# Enveloping body

A three-dimensional body and the corresponding cuboid enveloping body (in white)

A casing body ( English bounding volume ) is in the computational geometry , a simple geometrical body , which encloses a complex three-dimensional object or a complex body.

## Applications and variants

Enveloping bodies are mainly used to accelerate algorithms in algorithmic geometry or computer graphics, for example in ray tracing . They are often structured hierarchically (bounding volumes enclose other bounding volumes) in order to further increase efficiency.

The following envelopes are common:

• Balls (Bounding Spheres). This type of enveloping body is particularly widespread in collision detection , since collisions with spheres can be calculated very easily.
• Cuboid or cube (bounding boxes). Cuboid enveloping bodies often describe objects more precisely than spheres and are therefore advantageous in some applications such as ray tracing. Ray tracing can be accelerated via bounding volume hierarchies (BVH). Arbitrarily oriented cuboids are also referred to as Oriented Bounding Boxes (OBB) , cuboids aligned to the axes as Axis-Aligned Bounding Boxes (AABB) . AABBs are usually defined by two points that indicate the position of the corners on either side of a cuboid diagonal. A two-dimensional bounding box is called a minimally surrounding rectangle .
• Called k-DOP or k-Discretely Oriented Polytopes . In contrast to OBBs, k-DOPs allow multiple restriction areas, which means that they can enclose objects better (more closely). These restriction surfaces must always be parallel to each other in pairs so that a k -DOP can also be viewed as an intersection of k slabs. The overlap test (cut test) of two k -DOPs can be carried out in time .${\ displaystyle O (k)}$