BSP trees

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BSP Trees

Binary space partitioning trees.

Used to store a collection of objects in n-

dimensional space.

Tree recursively divides n-dimensional

space using (n-1)-dimensional hyperplanes.

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Space Partitioning

n-dimensional space

splitting hyperplane

(n-1)-dimensional

a1x1 + a2x2+ anxn + an+1 = 0

ax + by + c = 0 (2D)

ax+by+cz+d = 0 (3D)

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Space Partitioning

n-dimensional space +ve half spaceax + by + c > 0

ax+by+cz+d > 0

-ve half space

ax + by + c < 0

ax+by+cz+d < 0

coincident

ax + by + c = 0

ax+by+cz+d = 0

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Classifying Objectz

In 2D,ph is the line ax + by + c = 0.

Compute ax + by + c for all vertices ofz.

If all values are = 0;zis coincident to ph.

If all values are = 0;zis right ofph.

Otherwise,zspansph and is to be split byfinding intersection points withph.

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2D

a

b

c

d

g

e

f

h

Equation ofph is

x

6 = 06

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2D

a

b

c

d

g

e

f

h

Equation ofph is

y

x

2 = 0

2

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3D

x

z

y

Equation ofph is

z2 = 0

General: ax + by + cz+ d= 0

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Space Partitioning

n-dimensional space

-ve +ve

-ve

+ve

coincident list

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Objects in 2D

a

b

c

d

g

e

f

h

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Objects in 2D

a

b

c

d

g

e

f

h

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Objects in 2D

a

b

c

d

g

e

f

h

a-d e-h

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Objects in 2D

a

b

c

d

g

e

f

h

a-d e-h

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Objects in 2D

a

b

c

d

g

e

f

h

a-b e-fc-d g-h

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Objects in 2D

a

b

c

d

g

e

f

h

a bc de f

g h

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Collision Detection

a

b

c

d

g

e

f

h

a bc de f

g h

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Visibility Ordering

a

b

c

d

g

e

f

h

a bc de f

g h

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BSP Tree Construction

Select partitioning hyperplanes.

Partition objects.

Repeat on partitions.

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Partitioning Hyperplane Selection

Face of an object.

a

b

c

d

g

e

f

h

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Face of an object.

a

b

c

d

g

e

f

h

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Autopartition

Only object

faces are used assplitting

hyperplanes

a

b

c

d

g

e

f

h

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Axis-aligned orthogonalhyperplanes

a

b

c

d

g

e

f

h

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Axis-aligned orthogonalhyperplanes

a

b

c

d

g

e

f

h

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Balance # objects (pieces) oneach side of hyperplane

Minimize increase in numberof objects/pieces.

a

b

c

d

g

e

f

h

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3D Example

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3D Example

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3D Example

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3D Example

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Another 3D Example

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Another 3D Example

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BSP Tree of an Object

Each leaf represents a region that is either wholly inside oroutside the object.

Object surface is considered inside object.

Surface planes are used as partitioning planes.

Orient partitioning hyperplanes so that interior is to left.

a

a

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a

a

b

b

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a

a

b

b

c

c

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a

a

b

b

c

c

dd

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a

a

b

b

c

d

ed

e

c

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a

a

b

b

c

df

ed

c

fe

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With this orientation, left leaves are interior and

right leaves are exterior.

a

a

b

b

c

df

ed

ef

c

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BSP Tree Construction

Node structure:

ph = equation of partitioning hyperplane

cList= list of objects coincident withph

leftChild

rightChild

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BSP Tree ConstructionBSPtree(O)

// O is object set

ifO is empty, return null;

Create a new nodeN;

N.ph = partitioning hyperplane forO;

lList = rList = N.cList = null;

for each objectzin O do

ifzis coincident toph or|O| = 1, addztoN.cList;

ifzis left ofph, addzto lList;

ifzis right ofph, addzto rList;ifzspansph, splitzand add pieces to lListand rList;

N.leftChild = BSPTree(lList);

N.rightChild = BSPTree(rList);

return N;

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Basic Draw Back to Front

draw(N)

ifeye left ofN.ph

{draw(N.rightChild); draw N.cList; draw(N.leftChild)};

else ifeye right ofN.ph

{draw(N.leftChild); draw N.cList; draw(N.rightChild)};

else // eye coincident toN.ph{draw(N.leftChild); draw(N.rightChild)};

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Basic Draw Front to Back

draw(N)

ifeye left ofN.ph

{draw(N.leftChild); draw N.cList; draw(N.rightChild)};

else ifeye right ofN.ph

{draw(N.rightChild); draw N.cList; draw(N.leftChild)};

else // eye coincident toN.ph{draw(N.rightChild); draw(N.leftChild)};

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Randomization

Autopartition.

Splitting hyperplane is randomly selected to beone of the object faces.

Lines in 2D; planes in 3D.

a

b

g

e

h

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Randomization2D Analysis Start with n (nonintersecting) line segments.

Total number of line segments in autopartition bspis expected to be

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So, number of nodes in bsp is O(n log n).

Construction time at each node is O(n) as at eachnode O(n) segments need to be partitioned.

Time is O(n2log n) per construction round.

2 rounds expected.

Expected complexity is O(n2log n).

Randomization2D Analysis

8

3

8

4

6

n = 29

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Randomization3D Analysis

Start with n (nonintersecting) triangles. Total number of triangles in autopartition bsp is

O(n2).

There exist n-triangle examples for which every

autopartition has W(n2) triangles.

n = 24

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Free Partitions

One that does not split an object. Do a free partition whenever possible;

otherwise, randomly select a segment/face

as partitioning hyperplane.

8

3

8

4

6

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Free Partitions

Using an object face, that crosses a BSPregion, results in a free partition of that BSP

region.

Portion of black segment

bounded by red

(previous) splitting lines

may be used to partition

blue region. No segments

in blue region can be split

as line segments are non-

intersecting.

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Free Partitions

Using an object face, that crosses a BSPregion, results in a free partition of that BSP

region.

When green segment is

used to partition blue

region, segments in blue

region may be split.

BSP trees

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