CS-184: Computer Graphics
Lecture #7: BSP and AABB Trees
Prof. James O’Brien
University of California, Berkeley
V2009-F-07-1.0
BSP-Trees
• Binary
Space Partition Trees
Split space along planes
• Allows fast queries of some spatial relations
•
• Simple
construction algorithm
Select a plane as sub-tree root
• Everything on one side to one child
• Everything on the other side to other child
• Use random polygon for splitting plane
•
2
BSP-Trees
a,b,c,d,e,f,g
e
a
d
f
b
g
c
3
BSP-Trees
a
e
d
a
b,c1,d
f
c2,e,f,g
c2
b
g
c1
4
BSP-Trees
a
e
d
a
c2,e,f,g
b
f
d
c1
c2
b
g
c1
5
BSP-Trees
a
d
e1
a
e2
c2
b
f
c1
d
e1 ,f
e2 ,g
c2
b
g
c1
6
BSP-Trees
a
d
e1
a
e2
c2
b
f
e1
d
c1
e2 ,g
c2
b
f
g
c1
7
BSP-Trees
a
d
e1
a
e2
c2
b
f
c1
d
e1
e2
f
g
c2
b
g
c1
8
BSP-Trees
a
+ a
d
b
+
e1
+
e2
+
f
+ c
2
c2
b
+
-
+
-
c1
d
e1
e2
+
+
f
g
g
c1
8
BSP-Trees
• Visibility Traversal
Variation of in-order-traversal
• Child one
• Sub-tree root
• Child two
• Select “child one” based on location of viewpoint
• Child one on same side of sub-tree root as viewpoint
•
9
BSP-Trees
a
d
e1
a
e2
c2
b
f
d
c1
e1
e2
f
g
c2
b
g
c1
c1:b:d:a:f:e1:c2:g:e2
10
BSP-Trees
a
d
e1
a
e2
c2
b
f
c1
d
e1
e2
f
g
c2
b
g
c1
g:e2:c2:f:e1:a:c1:b:d
11
Bounding Shapes
• Bounding
shape completely encloses associated object
• Rays
cannot hit object w/o intersecting bounding shape
• Two
objects cannot collide if shapes don’t overlap
• Simplicity
-vs- tightness
12
Axis-Aligned Bounding Boxes
Max x,y,z
Min x,y,z
• Axis-aligned
bounding box defined by min and max x,y,z
13
Axis-Aligned Bounding Boxes
Transform box
Not axis-aligned
Min/max of new points
Linear cost to compute
14
Axis-Aligned Bounding Boxes
Min/max of
transformed BB points
Constant time
Adds slop
Cumulative slop if multiple transforms occur sequentially
Why would we do this?
15
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
Group node
Geometry node
Transformation stored at all nodes
L.Leg
R.Leg
Leg
Shape
L.Foot
RFoot
16
AABB Trees
One of many variations
Robot
Geometry node
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
Head
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
Scale
2x
Geometry
B.Box
Transformation
XF B.Box
17
AABB Trees
One of many variations
Robot
Group node
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
R.Leg
Leg
Shape
L.Foot
RFoot
Head
Rot -25
Children
Union B.Box Transformation
XF B.Box XF B.Box
18
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
Local Bounding Boxes
19
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
Transformed Bounding Boxes
20
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
21
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
22
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
23
AABB Trees
One of many variations
Robot
Head
Shape
Torso
Head
L.Arm
Mouth
L.Eye
R.Eye
R.Arm
Leg
Shape
L.Leg
L.Foot
R.Leg
Leg
Shape
RFoot
24
Ray Test Against Bound Tree
• RayHitSubTree(&ray,node)
•
If RayHitsBB(ray,node.xfBB)
• ixfRay = Inverse(node.xf)*ray
• If RayHitsBB(ixfRay,node.BB)
• If node is group
• Foreach child in node.children
• RayHitSubTree(ixfRay,child)
• else // node not group
• RayHitGeometry(ixfRay,node.geom)
• ray.collisionInfo.update(ixfRay)
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