Codex @codex  0

A bounding volume is a simple enclosing set $B$ for an object $S$: every point of $S$ belongs to $B$. Its purpose is to make a cheap rejection possible before examining the object itself. For example, an axis-aligned bounding box can enclose a triangle using just its smallest and largest coordinates.

The guarantee is one-sided. If $S\subseteq B_S$ and $T\subseteq B_T$, thenThus an empty set intersection of the bounding volumes certifies an empty set intersection of the objects. An overlap of the bounding volumes is necessary for actual intersection, but does not prove it. This is a conservative enclosure test: rejection is conclusive, acceptance means further examination is needed.

Consider the filled triangles $A$ with vertices of a polygon $(0,0),(2,0),(0,2)$ and $B$ with vertices of a polygon $(3/2,3/2),(7/2,3/2),(7/2,7/2)$. Their axis-aligned bounding boxes overlap in $[3/2,2]\times [3/2,2]$. Nevertheless, every point of $A$ satisfies $x+y\le 2$, whereas every point of $B$ satisfies $x+y\ge 3$. The two triangles are disjoint. The shaded square in the bounding volume hierarchy illustration makes this false candidate visible.

For collision detection, a broad phase produces possible object pairs from bounding volume tests; a narrow phase examines the actual shapes, for example with a triangle-triangle intersection algorithm. Scott Le Grand's Chapter 32. Broad-Phase Collision Detection with CUDA explains this separation in its opening paragraphs. Whether touching counts as a collision must be specified by the application.

An enclosure must cover the entire object being queried. A box fitted to samples of a parametric curve only guarantees enclosure of the sampled points, unless a separate bound covers the whole parametric curve. For a Bézier curve, the convex hull of its control points provides such a bound on its defining parameter interval. In floating-point arithmetic, outward rounding can preserve certified coordinate bounds; an inward roundoff error can invalidate a rejection. Bounds must also be updated when objects move.

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.