A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing

6 Analytical Geometry

parameter t) by t

b

±t

∗

, and the closest intersection is the smaller

value of t

b

± t

∗

.

The early termination is very effective, since in most applica-

tions the ray does not meet the given sphere in the majority of

cases. In other cases, only a binary test is required, for example,

if the sphere is used as a bounding volume, where the ray–sphere

intersection is just meant to test whether the ray should go farther

and intersect the object bounded by the sphere. Clearly, in such

situations, applying the analytical formula for ray–sphere inter-

section is overly expensive.

Analytical approach

As mentioned, not every problem can be approached ...