1. Maths & Technologies for Games Spatial Partitioning 2
CO3303
Week 9

2. Contents Introduction Grid-based Partitions Quad-Trees / Oct-Trees
Binary Space Partitioning

3. Introduction / Recap
Dividing a game world into partitions has many uses
Have looked at PVS & portal systems
To cull unseen partitions
Such visibility techniques also useful for AI entities
Here we introduce a range of tree-based partitions
BSP / Quad / Oct trees

4. Grids as Spatial Partitions
One of the most intuitive spatial partitions is a grid
Can collect local entities for visibility culling, AI etc.
Can be used to map terrain:
Height maps, influence maps…
Can be extended to 3D
Or 2.5D – 2D grid with heights
Disadvantages:
May have many empty nodes, wasting memory & reducing cache efficiency
Choice of partition size tricky – too small gives many empty nodes, too large and culling etc. is ineffective

5. Mapping a Grid to the World
A grid is an integer indexed structure (e.g. an array)
For a rectangle of world space
Need to map between world-space coords & grid indexes
(GridX,GridY)  (WorldX,WorldY)
Conversions for X dimension are (Y dimension similar):
GridX = (int)(GridWidth * (WorldX – MinX) / (MaxX – MinX))
WorldX = MinX + (float)GridX * (MaxX – MinX) / GridWidth
Second formula will give bottom-left of grid square
For centre of grid square add 0.5 to GridX

6. Quadtrees / Octrees
Quadtrees / Octrees are hierarchical partition systems
Use a tree structure to represent an area/volume of space
Formed by recursive subdivision
Use specific division scheme
Unlike portals where division into partitions was arbitrary
Quadtrees are in 2D, octrees in 3D
Otherwise the same
Some similarities with grid system

7. Creating a Quadtree Root node is entire space
Usually square
Divide into four equal quadrants
These are the children nodes
Repeat division with each quadrant
Until some condition is met
Maximum depth
An empty node
Only one entity contained
Etc…application dependent

8. Location in a Quadtree Easy to find which node point is in:
Start at root node
Find the quadrant the point is in
Just two comparisons (X and Y)
Repeat process on this quadrant
Can be optimised:
Have world space as square
Power of 2 in size (e.g. 2048)
Centred on origin
Can use bitwise integer math
[See Rabin]

9. Quadtrees for Visibility Culling
Use for frustum culling:
Traverse quadtree from root
Cull entire nodes outside frustum
Along with all entities and all child nodes in them
Reject many entities at once
Nodes are AABB
Viewing frustum is 6 planes
To test if a node is visible:
Six AABB <-> plane tests
Saw this idea in initial lecture on partitions. Several approaches, see:

10. Quadtree Problem Problem: Worst case: entity overlaps origin
Entities aren’t points
May overlap a node boundary
In this case the entity needs to be in a larger parent node
Worst case: entity overlaps origin
Doesn’t fit in any node except root
Even a tiny entity
Will never be culled
Hot-spots like this al the way around the boundaries of larger nodes

11. Loose Quadtrees Solution: Loose Quadtrees Few changes to algorithms
Have nodes overlap
Entities will then fit in original node area
Potentially with their bounding sphere overlapping adjacent nodes
Few changes to algorithms
Increase node size:
When inserting entities
When doing frustum culling
Removes hot-spot problem
At the expense of larger nodes at the same level

12. Quadtrees for Collision Detection
We saw intersection of viewing frustum with quadtree
Fairly easy to find intersection of other primitives
Spheres, cuboids, rays, capsules
Find the set of nodes intersected
And hence the set of entities that are candidates for collision
Basis for collision detection / ray casting / physics systems
Can help if we add adjacency information to the tree

13. Binary Space Partitioning
A Binary Space Partition (BSP) is also a hierarchical division of space
Another tree structure
This one represents all space
Quadtrees only cover root node
Partitions separated by:
Lines in 2D
Planes in 3D
Recursively divide each partition into two smaller ones with an arbitrary line / plane
Creates a binary tree

14. Creating a BSP Repeatedly divide space in two Stop when:
Use best line/plane to balance tree
Stop when:
Maximum of X elements in each partition (X = 1 here)
Partitions are small enough
Tree reaches certain depth
Choice depends on application

15. Locating a Point in a BSP
Given a point, easy to find which partition it is in:
Start at root of tree
Find which side of the dividing line/plane the point is on
Follow appropriate edge
Repeat until find leaf partition
Example here:
Check P against X
Not in A, follow other edge
Check P against Y
Not in B, follow other edge, etc.

16. Reminder: which Side of a Plane?
A plane is defined by:
A vertex on the plane P
A vector / normal N at right angles to the plane
To find which side of the plane a vertex A is on:
d = N • PA = |N| |PA| cos α
A will be on the same side pointed at by the vector N if:
N • PA is a simple operation
Don’t need to calculate α

17. BSP for Solid/Hollow Spaces
Can use the polygons in the scene as the division planes
Choose a polygon as a plane
All other polygons placed on one or other side of plane
Polygons crossing the planes are split
BSP splits space into hollow / solid volumes
All polygons / entities placed in hollow ones

18. BSP / Brush Modelling
This “traditional” style of BSP used for FPS games
First appeared in Doom (original version)
Often used in conjunction with a PVS
Can also be used to render partitions in a strict back to front order
Lends itself to a unique form of 3D modelling called brush modelling
Start with a entirely solid world
Cut out primitives (boxes, cylinders etc.)
Entities placed in hollowed out areas
Like "digging out" the level

19. BSP Pros / Cons BSP trees are a well established technique
Real-world apps tend to use refinements of the approach
Used for rendering / collision / ray-tracing
Can be generated automatically
Fully classify space
Easy to find which partition a point is in
Need good algorithm to choose dividing planes
Try to balance tree, examples were unbalanced
Hollow/solid BSP generates extra polygons due to splitting