?? ?? H/W? S/W? ??? ???? ??? ???. ?? ??? ?????.

1
? 7 ? ??

• ???????
• ? ??

2
??

• ??? ????? ????? ??? ??????
• ??? ??? ??? ??????
• ???? ??? ??????
• ??? ?? ?? ??? ?? ???? ??? ?????
• ???? ???? ????? ??????
• ?? ?? H/W? S/W? ??? ???? ??? ???. ?? ??? ?????.
• ??? ??
• ?? ??, ?? ???? ?? ??? ??? ???? ??? ??

3
7.1 4?? ?? ??
??? ?????? ???? ??
?? 7.1 ?? ??
4
7.1.1 ???

• ???? ?? ???? ???? ??? ???? ???? ??
• ???
• ???? ??? ??
• ???? ???? ?? ??. ?) ??
• ???? ??? ?? ??????? ???? ??? ??? ???? ??? ??.

5
7.1.2 ???? ??

• ???? ???
• ?? ??? ??? ??? ???? ??
• ???? ?? ?? ?? ??
• ???, ??, ????, ???, ??
• ???? ?? ??
• ?? ?? ??
• ?? ??? ???

6
7.1.2 ???? ??

• ??(Clipping)
• ??? ?? ?? ?? ??? ?? ??? ???? ? ?? ??? ???.
• ?? ??(Hidden Surface Removal)
• ??? ?? ?? ??? ?????? ?? ??? ??? ??? ? ??.
• ??? ?? (Visible-surface determination)
• ???
• ?? ??? ??? ???
• 6.3.3 ?? (p231).

7
7.1.3 ????

• ????(rasterization) ?? ????(scan conversion)
• 2?? 3?? ??? ???? 3?? ??? ??? ? ? 2??? 2?? ???
  ????. ????? ??? ???? ??? ???? ??
• ??? ???? ???? ????? ??? ?.
• Frame Buffer ? ??? ???? ???. OpenGL? ???? ???
  ???? ????? ?? ??? ??? ? ??. (10?? ??)

8
7.1.4 ??(Display)

• Frame Buffer? ?? ???? ???? CRT? ???? ?? (?????
  ????)
• ?? ?? (Aliasing)
• Anti-aliasing ???? ??
• ??? ? ??

9
7.1.5 ?? ?? ??

• ????(image-oriented)? ?? ??(object-oriented)
• ?? ??
• For (each object) render(object)
• ??? ??? ??
• ?? 7.2
• ?? ?? ???? ? ??? ????? ???? ??? ??
• ?? ??? ??, ?? ??? ??? ?? ?????? ?? ?? ????? ??.
• ?? ?? ??? ?? ??? ??? ?? ?? ?(chip)? ???? ???,

10
?? 7.2 ?? ?? ??
11
7.1.5 ?? ?? ??

• ????(image-oriented)
• For (each pixel) assign_a_color(pixel)
• ? ??? ???, ?? ???? ?? ??? ? ??? ?? ??? ? ? ?? ??
  ???? ?? ????? ????.
• ???(Coherence)? ???? ???? ??.

12
7.2 ??? ??

• ??? ??? ??? ??? ???? ??? ???.
• ???
• ??(??) ???
• ?(???) ???
• ?? ???
• ??? ?? ???
• ???(??) ??
• ?? 7.3

??-????, ????
13
?? 7.3 ?? ???
??
?????
??
?
??
14
??? ??

• (1) ??? ??
• ?? ??? ??
• ???? ???
• ???? ?????
• ?? ??? ?? ?? ???? ??
• ?? ??? 2? ???? ??? ?? ???
• (2) ?? ??? ?? ???
• ??? ?? ?? ???? ?? ??? ??

15
??(Viewport)??

• (3) ?? ???? ?? (Viewing Transformation)
• ?? 7.4
• Larry ? ??

16
?? 7.4 ?? ??
17
???? ?
18
Windows, Viewports, and Clipping
19
Windows and Viewports
Window
Interface Window
Viewport
Information outside the viewport is clipped away
20
Terminology

• World Coordinate System (Object Space) -
  Representation of an object measured in some
  physical units.
• Window - The rectangle defining the part of the
  world we wish to display.
• Screen Coordinate System (Image Space) - The
  space within the image is displayed
• Interface Window - The visual representation of
  the screen coordinate system for window
  displays (coordinate system moves with interface
  window).
• Viewport - The rectangle on the raster graphics
  screen (or interface window for window
  displays) defining where the image will appear.
  (Default is usually entire screen or interface
  window.)
• Viewing Transformation - The process of going
  from a window in world coordinates to a viewport
  in screen coordinates.

21
Viewing Transformation
Choose Window in World Coordinates
Clip to size of Window
Translate to origin
Scale to size of Viewport
Translate to proper position on screen (Interface
Window)
22
Notes on Viewing Transformation

• Panning - Moving the window about the world
• Zooming - Reducing the window size
• As the window increases in size, the image in the
  viewport decreases in size and vice versa
• Beware of aspect ratio.

23
Viewing Transformation Example
(10, 30)
(50, 30)
(0, 1)
(0.5, 1)
Viewport wanted
(0, 0.5)
(0.5, 0.5)
(10, 5)
(50, 5)
X scale 0.5/40 1/80
(0,0)
(1, 0)
1 0 0 0 1 0.5 0 0 1
1/80 0 0 0 1/50 0 0
0 1
1 0 -10 0 1 -5 0 0 1
1) Translate window to origin
2) Scale to correct size
3) Translate to proper coordinates
24
7.3 ?? ??(Clipping)

• ???(clipper) ?? ?? ?? ?? ? ??? ??? ?????? ????.
• ??(accepted), ??(rejected, culled)
• ??? ??? ?? ? ? ?? ? ??? ???? ??? ? ??.
• ???? ?? ??? ? ??.
• 3?? ?? ??? ?? ??
• 2?? ???? ?? ??

25
7.3.1 Cohen-Sutherland ??

• ??? ?? ?, ?? ??? ???? ????. ???? ?? ??? ???? ????
  ???. ?? ??? ???
• 3???? ?? ?? ??
• 2???? ??
• Larry ?? ??

26
?? 7.5 ??? ??
27
?? 7.6 ??? ??? ?? ??
28
?? 7.7 Cohen-Sutherland ?????? ????? ??
29

• o1outcode(x1,y1), o2outcode(x2,y2)
• Case 1 o1o20,
• trivially accept, ?? AB
• Case 2 o1?0, o20
• ?? CD, ??? ?? ? ?? ?? ?? ??
• Case 3 o1o2 ? 0
• Trivially reject
• ?? EF
• Case 4 o1o2 0
• ? ?? ?? ??
• ?? GH, ?? IJ ??, ??? ??? ???.

30
Clipping
(xl, yt)
(xr, yt)
A point is visible if xl lt X lt xr and yb lt Y lt yt
(xl, yb)
(xr, yb)

• A line is visible if both of its end points are
  in the window.
• Brute Force Method - Solve simultaneous equations
  for intersections of lines with window edges.

31
Cohen-Sutherland Algorithm

• Region Checks Trivially reject or accept lines
  and points.
• Fast for large windows (everything is inside) and
  for small windows (everything is outside).
• Each vertex is assigned a four-bit outcode.
• Bit 1 lt-- sign bit of (yt-Y) -- point is
  above window
• Bit 2 lt-- sign bit of (Y-yb) -- point is
  below window
• Bit 3 lt-- sign bit of (xr-X) -- point is
  to right of window
• Bit 4 lt-- sign bit of (X-xl) -- point is
  to left of window

32
Cohen-Sutherland Clipping (cont.)
1001
1000
1010
Bit 1 Above Bit 2 Below Bit 3 Right Bit 4
Left
0001
0000
0010
0101
0100
0110

• A line can be trivially accepted if both
  endpoints have an outcode of 0000.
• A line can be trivially rejected if any
  corresponding bits in the two outcodes are both
  equal to 1. (This means that both endpoints are
  to the right, to the left, above, or below the
  window.)

33
Clipping Lines Not Accepted or Rejected

• In the case where a line can be neither trivially
  accepted nor rejected, the algorithm uses a
  divide and conquer method.

D
Example
C
B
A
H
E
F
G
Line AD 1) Test outcodes of A and D --gt
cant accept or reject. 2)
Calculate intersection point B, which is
conceptually on the
window side of the dividing line. Form new line
segment AB and discard
the rest of the line because it is
above the window.
3) Test outcodes of A and B. Reject.
34
7.3.2 Liang_Barsky ??

• ??

35
?? 7.8 ?? ??? ??? ?? ???? ??? ??
36
7.4 ??? ??

• 10??? ??? Antialiasing? ??(Compositing)? ???? ??
  ???? ??? ???? ??? ??
• Sutherland-Hodgeman ???? ??

37
?? 7.9 ??? ?? ?? ??
??? ?? ???? ?? ??? ???? ??? ? ??? ???? ???
??????? ??? ??? ???
38
?? 7.10 ??? ???? ??
?? ???(Concave)? ?? ??? ??? ??? ?? ??? ? ??.
39
?? 7.11 ??? ??? ??
40
?? 7.12 ??? ???? ??
?? ???(convex)? ??? ???? ??? ??? ??? ??? ?? ????
???. ?? ???? ?? ?? ?? ????? ??(tessellate) ? ?
??.
41
?? 7.13 Blackbox??? ???
?? ??
42
?? 7.14 ??? ?? ??
??? ?? ??
????
43
?? 7.15 ??? ???? ???
44
?? 7.16 ??? ???
??? ?? ??? ?
45
? ?? ?? ??? ??
?? ????? ??
Out-gtin I, N
In-gtin N
In-gtout I
Out-gtout Nothing
Intersection I, Previous endpoint P, Next
Point N
46
?? 7.17 ??? ?? ??? ?
47
Polygon Clipping

• Polygons can be clipped against each edge of the
  window one edge at a time. Window/edge
  intersections, if any, are easy to find since the
  X or Y coordinates are already known.
• Vertices which are kept after clipping against
  one window edge are saved for clipping against
  the remaining edges. Note that the number of
  vertices usually changes and will often increase.

48
Polygon Clipping Algorithm
P4
P4
I2
I2
P3
I1
I1
P1
P2
P1

• The window boundary determines a visible and
  invisible region.
• The edge from vertex i to vertex i1 can be one
  of four types
• Exit visible region - save the intersection (I2)
• Wholly outside visible region - save nothing
• Enter visible region - save intersection and
  endpoint(I1,P1)
• Wholly inside visible region - save endpoint(P4)

49
Polygon clipping issues

• The final output, if any, is always considered a
  single polygon.
• The spurious edge may not be a problem since it
  always occurs on a window boundary, but it can be
  eliminated if necessary.

50
Pipelined Polygon Clipping
Clip Top
Clip Right
Clip Bottom
Clip Left

• Because clipping against one edge is independent
  of all others, it is possible to arrange the
  clipping stages in a pipeline. the input polygon
  is clipped against one edge and any points that
  are kept are passed on as input to the next stage
  of the pipeline.
• This way four polygons can be at different stages
  of the clipping process simultaneously. This is
  often implemented in hardware.

51
7.5 ?? ?? ??? ??
52
7.5.1 ?? ???

• ?? ???(bounding box) ?? ??(extent)
• ?? ???? ???? ???? ???? ??? ?? ?? ????.
• ?? ???? ???? ??? ???? ?? ? ??.

53
?? 7.18 ???, ?? ???, ?? ???
54
?? 7.19 ?? ????? ??
?? ???? y ???? ???? ??? ?? ??? ?? ??? ???? ??? ??
??
?? ???? ?? ?? ???? ?? ??? ???? ??? ?? ??? ????.
55
7.5.2 ??, ??, ???
?? 7.20 ?? ??
56
7.5.2 ??, ??, ???

• 2? ??? ?? ??? ???? ??? ????? ?? ??? ? ???
• ??? ?? ??? ???? ??? ??? ??? ?? ????, ???? ???
  ? ? ?? ??? ??
• ??? ???? ????, ??? ?? ????? ??? ?? ?? ?? ??? ??
  ? ??.

57
7.5.2 ??, ??, ???

• ???? ?? ? ?? ?? ??? ??
• ?? ??, ???? ??
• ?? ??? ??
• ???? ?? ?? ????? ??? ?? ???
• ??? frame buffer?? ??.
• ???? ??? ??
• OpenGL API? ?? ? ?? ??? ???.

58
7.5.3 ??? ???? ??

• ??? ????, ?? ??? ??? ??? ??? ???? ??.
• ??? ?? ?? raster ??? ??? ????.
• ??? ??? ??? ???? ?? ???? ??? ???? ?? ?? ??.

59
7.6 ?????? ??

• ?? ??? ?? ?? ??(bound volume)? ??? ???.
• 2?? ?? ????( Cohen-Sutherland, Liang-Barsky,
  Sutherland-Hodgeman)? 3???? ?? ??? ? ??.

60
?? 7.21 ????? ?? ??? ??
61
?? 7.22 ????? Cohen-Sutherland ??
2???? 4 ?? ??? 3???? 6 ??? ????.
62
?? 7.23 ??-????
2?? ??? ??? ??? ??. ??? ?? 3?? ?? ??? ??? ??
??. ??? ??? ?? ??
63
??? ??? 6?? ??? 1?? ????? ????. ??? ?? ?? ??? ??
?? ? ?? ????? ?? ? ??.
64
?? 7.24 ?? ??? ?? ??
65
?? 7.25 ?? ??? ?? ??
66
?? 7.26 ??? ?? ?? ??? ??

• ??? ?? ?, ?? ?? ??? ???? ??? ?? ??? ?? ??? ??
  ???? ???? ??? ?? ?? ??? ????.
• ???, ??? ?? ??? ?, ????? ??? ???? ??(??? ??) ??
  ????? ??? ?? ??? ????.
• ?? ??? ???? ?? ??? ? ? ??.

67
7.7 ?? ??

• ???? ?? ???? ???? ??? ??? ??? ?? ??? ??, ?? ??
  ???? ??? ???? ???? ??(?,??,???)? ?? ??.
• ??? ???? ???? ??? ??? ???? ?? ?? ?? ??? ???? ??
  ??? ?? ?? ?? ???? ??? ? ??.

68
7.7.1 ?? ??? ???? ??

• Object space ? image space
• K?? ????? ??? ??? ????? ??? ??? ??
• ???? ? ?? ??? ?, ? ??? A? B? ????. (?? 7.27 ??)
• A? B? ?????? ??? ??? A? ????.
• B? A? ??? B? ????.
• A? B ?? ??? ??? A? B ?? ????.
• A? B? ????? ??? ? ???? ??? ??? ??? ???? ??.
• ?? ????
• ????? ?? ?? ???? ?? ??? ? ???.

69
?? 7.27 ? ???
(a) B? A? ????? ??? (b) A? B? ????? ??? (c) A? B
?? ???. A? B ?? ????. (d) B? A? ??? ???.
70

• ??? ??(ray)?? ??? ???.
• ??? ?? ??? ??? ??? ???.
• ?? 7.28 ??
• k? ???? ??? ???? ????? ??? ???? ????? ????,
• ???? ??? ?? ???? ?? ??? ???? ???. ????? ???? ???
  ???? ????.
• ??? ???, ?? ??? nmk ? ???.
• ??? ?? ???? ?.
• ?? ???? ?? ??? ??? ???.

71
?? 7.28 ???? ?? ??
72
7.7.2 ?? ??

• Back face culling
• ?? ?? ????? ???? ?? ?? ?? ????? ?????? ?? ??? ??
  ? ??. (? 50 ??)

73
????(Backface culling)
74
?? 7.29 ?? ??
V ? ???? ?? N ? ?? T? ??? ??? ???
??? ? ???? ??? ???. ??? ??
???? ??? ???.
75
?? (front face ? ?? )
??? ?? ???? ?? ??? ? ??? ?? ??? ??.
??? ??
?? ???
???
C? ???? ??, ???? ???.
76
7.7.3 z-?? ????(1)

• ?? ????? ???? ?? ?? ????
• H/W ?? S/W? ???? ??, ????? ??? ???? ??
• ????? ???? ?????, ??? ?? ???? ??? ????? ?? ????.
• 7.9??? ???? ???? ??? ????? ??? ? ??.
• ??? ??(Frame Buffer)? ?? ??? ??(depth)? ????
  buffer(z-buffer)? ????.

77
7.7.3 z-?? ????(2)

• ??? ??? ???? ??? ?? ??? ???? ?? ????.
• ? ?? ???? ???? ???? ??. ??? ?? ?? ??? ?????? ? ??
  ?? ???.
• ??7.30 ??
• B? ???? ??? ??? ?? Z2? ??? A? ?? Z1 ?? ?? ??? ???
  ????. ??? A? ???? ??? ??? ???? ???.

78
?? 7.30 z-?? ????
79
7.7.3 z-?? ????(3)

• ?? ??? ?? ??? ?????.
• ??? ??? ????? ??? ??.
• ??? ??? ??? ???? ??.
• ??? ????? ???? ???? ???? ?? ??? ??? ?????? ???
  ??? ? ??. ? ??? dl ?? ???? z-buffer ?? ????.
• ? ??? z-buffer ??? ??, ? ?? ??? ?? ???
• ? ??? z-buffer ??? ???, z-buffer ? ?? ?? ????,
  ??? ??? ???? ??? ?? ??? ??? ???? ????.
• Foley ?? ?? ??

80

• Z-??.
• ???? z-buffer? ?? z?? ?? ??? ??
• ??? ???? ???? ?? ??? ??

81
Depth Coherence(?? ???)

• 7.9?? ??? ????? ?????? ???? ???? ??????? ????.

• z1 ?? ?? ???? ????? ??, (x?x,y)??? z ?? z2 ?
  z1? ?? 1?? ???? ?? ? ??. ?? -A/C ?? ??? ?????
  ??.

82
?? 7.31 ??? z-?? ????
83
7.7.4 ?? ??? ?? ????

• Depth Sort Algorithm, Painters Algorithm
• ???? ?? ??
• ?? ?? ????
• ?? ???? ??? ??(?????? ?? ?) z ??? ?? ????.
• ??? ???? z ?? ??? ?? ? ???? ???? ???? ?? ?????
  ???? ????.
• ?? z ??? ?????? (?, ??? ???) ? ???? ????.
• ?? ????? ? ???? ??? z?? ?? ??? ??? ? ?? ???? ??
  ?? ????? ??? ???? .

84
?? 7.32 ?? ????
?? ??? ??? ??? ??? ????. ??? ?? ?? ?? ???? ??
???, ????? ??? ?? ?? ???? ?? ?? ???? ???? ???
???? ???. ?? ??? (back-to-front rendering)
85
?? 7.33 ??? ??? z ??
86
?? 7.34 x? y ???? ??? ?? ??

1. ??? ?? x ??
2. ??? ?? y ??

87
?? 7.35 ?? ??? ?? ???
88

• ??? P? ?? ???? ?? ??? ?? ??? ????. P? z-????? ???
  z-?? ??? ?? ?? ??? Q? ????, P? Q? ?? ? ?? ???, Q?
  ??? ?? P? ?? ? ??? ???? ??.
• 5?? ??? ??? ???? ?, ? ? ????? ???? P? Q? ?? ? ???
  ???? ?? ??.
• ????? x-????? ??? ????
• ????? y-????? ??? ????
• Q ??? ??? P? ??? ?????
• P? ??? ??? Q? ??? ?????
• (x,y) ????? ????? ??? ?? ??? ????
• ? Foley ? ??

89

• 5?? ??? ?? ???? ??? P? Q? ???? ???? ??? Q? P???
  ?? ??? ? ???? ????. ?? 1,2, 5? ?? ??? ??? ??
• 3 P ??? ??? Q? ??? ?????
• 4 Q? ??? ??? P? ??? ?????
• ?? 7.36? ?? 3.37? ???? ?? ??????? ?????. ??? ???
  ???.

90
?? 7.36 ??? ??
91
?? 7.37 ???? ???
92
7.7.5 ??? ????

• ????? ????? ???? ????? ?? ??? ??? ??? ?? ???? ??
  ?? ????
• ??? ????? ?? ?? ????? ??? ??
• ?? 7.38? ??

93
?? 7.38 ??? ????(1)
??? i ??? A? ? a ?? ??. ? ?? ????? ??? ?? ???
????. ? b ?? ?? ???? ????? ??? ???. ??? B? ? c??
??. ? ?? ?????? ?? ??? ???. ? d?? ???? ??? ??
94
?? 7.38 ??? ????(2)
??? j ??? A? ? a ?? ??. ? ?? ????? ??? ?? ???
????. ??? B? ? c ?? ???? ??? ????? ????? ? d? ??
??? ??? ????? ??.
95
7.8 ?? ??

• ???? ???? ??? ????? ??? ???? ??? ??
• ??
• ??? ???? ???.
• ? ?? ??? ??? ??? ? ??? ?? ???? ??.
• ?? ???? 2???? ?????, ?? ????? ???.

96
Write_pixel(int ix, int iy, int value) value
index mode?? index, RGB mode??? 32 bit ?
(???) ix, iy ???
97
?? 7.39 ??? ?????? ??
98
DDA(Digital Differential Analyzer) ????
?? 7.40 DDA ????? ?? ??? ??
99
DDA(Digital Differential Analyzer) ????
m? 1?? ? ??? ???? ??
m? ?? ?????? ???? ????.
for (ixx1, ixltx2, ix) ym write_pixel(x,
round(y), line_color)
100
?? 7.41 ???? ? ??? ?? ??? ?? ??? ???
101
?? 7.42 ??? DDA ????? ?? ??? ??
102
7.9 Bresenham ????

• DDA? ??? ????? ??? ??? ??? ????. ??? ???? ??? ???
  ??? ??? ????

103
?? 7.43 Bresenham ????? ?? ??
104
?? 7.44 Bresenham ????? ?? ??
???? da-b dgt0 ???? ?? dgt0 ??? ??
105
if
otherwise
What is ?
?? ?? ??? ??? ?? ??? ???? ????.
106
?? 7.45 a ? b ? ??
107
Design of Line and Circle Algorithms
108
Basic Line and Circle Algorithms

• 1. Must compute integer coordinates of pixels
  which lie on or near a line or circle.
• 2. Pixel level algorithms are invoked hundreds
  or thousands of times when an image is created or
  modified.
• 3. Lines must create visually satisfactory
  images.
• Lines should appear straight
• Lines should terminate accurately
• Lines should have constant density
• Line density should be independent of line length
  and angle.
• 4. Line algorithm should always be defined.

109
Simple DDA Line AlgorithmBased on the
parametric equation of a line

• Procedure DDA(X1,Y1,X2,Y2 Integer)
• Var Length, I Integer
• X,Y,Xinc,Yinc Real
• Begin
• Length ABS(X2 - X1)
• If ABS(Y2 - Y1) gt Length Then
• Length ABS(Y2-Y1)
• Xinc (X2 - X1)/Length
• Yinc (Y2 - Y1)/Length
• X X1
• Y Y1
• DDA creates good lines but it is too time
  consuming due to the round function and long
  operations on real values.

For I 0 To Length Do Begin Plot(Round(X),
Round(Y)) X X Xinc Y Y Yinc End
For End DDA
110
DDA Example

• Compute which pixels should be turned on to
  represent the line from (6,9) to (11,12).
• Length Max of (ABS(11-6), ABS(12-9)) 5
• Xinc 1
• Yinc 0.6
• Values computed are
• (6,9), (7,9.6),
• (8,10.2), (9,10.8),
• (10,11.4), (11,12)

111
Fast Lines Using The Midpoint Method

• Assumptions Assume we wish to draw a line
  between points (0,0) and (a,b) with slope m
  between 0 and 1 (i.e. line lies in first octant).
• The general formula for a line is y mx B
  where m is the slope of the line and B is the
  y-intercept. From our assumptions m b/a and B
  0.
• y (b/a)x 0 --gt f(x,y) bx - ay 0 is an
  equation for the line.

112
Fast Lines (cont.)

• For lines in the first octant, the next
• pixel is to the right or to the right
• and up.
• Assume
• Distance between pixels centers 1
• Having turned on pixel P at (xi, yi), the next
  pixel is T at (xi1, yi1) or S at (xi1, yi).
  Choose the pixel closer to the line f(x, y) bx
  - ay 0.
• The midpoint between pixels S and T is (xi 1,yi
  1/2). Let e be the difference between the
  midpoint and where the line actually crosses
  between S and T. If e is positive the line
  crosses above the midpoint and is closer to T.
  If e is negative, the line crosses below the
  midpoint and is closer to S. To pick the correct
  point we only need to know the sign of e.

113
Fast Lines - The Decision Variable

• f(xi1,yi 1/2 e) b(xi1) - a(yi 1/2 e)
  b(xi 1) - a(yi 1/2) -ae f(xi 1, yi
  1/2) - ae 0
• Let di f(xi 1, yi 1/2) ae di is known
  as the decision variable.
• Since a ? 0, di has the same sign as e.
• Algorithm
• If di ? 0 Then
• Choose T (xi 1, yi 1) as next point
• di1 f(xi1 1, yi1 1/2) f(xi 11,yi
  11/2)
• b(xi 11) - a(yi 11/2) f(xi 1, yi
  1/2) b - a
• di b - a
• Else
• Choose S (xi 1, yi) as next point
• di1 f(xi1 1, yi1 1/2) f(xi 11,yi
  1/2)
• b(xi 11) - a(yi 1/2) f(xi 1, yi
  1/2) b
• di b

114
Fast Line Algorithm
The initial value for the decision variable, d0,
may be calculated directly from the formula at
point (0,0). d0 f(0 1, 0 1/2) b(1) -
a(1/2) b - a/2 Therefore, the algorithm for a
line from (0,0) to (a,b) in the first octant
is x 0 y 0 d b - a/2 For i 0
to a do Begin Plot(x,y) If d gt 0 Then
Begin x x 1 y y 1 d d b
- a End

• Else Begin
• x x 1
• d d b
• End
• End

Note that the only non-integer value is a/2. If
we then multiply by 2 to get d' 2d, we can do
all integer arithmetic using only the operations
, -, and left shift. The algorithm still works
since we only care about the sign, not the value
of d.
115
Bresenhams Line Algorithm

• We can also generalize the algorithm to work for
  lines beginning at points other than (0,0) by
  giving x and y the proper initial values. This
  results in Bresenham's Line Algorithm.
• Begin Bresenham for lines with slope between 0
  and 1
• a ABS(xend - xstart)
• b ABS(yend - ystart)
• d 2b - a
• Incr1 2(b-a)
• Incr2 2b
• If xstart gt xend Then Begin
• x xend
• y yend
• End
• Else Begin
• x xstart
• y ystart
• End

For I 0 to a Do Begin Plot(x,y) x x
1 If d ? 0 Then Begin y y 1 d d
incr1 End Else d d incr2 End For
Loop End Bresenham
116
Optimizations

• Speed can be increased even more by detecting
  cycles in the decision variable. These cycles
  correspond to a repeated pattern of pixel
  choices.
• The pattern is saved and if a cycle is detected
  it is repeated without recalculating.
• di 2 -6 6 -2 10
  2 -6 6 -2 10

117
Fast Circles

• Consider only the first octant of a circle of
  radius r centered on the origin. We begin by
  plotting point (r,0) and end when x lt y.
• The decision at each step is whether to choose
  the pixel directly above the current pixel or the
  pixel which is above and to the left (8-way
  stepping).
• Assume Pi (xi, yi) is the current pixel.
• Ti (xi, yi 1) is the pixel directly above
• Si (xi -1, yi 1) is the pixel above and to
  the left.

118
Circle Drawing Algorithm

• We only need to calculate the values on the
  border of the circle in the first octant. The
  other values may be determined by symmetry.
  Assume a circle of radius r with center at (0,0).
• Procedure Circle_Points(x,y Integer)
• Begin
• Plot(x,y)
• Plot(y,x)
• Plot(y,-x)
• Plot(x,-y)
• Plot(-x,-y)
• Plot(-y,-x)
• Plot(-y,x)
• Plot(-x,y)
• End

119
7.10 ???? ?? ??

• ??? ??? ???? ?? ??
• ??? ?? ???? ??, ?? ???? ???????, ??? ???? ?? ??
  ??? ????.

120
7.10.1 z-buffer? ??? ????

• ???? ?? ?? ????? ????. ?? ?? ???? ?????, ?? ??,
  ?? ??, ??? ?? ????? ??.
• Z-buffer ????? ? ?? ??? ??? ?? ? ??.
• ?? ???? ???? ??? ?? ????? ?? y? ??? ????. (??
  7.47, ????)
• ??? ???? ??? ???? ??? ????? ????, ?? ???? ??? ???
  ???? ??? ????.(?? ??)
• ???? ?? ?? ??? ????, ??? ????. ???, ??? ?? ???
  ?? ???? ???? ??? ??? ??? ???? ????(???)

121
?? 7.46 ???? ? ?? ??
??? ?? ???
?? ???
122
?? 7.47 ???
?? ???
????? ???
123
7.10.2 ???? ??

• ??? ??? ???? ???. ?? ??? ??? ???, ??? ??(???)???
  ????.
• ??? ??? ?? ??? ??? ??? ?? ??? ??? ?? ??? ????
  ????.
• Filling
• ?? ???(flood fill)
• ??? ??? (scan line fill)
• ??-?? ??? (even-odd fill)

124
7.10.3 ?? ???

• Bresenham ????? ????, ???? ??? ?????? ???? ????
  ???? ?? ???? ????. (?? 7.48)
• ??? ???? ???(?? ???)? ??, ? ??? ????? ????. ???
  ?? ?? ????(??? ??????) ????? ???.

125
flood_fill (int x, int y) if (read_pixel(x,y)
WHITE) write_pixel(x,y,BLACK) flood_fil
l(x-1,y) flood_fill(x1,y) flood_fill(x,y-1)
flood_fill(x,y1)
126
?? 7.48 ?? ?? ??? ???
???
???
127
7.10.4 ??? ????

• ???? ??? ???? ??? ??-?? ??? ?????, ??? ??? ???
  ??? ?? ??? ??? ??(span)? ????.
• ???? ???? ???? ??? ???? ????. ??? ??? ????? ????.
  (Edge Coherence)
• ?? 7.50? ?? 7.51? ??
• Y-x ???? ??? ??? ????? ????? ??? ? ? ??. ??
  7.52 ??

128
?? 7.49 ??? ??? ???
129
?? 7.50 ?? ???? ?? ??? ???
130
?? 7.51 ??? ??? ??
131
?? 7.52 y-x ????? ?? ????
132
?? 7.53 ???
??-?? ??? ??? ?????, ?? ??? ??? ??? ????? ??. ?,
???? ??? ??? ?, (a)? ??? 0??? 2??? ????, (b)?
???? 1??? ????? ??.
133
7.11 ??????

• 7.54 ???? Raster ??
• ?? ?? ?? ??? ???? ????? ?? ??

134

• ??? ??? ??? ??? ??
• ??? ??? ??? ??? ??
• ?? ?? ?? ???? ??? ??? ???? ???.??? ?? ??? ???
  ??? ?? ???????.

• Raster? ? ??? ??? ?? ??? ??? ???? ???? ???? ???
  ???.

135
Anti-aliasing

• ?? ??(area averaging)
• ??? ???? ?? ??? ???? ? ??? ?? ????.( Bresenhams
  algorithm? ? x?? ??? ???? ? ?? ?? ???? ?? ??? )
• ??? ? ? ???? ??
• ? ??? 1? 2? ??? ?? ??? ?
• ?? ??
• ?? ????? ???? ?? ??? ??? ???? ?? ????? ? ??? ??
  ?? ??.

136

• ?? 7.55 Aliasing ??? Anti-aliasing? ??
• ????? ?? (b) ??????? ??
• (c) ??? ????? ?? (d) ??? ????????

137
?? 7.56 ? ??? ???? ???
? ???? ?? ?? ?? ???, ??? ??? ?? ????? ?? ??? ????
??? ?? ???. ? ???? ?? ??? ?? ??? ??? ?? ??? ? ???
? ? ??? ??? ?? ? ??. (10?? ?? ??? ??)
138
?? 5.57 ?? ?? ???? (time-domain aliasing)

• ?? ??? ???? ?? ??? ??
• ? ???? ?????? ??? ? ???? ????? ??? ?? ??.

139
7.12 ????? ?? ??

• ?? ?????? frame buffer? ??? ??? display?? ? ????
  ???.
• ??? ???? (flickering)
• Frame buffer? ???? ??? display? ??? frame buffer?
  ?? ??? ??? ???? (2 ?? ???)
• ?? 60?? 75 Hz ?? ???
• Double Buffering

140
7.12.1 ???
??? ??
RGB ?? ??? ???? ? ? ??
?? 7.58 ??-??-?? ? (a) RGB ??? (b) HLS ???
141
? ?? ??

• OpenGL? 3?? (0.8, 0.6, 0.0)?
• CRT? ?? ?? ????? ??????
• ?? ?? ?? ??. ?? ??? CRT ?? ??? ?? ? ??? ??? ???.
• ? ????? ? ??? ????.

• M? 33 ???? ??? ?? ???.
• ??? ???? ?? ?? ???.

142

• CMYK ??
• ?? ??
• ??? ??? ?? ??? ?? ??
• HLS ??
• ?? (Hue) ?? ???? ??
• ?? (Lightness) ?? ??? ?? ????.
• ?? (Saturation) ??? ????. ????
• RGB ??? ?? ??? ??? ?????? ??? ? (?? ??? ??????
  ???)
• - ??? ????? ???? ??? ??

143
7.12.2 ?? ??
(??)
?? 7.59 ??? ?? (brightness)

• ??? ??? ?? ???? ???

144

• ??(brightness)? ??? ?? (intensity)
• ??? ??? ?? ???? ???.
• ? ??? ?? ??? ???? ??? ????? ????? ?.
• I ? CRT? ??, V? ???? ??,
• ? ??? CRT ? ??
• frame buffer? ?? ?? ??? ??? ?? ?? ???? ?? ???
  ??? ? ??. ??? ??? ???? ??? ?? ????? ??.

145
7.12.3 ????? ????
( Dithering and Halftoning)

• ???? (Halftoning)
• ?? ???(black and white)?? ??? ??? ?? ? ??? ????
  ?? ??? ??? ?.
• ?? ???? ???? ???? ???? ??? ???.
• ??? ???? ?? ?? ???? ???. ?? ?? ?? ?? ??? ???? ?
  ??.
• OpenGL?? ???.
• glEnable(GL_DITHER)

146
?? 7.60 ??? ???(digital halftone) ??

• ??? ??? ??, ??? ??? ??, ?? ??? ??? ???? ???? ??
• 0?? 16?? ?? ??? ???? 17?? ??? ??? ???. ?????? ¼?
  ?? ??? 2???? 17??? ?????? ???.
• ??? ?? ?? ????(dither) ??? ??? ?? ?? ????? ??.