Here, we determine efficient methods for tracing particles.
In surface-to-surface transport, we trace particles from their surface of
origin (``birth'') to their eventual absorption (``death'') through possibly
many intermediate reflections.
Mathematically, this reduces to tracing particles from their point of emission,
** **,
to their point of intersection with a surface,
** **.
Given that a particle is travelling from emission (or origination) point
** **
in direction ** E** (** E** is determined from the emission or
reflection properties to be covered in the next section), the equation of a
line in 3-space is:

and the equation describing surface **L** is:

Solving equations (1) and (2) simultaneously:

and

where:

Equations (3) and (4) yield a point that is at the intersection of the two infinite lines defined by the equations of the surface and the particle trajectory. Except where these lines are parallel (rarely, if ever, encountered in practice), there is one and only one intersection. Mathematically stated, if the denominator of equations (3) and (4) is zero, there is no intersection. Although it is possible to implement a check for parallel lines, one can usually avoid so doing (especially if your computer does hardware recovery from overflow as is common on many current RISC computers).

(See exercise 7.)