 Bounding volume hierarchy

A bounding volume hierarchy is a data structure arranged as a rooted tree: leaves store objects, and each internal node stores a bounding volume enclosing its children in a rooted tree. Consequently, every node encloses all objects in its subtree. A common choice is a binary tree of axis-aligned bounding boxes. The objects assigned to sibling subtrees can be different even when their bounding volumes overlap.^[ref]

Here is an exact two-dimensional example. Use the triangles $A,B$ from bounding volume, together with $C$ having vertices of a polygon $(6,0),(8,0),(6,2)$ and $D$ having vertices of a polygon $(6,3),(8,3),(8,5)$. Group $A,B$ under $L$, and $C,D$ under $R$. Their parent axis-aligned bounding boxes and the box at the root of a tree areThe diagram shows all seven nodes. Taking the coordinate-wise minimum and maximum of the two child boxes constructs each parent box; the resulting box contains the set union of the children, and can also contain empty space between them.

A bounding volume hierarchy traversal starts at the root of a tree and discards a subtree when its bounding volume misses the query. Accepted internal nodes expose their children; accepted leaves supply candidates for actual shape tests. Box2D's official Collision: Dynamic Tree describes axis-aligned bounding box region queries that return overlapping leaf boxes. A query box that bounds a different query shape can produce additional false candidates.

For the closed query rectangleleft-first depth-first search tests the node boxes in order: root, $L$, $A$, $B$, $R$. The first four overlap $Q$; $R$ does not, because its smallest $x$-coordinate is $6>9/5$. Pruning $R$ skips both $C$ and $D$. Five box tests leave two broad phase candidates. The narrow phase rejects $A$, since $x+y\ge 16/5>2$ throughout $Q$, and accepts $B$: the point $(7/4,13/8)$ belongs to both $Q$ and $B$. There are two detailed shape tests and one true hit.

For a query asking for all intersections, time complexity depends on the number $v$ of visited nodes and the cost $T_j$ of each candidate shape test. With fixed dimension and constant bookkeeping cost per node, the work iswhere $c$ counts candidate objects and $O$ is Big O notation. A balanced binary tree has small height of a binary tree, but a query can enter both children at every level. In this example a query containing $[0,8]\times [0,5]$ visits all seven nodes and tests all four triangles. More generally, a full binary tree with $n$ single-object leaves has $2n-1$ nodes, so such a query takes $O(n)$ work for constant-cost shape tests even when its height of a binary tree is $O(\log n)$. Listing $n$ hits itself requires $n$ outputs. Small height alone gives no unconditional logarithmic query bound.

For collision detection between two bounding volume hierarchies, traverse pairs of nodes: reject disjoint pairs of bounding volumes, expand a nonleaf member of an accepted pair, and apply the narrow phase at accepted leaf pairs. A single collection's self-query must exclude pairing an object with itself and avoid duplicate unordered pairs. If every object overlaps every other object, reporting all pairs already requires $n(n-1)/2$ results. Construction quality, empty space in boxes and the queries determine how much pruning is useful.^[ref]

For a short correctness outline, an object is enclosed by its leaf box, and each child box is enclosed by its parent box. Repeated subset relations carry this property to every ancestor. A true query intersection therefore prevents rejection at every ancestor of the object's leaf. Updating leaf boxes after motion also requires restoring all affected ancestor bounds before querying.