To test for this the ray tracer must compute and keep the product of the global and reflection coefficients as the rays are traced. Example: let Kr = 0.5 for a set of surfaces. Then from the first surface the maximum contribution is 0.5, for the reflection from the second: 0.5 × 0.5 = 0.25, the third: 0.25 × 0.5 = 0.125, the fourth: 0.125 × 0.5 = 0.0625, the fifth: 0.0625 × 0.5 = 0.03125, etc. In addition we might implement a distance attenuation factor such as 1/D2, which would also decrease the intensity contribution. For a transmitted ray we could do something similar but in that case the distance traveled through the object would cause even faster intensity decrease. As an example of this, Hall & Greenberg found that even for a very reflective scene, using this with a maximum depth of 15 resulted in an average ray tree depth of 1.7. Bounding volumes Enclosing groups of objects in sets of hierarchical bounding volumes decreases the amount of computations required for ray tracing. A cast ray is first tested for an intersection with the bounding volume, and then if there is an intersection, the volume is recursively divided until the ray hits the object. The best type of bounding volume will be determined by the shape of the underlying object or objects. For example, if the objects are long and thin, then a sphere will enclose mainly empty space compared to a box. Boxes are also easier to generate hierarchical bounding volumes. Note that using a hierarchical system like this (assuming it is done carefully) changes the intersection computational time from a linear dependence on the number of objects to something between linear and a logarithmic dependence. This is because, for a perfect case, each intersection test would divide the possibilities by two, and result in a binary tree type structure. Spatial subdivision methods, discussed below, try to achieve this. Kay & Kajiya give a list of desired properties for hierarchical bounding volumes: Subtrees should contain objects that are near each other and the further down the tree the closer should be the objects. The volume of each node should be minimal. The sum of the volumes of all bounding volumes should be minimal. Greater attention should be placed on the nodes near the root since pruning a branch near the root will remove more potential objects than