Computer Graphics

1
Computer Graphics

• 2D GraphicsPolygons

2
Polygons

• Generally represented as vertex and edge data
• Edge data can be explicitly or implicitly
  represented in a data structure
• Explicit - list of vertices (x,y) and a list of
  edges (v1,v2) no order required
• Implicit - list of vertices in either clockwise
  or counter-clockwise order

3
PolygonsExplicit data structure

• Edges
• 1, 4
• 4, 3
• 3, 2
• 2, 5
• 5, 1

V3

• Vertices
• 4, 3
• -2, -4
• -4, 4
• 0, 2
• 2, -4

V1
V4
V2
V5
4
PolygonsImplicit data structure
V3

• Vertices
• 4, 3
• 0, 2
• -4, 4
• -2, -4
• 2, -4

V1
V2
V4
V5
5
Filling shapes

• Shapes are generally filled by setting each
  pixel, for a given scan-line, from xmin to xmax
• Spans reflect spatial coherence - primitives
  often do not change from pixel to pixel or from
  scan line to scan line

6
Filling shapes
Span coherence - all pixels on a span are set to
the same value
Edge coherence - all pixels on an edge are set to
the same value
7
Filling polygons

• Scanline algorithm - How to ...
• Find the intersections between each scan lines
  and all of the edges
• Sort the intersections by Y, and then by X
• Set all pixels between pairs of intersections
  that lie interior to the polygon

8
Filling polygons

• In order to insure that pixels on shared edges
  are not often set twice
• Only choose pixels which are found inside a
  polygon

9
Filling polygons

• Horizontal edges
• Ignored as they are automatically drawn due to
  the fact that there are included via other edges

10
Filling polygons

• A vertex shared by two edges, where the dY does
  not change direction from edge to edge,
• Need to be addressed only once

11
Filling polygons

• To accomplish this, one of the two edges needs to
  be foreshortened before being added to the Edge
  Table

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Edge table
y min
y max
A
x min
6
1/m
5
DA
4
D
3
BA
2
B
1
CD
CB
0
C

• For each edge, create an edge record consisting
  of ymax, xmin (value for x corresponding to ymin)
  and 1/m
• Place the each edge record into the sublist
  (bucket) corresponding to the edges ymin

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Active edge table
y max
x
1/m

• Set y to the smallest y coordinate with a list in
  the ET
• Initialize the AET to be empty
• Repeat until AET and ET are both empty
• Move edge records from ET to AET where yminy
• Draw spans using two edge records at a time
• Remove from AET all edges where ymax y
• Increment y
• Increment each x by 1/m

14
Clipping

• Simultaneous equations
• Cohen-Sutherland Line-Clipping Algorithm
• Parametric Line-Clipping Algorithm
• Circles
• Sutherland-Hodgman Polygon-Clipping Algorithm

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Clipping

• Lines within the clipping rectangle are trivially
  accepted
• Lines outside the rectangle with respect to one
  halfplane are trivially rejected

Accepted
Rejected
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Clipping

• Lines that are neither accepted nor rejected
• are clipped against the extended edges of the
  clipping rectangle
• thus creating new endpoints

Accepted
Rejected
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Simultaneous equations

• For a given line segment (x0,y0,x1,y1), each
  point in the edge can be described by...
• x x0 t (x1 - x0)
• y y0 t (y1 - y0)
• ...where 0 ? t ? 1

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Simultaneous equations

• First, the intersection of the extended clipping
  rectangle edge and the extended line segment is
  determined.
• Then, this point (x,y) is placed into two pairs
  of simultaneous equations to compute tedge and
  tline
• If tedge and tline and both in the range 0,1,
  then this is an intersection we wish to clip to

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Cohen-SutherlandLine-Clipping Algorithm

• Uses a four-bit code for each of the nine regions
• 1 0 0 1top-bottom-left-right
• 1 outside0 inside

1001
1000
1010
0001
0000
0010
0101
0100
0110
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Cohen-SutherlandLine-Clipping Algorithm
if outCode.top 1 then clip against
top else if outCode.bottom 1 then clip
against bottom else if outCode.right 1
then clip against right else if outCode.left
1 then clip against left code endpoint
p0 if (accept) then draw P0 to P1

• accept ? false
• done ? false
• code endpoints P0 and P1
• while (not done)
• if (outCode0 0) and (outCode1 0) then
• accept ? true
• done ? true
• else if (outCode 0 and outCode1) ? 0 then
• done ? true
• else if outCode0 0 then
• swap points and codes

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ParametricLine-Clipping Algorithm

• Cyrus-Beck (1978)
• Somewhat different from Cohen-Sutherland
• First, pick a point PE somewhere on the edge
  being tested

N? P(t) - PE lt 0
N? P(t) - PE 0
N? P(t) - PE gt 0
Ni
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ParametricLine-Clipping Algorithm

• Next, since the dot product of the edges normal
  with P(t)-PE is zero when P(t) is on the edge
• with some rearrangement and substitution we get

N? P(t) - PE lt 0
N? P(t) - PE 0
N? P(t) - PE gt 0
Ni
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ParametricLine-Clipping Algorithm

• If t is in the range 0,1
• then the edge intersects our line segment (P0,P1)
• And so, we clip against the edge using t to
  compute the new endpoint

N? P(t) - PE lt 0
N? P(t) - PE 0
N? P(t) - PE gt 0
Ni
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Antialiasing

• Increasing resolution
• Area sampling