3D graphics eBook - Course Materials Repository

Bounding volume 24
bounding boxes become more sophisticated.
A bounding sphere is a sphere containing the object. In 2-D graphics, this is a circle. Bounding spheres are
represented by centre and radius. They are very quick to test for collision with each other: two spheres intersect
when the distance between their centres does not exceed the sum of their radii. This makes bounding spheres
appropriate for objects that can move in any number of dimensions.
A bounding ellipsoid is an ellipsoid containing the object. Ellipsoids usually provide tighter fitting than a sphere.
Intersections with ellipsoids are done by scaling the other object along the principal axes of the ellipsoid by an
amount equal to the multiplicative inverse of the radii of the ellipsoid, thus reducing the problem to intersecting the
scaled object with a unit sphere. Care should be taken to avoid problems if the applied scaling introduces skew.
Skew can make the usage of ellipsoids impractical in certain cases, for example collision between two arbitrary
ellipsoids.
A bounding cylinder is a cylinder containing the object. In most applications the axis of the cylinder is aligned with
the vertical direction of the scene. Cylinders are appropriate for 3-D objects that can only rotate about a vertical axis
but not about other axes, and are otherwise constrained to move by translation only. Two vertical-axis-aligned
cylinders intersect when, simultaneously, their projections on the vertical axis intersect – which are two line
segments – as well their projections on the horizontal plane – two circular disks. Both are easy to test. In video
games, bounding cylinders are often used as bounding volumes for people standing upright.
A bounding capsule is a swept sphere (i.e. the volume that a sphere takes as it moves along a straight line segment)
containing the object. Capsules can be represented by the radius of the swept sphere and the segment that the sphere
is swept across). It has traits similar to a cylinder, but is easier to use, because the intersection test is simpler. A
capsule and another object intersect if the distance between the capsule's defining segment and some feature of the
other object is smaller than the capsule's radius. For example, two capsules intersect if the distance between the
capsules' segments is smaller than the sum of their radii. This holds for arbitrarily rotated capsules, which is why
they're more appealing than cylinders in practice.
A bounding box is a cuboid, or in 2-D a rectangle, containing the object. In dynamical simulation, bounding boxes
are preferred to other shapes of bounding volume such as bounding spheres or cylinders for objects that are roughly
cuboid in shape when the intersection test needs to be fairly accurate. The benefit is obvious, for example, for objects
that rest upon other, such as a car resting on the ground: a bounding sphere would show the car as possibly
intersecting with the ground, which then would need to be rejected by a more expensive test of the actual model of
the car; a bounding box immediately shows the car as not intersecting with the ground, saving the more expensive
test.
In many applications the bounding box is aligned with the axes of the co-ordinate system, and it is then known as an
axis-aligned bounding box (AABB). To distinguish the general case from an AABB, an arbitrary bounding box is
sometimes called an oriented bounding box (OBB). AABBs are much simpler to test for intersection than OBBs,
but have the disadvantage that when the model is rotated they cannot be simply rotated with it, but need to be
recomputed.
A bounding slab is related to the AABB and used to speed up ray tracing [1]
A minimum bounding rectangle or MBR – the least AABB in 2-D – is frequently used in the description of
geographic (or "geospatial") data items, serving as a simplified proxy for a dataset's spatial extent (see geospatial
metadata) for the purpose of data search (including spatial queries as applicable) and display. It is also a basic
component of the R-tree method of spatial indexing.
A discrete oriented polytope (DOP) generalizes the AABB. A DOP is a convex polytope containing the object (in
2-D a polygon; in 3-D a polyhedron), constructed by taking a number of suitably oriented planes at infinity and
moving them until they collide with the object. The DOP is then the convex polytope resulting from intersection of
the half-spaces bounded by the planes. Popular choices for constructing DOPs in 3-D graphics include the