Clipping
Clipping and
Hidden Surfaces
•
CS-184: Computer Graphics
Prof. James O’Brien
Don’t want to draw stuff that is not in screen area
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Waste of time
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Bogus memory access
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Looks ugly
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Clipping removes unwanted stuff
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Can also clip for other reasons
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Cut away views
1
Basic Line Clipping
2
Cohen-Sutherland Line Clip
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•
Clip line against each edge of clip region in turn
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If both endpoints outside, discard line and stop
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If both endpoints in, continue to next edge (or finish)
–
If one in, one out, chop line at crossing pt and
continue
Works in both 2D and 3D for convex clipping
regions
3
Cohen-Sutherland Line Clip
1
1
2
1
Cohen-Sutherland Line Clip
2
3
3
4
4
2
1
4
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Trivial reject and accept (speed things up…)
2
3
3
4
4
Trivial Keep
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Trivial Reject
NOT Trivial Reject
6
1
Cohen-Sutherland Line Clip
Liang-Barsky Clipping
•
•
Out codes
1
0010
2
•
3
•
4
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Parametric clipping - view line in parametric
form and reason about the parameter values
More efficient, as not computing the coordinate
values at irrelevant vertices
Works for rectilinear clip regions in 2D or 3D
Clipping conditions on parameter: Line is inside
clip region for values of t such that (for 2D):
0101
xmin≤ x1 +t∆x ≤ xmax ∆x = x2 − x1
ymin≤ y1 +t∆y ≤ ymax ∆y = y2 − y1
7
Liang-Barsky Clipping
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Liang-Barsky Clipping
Infinite line intersects clip region edges at:
tk =
qk
pk
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p1 = −∆x q1 = x 1 − x min
where
p 2 = ∆x
q 2 = x max − x 1
p 3 = −∆y q 3 = y1 − y min
p 4 = ∆y
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When pk<0, as t increases line goes from outside to
inside - entering
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When pk>0, line goes from inside to outside – leaving
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When pk=0, line is parallel to an edge (clipping is easy)
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q 4 = y max − y1
If there is a segment of the line inside the clip region,
sequence of infinite line intersections must go: enter,
enter, leave, leave
Note: Left edge is 1, right edge is 2, top edge is 3, bottom is 4
9
Liang-Barsky Clipping
Enter
10
Liang-Barsky Clipping
Leave
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Compute entering t values, which are qk/p k for each p k<0
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Compute leaving t values, which are qk/pk for each pk>0
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Leave
Leave
Leave
•
•
Enter
Enter
•
Enter
Parameter value for small t end of line is:tsmall= max(0, entering
t’s)
parameter value for large t end of line is: tlarge=min(1, leaving t’s)
if tsmall<tlarge, there is a line segment - compute endpoints by
substituting t values
Improvement (and actual Liang-Barsky):
–
11
compute t’s for each edge in turn (some rejects occur earlier like this)
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2
Sutherland-Hodgman Polygon Clip
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Sutherland-Hodgman Polygon Clip
Clip the polygon against each edge of the clip
region in turn
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Clip polygon each time to line containing edge
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Only works for convex clip regions (Why?)
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To clip a polygon to a line/plane:
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Consider the polygon as a list of vertices
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One side of the line/plane is considered inside the clip region,
the other side is outside
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We are going to rewrite the polygon one vertex at a time – the
rewritten polygon will be the polygon clipped to the line/plane
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Check start vertex: if “inside”, emit it, otherwise ignore it
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Continue processing vertices as follows…
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Sutherland-Hodgman Polygon Clip
Inside
Outside
Inside
Outside
s
p
Output p
Inside
Outside
p
s
i
p
s
Output i
No output
Inside
p
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General Clipping (Weiler Algorithm)
Outside
i
s
Output i and p
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General Clipping (Weiler Algorithm)
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Inside-Outside Testing
Rearranging pointers
makes it possible to
enumerate all components
of the intersection
Changes to
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18
3
Inside/Outside in Screen Space
Finding Intersection Points
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Backface Culling
20
Z-Buffer ( a.k.a. Depth Buffer )
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Use additional buffer to store a Z value at each pixel
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When filling pixel write Z value to buffer also
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Don’t fill if new Z value is larger than old
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Quantization and aliasing problems sometimes occur
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Very commonly used
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Standard feature in hardware
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Interpolating Z values over polygons
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Z-Buffer and Transparency
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A-Buffer
Must render in order back to front
Front
Partially
transparent
3rd
Partially
transparent
1st
Opaque
2nd
Opaque
3rd
Opaque
1st
Opaque
2nd
Good
Talk about compositing later…
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Not Good
23
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Store list of polygons at each pixel (fragments)
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Draw opaque stuff first,only keep closest frag
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Second pass for transparent stuff
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Allows antialiasing…
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Not good for hardware implementation
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4
Scan Line Algorithm
Scan Line Algorithm
Assume polygons don’t intersect each other
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Advantages:
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As scanning keep list of active edges
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Decide who’s on top when crossing edges
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–
–
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Simple
Potentially fewer quantization errors (more bits available for
depth)
Don’t over-render (each pixel only drawn once)
Filter anti-aliasing can be made to work (have information
about all polygons at each pixel)
Disadvantages:
–
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Invisible polygons clog AEL, ET
Non-intersection criteria may be hard to meet
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Painters Algorithm
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26
BSP-Trees (Object Precision)
Sort based on depth and draw back to front
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Construct a binary space partition tree
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Tree splits 3D world with planes
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How dow we sort quickly?
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What about intersecting polygons?
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Depth-first traversal gives a rendering order (varient)
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The world is broken into convex cells
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Each cell is the intersection of all the half-spaces of splitting
planes on tree path to the cell
Also used to model the shape of objects, and in other
visibility algorithms
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BSP visibility in games does not necessarily refer to this
algorithm
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BSP-Tree Example
Building BSP-Trees
A
C
4
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A
-
3
-
Choose polygon (arbitrary)
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Split its cell using plane on which polygon lies
+
B
B
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–
C
•
+
-
+
2
4
1
•
1
3
Splitting planes could be chosen in other ways, but there
is no efficient optimal algorithm for building BSP trees
–
2
29
May have to chop polygons in two (Clipping!)
Continue until each cell contains only one polygon
fragment
Optimal means minimum number of polygon fragments in a
balanced tree
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5
Building Example
Building Example
5
• We will build a BSP
tree, in 2D, for a 3
room building
– Ignoring doors
• Splitting edge order
is shown
-
1
3a, 4a, 6
2
3
5
+
2, 3b, 4b, 5
3b
2
4b
4
1
1
4a
3a
– “Back” side of edge
is side with the
number
6
6
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Building Example
1
3a, 4a, 6
5b
+
2
-
Building Example
3a
+
4b, 5a
5a
3b
2
4a, 6
5b
+
-
+
4b
3b, 5b
1
4b, 5a
5a
2
+
3b
2
4b
3b
+
5b
1
4a
3a
1
3a
6
4a
6
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Building Example (Done)
1
3a
-
+
4a
6
5a
•
2
+
4b
+
BSP-Tree Rendering
5b
+
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3b
2
4b
3b
+
5a
+
5b
1
3a
Observation: Things on the opposite side of a splitting plane from
the viewpoint cannot obscure things on the same side as the
viewpoint
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Rendering algorithm is recursive descent of the BSP Tree
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At each node (for back to front rendering):
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Recurse down the side of the sub-tree that does not contain the viewpoint
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Draw the polygon in the splitting plane
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Recurse down the side of the tree containing the viewpoint
Test viewpoint against the split plane to decide which tree
4a
Paint over whatever has already been drawn
6
35
36
6
BSP-Tree Rendering Example
A
C
4
BSP-Tree Rendering
A
-
3
•
+
B
-
B
1
2
C
+
-
Advantages:
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One tree works for any viewing point
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Filter anti-aliasing and transparency work
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Can also render front to back, and avoid drawing back
polygons that cannot contribute to the view
Have back to front ordering for compositing
+
3
2
4
1
3rd
4th
1st
2nd
User two trees – an extra one that subdivides the window
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Disadvantages:
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Can be many small pieces of polygon
37
Cells and Portals
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Simple shapes
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Rooms in a building, for instance
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Doorways between rooms, windows, etc
In a world like this, can determine exactly which parts of
which rooms are visible
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•
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Define portals to be the transparent boundaries between
cells
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Cell and Portal Visibility
Assume the world can be broken into cells
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Start in the cell containing the viewer, with the
full viewing frustum
Render the walls of that room and its contents
Recursively clip the viewing frustum to each
portal out of the cell, and call the algorithm on the
cell beyond the portal
Then render visible rooms plus contents
An Exact Visibility Algorithm -- sometimes
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Cell-Portal Example
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Cell-Portal Example
View
View
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42
7
Cell-Portal Example
View
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Acknowledgments
•
Some of these slide contents were based on
original materials from
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Prof. David Forsyth of U.C. Berkeley
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Prof. Stephen Chenney of U.W. Madison
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