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Bounding volume 25
axis-aligned bounding box, made from 6 axis-aligned planes, and the beveled bounding box, made from 10 (if
beveled only on vertical edges, say) 18 (if beveled on all edges), or 26 planes (if beveled on all edges and corners). A
DOP constructed from k planes is called a k-DOP; the actual number of faces can be less than k, since some can
become degenerate, shrunk to an edge or a vertex.
A convex hull is the smallest convex volume containing the object. If the object is the union of a finite set of points,
its convex hull is a polytope.
Basic intersection checks
For some types of bounding volume (OBB and convex polyhedra), an effective check is that of the separating axis
theorem. The idea here is that, if there exists an axis by which the objects do not overlap, then the objects do not
intersect. Usually the axes checked are those of the basic axes for the volumes (the unit axes in the case of an AABB,
or the 3 base axes from each OBB in the case of OBBs). Often, this is followed by also checking the cross-products
of the previous axes (one axis from each object).
In the case of an AABB, this tests becomes a simple set of overlap tests in terms of the unit axes. For an AABB
defined by M,N against one defined by O,P they do not intersect if (M x >P x ) or (O x >N x ) or (M y >P y ) or (O y >N y ) or
(M z >P z ) or (O z >N z ).
An AABB can also be projected along an axis, for example, if it has edges of length L and is centered at C, and is
being projected along the axis N:
, and or , and
where m and n are the minimum and maximum extents.
An OBB is similar in this respect, but is slightly more complicated. For an OBB with L and C as above, and with I, J,
and K as the OBB's base axes, then:
For the ranges m,n and o,p it can be said that they do not intersect if m>p or o>n. Thus, by projecting the ranges of 2
OBBs along the I, J, and K axes of each OBB, and checking for non-intersection, it is possible to detect
non-intersection. By additionally checking along the cross products of these axes (I 0 ×I 1 , I 0 ×J 1 , ...) one can be more
certain that intersection is impossible.
This concept of determining non-intersection via use of axis projection also extends to convex polyhedra, however
with the normals of each polyhedral face being used instead of the base axes, and with the extents being based on the
minimum and maximum dot products of each vertex against the axes. Note that this description assumes the checks
are being done in world space.
References
[1] POV-Ray Documentation (http:/ / www. povray. org/ documentation/ view/ 3. 6. 1/ 323/ )
External links
• Illustration of several DOPs for the same model, from epicgames.com (http:/ / udn. epicgames. com/ Two/ rsrc/
Two/ CollisionTutorial/ kdop_sizes. jpg)