# lec39 - BSP Trees Binary space partitioning trees Used to...

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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 a 1 x 1 + a 2 x 2 + … a n x n + a n+1 = 0 ax + by + c = 0 (2D) ax+by+cz+d = 0 (3D)
Space Partitioning n-dimensional space +ve half space ax + 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 Object z In 2D, ph is the line ax + by + c = 0. Compute ax + by + c for all vertices of z . If all values are = 0; z is coincident to ph . If all values are <= 0; z is left of ph . If all values are >= 0; z is right of ph . Otherwise, z spans ph and is to be split by finding intersection points with ph .
2D a b c d g e f h Equation of ph is x – 6 = 0 6

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2D a b c d g e f h Equation of ph is y –x – 2 = 0 2
3D x z y Equation of ph is z – 2 = 0 General: ax + by + cz + d = 0

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Space Partitioning n-dimensional space -ve +ve -ve +ve coincident list
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
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
Objects in 2D a b c d g e f h a-b e-f c-d g-h

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Objects in 2D a b c d g e f h a bc d e f g h
Collision Detection a b c d g e f h a bc d e f g h

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Visibility Ordering a b c d g e f h a bc d e f g h
BSP Tree Construction Select partitioning hyperplanes. Partition objects. Repeat on partitions.

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Partitioning Hyperplane Selection Face of an object.
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• Spring '10
• sartajsahani
• 3D computer graphics, Collision Detection, Geometric algorithms, Binary space partitioning

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