PENELOPE 2003 - OECD Nuclear Energy Agency

160 Chapter 5. Constructive quadric geometry
where r and A ≡ (A x , A y , A z ) are considered here as one-column matrices. Notice that
the matrix A is symmetric (A T = A). Expressing the scaled quadric (5.14) in the form
(5.16), the equation for the rotated and shifted quadric is [see eq. (5.11)]
(r − t) T RAR T (r − t) + (RA) T (r − t) + A 0 = 0, (5.17)
which can be written in the generic form (5.16)
with
r T A ′ r + A ′T r + A ′ 0 = 0 (5.18)
A ′ = RAR T , A ′ = RA − 2A ′ t, A ′ 0 = A 0 + t T (A ′ t − RA). (5.19)
From these relations, the parameters of the implicit equation (5.12) are easily obtained.
A quadric surface F (x, y, z) = 0 divides the space into two exclusive regions that are
identified by the sign of F (x, y, z), the surface side pointer. A point with coordinates
(x 0 , y 0 , z 0 ) is said to be inside the surface if F (x 0 , y 0 , z 0 ) ≤ 0 (side pointer = −1), and
outside it if F (x 0 , y 0 , z 0 ) > 0 (side pointer = +1).
5.3 Constructive quadric geometry
A body is defined as a space volume limited by quadric surfaces and filled with a
homogeneous material. To specify a body we have to define its limiting quadric surfaces
F (r) = 0, with corresponding side pointers (+1 or −1), and its composition (i.e. the
integer label used by penelope to identify the material). It is considered that bodies
are defined in “ascending”, exclusive order so that previously defined bodies effectively
delimit the new ones. This is convenient e.g. to describe bodies with inclusions. The
work of the geometry routines is much easier when bodies are completely defined by their
limiting surfaces, but this is not always possible or convenient for the user. The example
in section 5.7 describes an arrow inside a sphere (fig. 5.2); the arrow is defined first so
that it limits the volume filled by the material inside the sphere. It is impossible to define
the hollow sphere (as a single body) by means of only its limiting quadric surfaces. It
is clear that, by defining a conveniently large number of surfaces and bodies, we can
describe any quadric geometry.
The subroutine package pengeom contains a subroutine, named LOCATE, that “locates”
a point r, i.e. determines the body that contains it, if any. The obvious method
is to compute the side pointers [i.e. the sign of F (r)] for all surfaces and, then, explore
the bodies in ascending order looking for the first one that fits the given side pointers.
This brute force procedure was used in older versions of pengeom; it has the advantage
of being robust (and easy to program) but becomes too slow for complex systems. A
second subroutine, named STEP, “moves” the particle from a given position r 0 within
a body B a certain distance s in a given direction ˆd. STEP also checks if the particle
leaves the active medium and, when this occurs, stops the particle just after entering