OBBs

An oriented bounding box (OBB) is a bounding volume that can be rotated with respect to a fixed world coordinate system. Unlike an axis-aligned bounding box (AABB), an OBB is defined by its center, three mutually orthogonal local axes, and three half-extents along those axes. In 3D, this yields a box rotated arbitrarily in space; in 2D, two axes and two extents define a rotated rectangle. OBBs are commonly used in computer graphics, collision detection, physics engines, and visibility tests because they can tightly fit objects with arbitrary orientation compared to AABBs.

With a 3D OBB, the center is c, the axes are unit vectors u0, u1, u2 forming an orthonormal basis, and the half-extents are e0, e1, e2. The orientation can be stored as a rotation matrix whose columns are the axes, or as a quaternion. An OBB can be transformed from local space to world space by rotating with the matrix and translating by c; the local coordinates (x, y, z) map to world coordinates as x*u0 + y*u1 + z*u2 + c.

Collision tests between OBBs commonly use the separating axis theorem (SAT). For two 3D OBBs there are 15 candidate axes: the three axes of each box and the nine cross products. Each axis is used to project the boxes; if a separating gap is found on any axis, the boxes do not intersect. To improve robustness, a small epsilon is often added to the projection terms. Tests for OBB–AABB or OBB–sphere are special cases with fewer axes.

OBBs can be constructed by fitting to a set of points (e.g., via principal component analysis or rotating calipers) or by aligning with an object's orientation. They are used in broad-phase collision detection, culling, and bounding-volume hierarchies, offering a tighter fit than AABBs while keeping computations efficient.