CS 551 / 645:
Introductory Computer Graphics
Rasterizing General Polygons
Clipping Lines
David Luebke
7/27/2016
Recap: Rasterizing Triangles
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Interactive graphics: polygons rule the world
In practice, polygons are often decomposed
into triangles (Why?)
Different methods for rasterizing triangles:
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David Luebke
Screen-space subdivision (Warnock’s Algorithm)
Primitive subdivision (REYES, Renderman)
Edge walking
Edge equations
7/27/2016
Recap: Edge Walking
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Basic idea:
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David Luebke
Draw edges vertically
Fill in horizontal spans for each scanline
Interpolate colors down edges
At each scanline, interpolate
edge colors across span
7/27/2016
Recap: Edge Equations
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Edge equations thus define two half-spaces:
David Luebke
7/27/2016
Recap: Edge Equations
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And a triangle can be defined as the
intersection of three positive half-spaces:
David Luebke
7/27/2016
Recap: Edge Equations
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We discussed methods for optimizing
rasterization with edge equations:
– Early termination
– Incremental computation of edge equations
– Fast test for negative edge equation values
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We discussed computing the coefficients
{A, B, C} of each edge equation
David Luebke
7/27/2016
Recap: Interpolating Parameters
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Given colors (and later, other parameters) at
the vertices, how to interpolate across?
Idea: triangles are planar in any space:
– This is the “redness”
parameter space
– Note:plane follows form
z = Ax + By + C
– Can use same methods!
– Setup cost: matrix mult
– Inner loop cost: 1 add
David Luebke
7/27/2016
General Polygon Rasterization
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Now that we can rasterize triangles, what
about general polygons?
We’ll take an edge-walking approach (why?)
David Luebke
7/27/2016
General Polygon Rasterization
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Consider the following polygon:
D
B
C
A
E
F
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How do we know whether a given pixel on
the scanline is inside or outside the polygon?
David Luebke
7/27/2016
General Polygon Rasterization
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Does it still work?
D
B
H
C
A
G
I
E
F
David Luebke
7/27/2016
General Polygon Rasterization
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Basic idea: use a parity test
for each scanline
edgeCnt = 0;
for each pixel on scanline (l to r)
if (oldpixel->newpixel crosses edge)
edgeCnt ++;
// draw the pixel if edgeCnt odd
if (edgeCnt % 2)
setPixel(pixel);
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Why does this work?
What assumptions are we making?
David Luebke
7/27/2016
Faster Polygon Rasterization
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How can we optimize the code?
for each scanline
edgeCnt = 0;
for each pixel on scanline (l to r)
if (oldpixel->newpixel crosses edge)
edgeCnt ++;
// draw the pixel if edgeCnt odd
if (edgeCnt % 2)
setPixel(pixel);
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Big cost: testing pixels against each edge
Solution: active edge table (AET)
David Luebke
7/27/2016
Active Edge Table
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Idea:
– Edges intersecting a given scanline are likely to
intersect the next scanline
– Within a scanline, the order of edge intersections
doesn’t change much from scanline to scanline
David Luebke
7/27/2016
Active Edge Table
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Algorithm:
– Sort all edges by their minimum y coord
– Starting at bottom, add edges with Ymin= 0 to AET
– For each scanline:
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Sort edges in AET by x intersection
Walk from left to right, setting pixels by parity rule
Increment scanline
Retire edges with Ymax < Y
Add edges with Ymin > Y
Recalculate edge intersections and resort (how?)
– Stop when Y > Ymax for last edges
David Luebke
7/27/2016
Next Topic: Clipping
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We’ve been assuming that all primitives
(lines, triangles, polygons) lie entirely within
the viewport
In general, this assumption will not hold:
David Luebke
7/27/2016
Clipping
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Analytically calculating the portions of
primitives within the viewport
David Luebke
7/27/2016
Why Clip?
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Bad idea to rasterize outside of framebuffer
bounds
Also, don’t waste time scan converting pixels
outside window
David Luebke
7/27/2016
Clipping
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The naïve approach to clipping lines:
for each line segment
for each edge of viewport
find intersection points
pick “nearest” point
if anything is left, draw it
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What do we mean by “nearest”?
How can we optimize this?
David Luebke
7/27/2016
Trivial Accepts
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Big optimization: trivial accept/rejects
How can we quickly determine whether a line
segment is entirely inside the viewport?
A: test both endpoints.
David Luebke
7/27/2016
Trivial Rejects
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How can we know a line is outside viewport?
A: if both endpoints on wrong side of same
edge, can trivially reject line
David Luebke
7/27/2016
Cohen-Sutherland Line Clipping
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Divide viewplane into regions defined by
viewport edges
Assign each region a 4-bit outcode:
David Luebke
1001
1000
1010
0001
0000
0010
0101
0100
0110
7/27/2016
Cohen-Sutherland Line Clipping
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How do we set the bits in the outcode?
How do you suppose we use them?
David Luebke
1001
1000
1010
0001
0000
0010
0101
0100
0110
7/27/2016
Cohen-Sutherland Line Clipping
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Set bits with simple tests
x > xmax
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y < ymin
etc.
Assign an outcode to each vertex of line
– If both outcodes = 0, trivial accept
– bitwise AND vertex outcodes together
– If result  0, trivial reject
David Luebke
7/27/2016
Cohen-Sutherland Line Clipping
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If line cannot be trivially accepted or rejected,
subdivide so that one or both segments can
be discarded
Pick an edge that the line crosses (how?)
Intersect line with edge (how?)
Discard portion on wrong side of edge and
assign outcode to new vertex
Apply trivial accept/reject tests; repeat if
necessary
– How did the programmer get stuck in the shower?
David Luebke
7/27/2016
Cohen-Sutherland Line Clipping
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Outcode tests and line-edge intersects are
quite fast (how fast?)
But some lines require multiple iterations
(see F&vD figure 3.40)
Fundamentally more efficient algorithms:
– Cyrus-Beck uses parametric lines
– Liang-Barsky optimizes this for upright volumes
David Luebke
7/27/2016
Coming Up…
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We know how to clip a single line segment
– How about a polygon in 2D?
– How about in 3-D?
David Luebke
7/27/2016