 12 (yk1 - 0.5)2 - r2
• pk1 pk 2 (xk 1) (yk12 - yk2) - (yk1
  -yk) 1

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MID POINT CIRCLE (contd)

• Pk1 - pk 2 xk1 1 OR 2 xk1 1 - 2yk1
• p0 5/4 - r (PLEASE DERIVE)
• INITIAL POINT (0, r)
• GET ONE POINT AND DUPLICATE IN 8 OCTANTS

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ELLIPSE DRAWING

• SLOPE BASED (DDA)
• MIDPOINT ALGO
• (x - a) / rx2 (y - b) / ry2 1
• f (x, y) (x - a)2 ry2 (y - b)2 rx2 -rx2 ry2
• dy / dx ????
• If f(x, y) lt 0, point is inside the ellipse

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MIDPOINT ELLIPSE (contd)

• Where slope lt 1, xk1 xk 1
• Where slope gt 1, yk1 yk - 1
• pk ????
• Pk1 - pk ???? (For Regions 1 and 2)

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OTHER CURVES

• FOR IMPLICIT CURVES F (X, Y) 0, DEVELOP
  MIDPOINT ALGO (eg, hyperbola, parabola)

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OTHER CURVES

• FOR EXPLICIT CURVES Y F(X), OR PARAMETRIC
  CURVES - APPROXIMATE WITH STRAIGHT LINE SEGMENTS
• Generate points at constant parameter values
• Join them by straight lines
• Closer points for larger curvatures

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ADDRESSING PIXELS

• BY PIXEL CENTERS
• BY GRID OF PIXEL BOUNDARY LINES (H V)
• PREFERRED, AS
• NO HALF INTEGER BOUNDARIES
• PRECISE OBJECT REPRESENTATIONS
• CONVENIENT IN RASTER ALGOS

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SIZE OF OBJECTS

• RECTANGLE (0, 0) --- (4, 3)

?
?
?
?
(4, 3)
?
?
?
(0, 0)