PENELOPE 2003 - OECD Nuclear Energy Agency

5.2. Quadric surfaces 159
Table 5.1: Reduced quadrics.
Reduced form Indices Quadric
z − 1 = 0 0 0 0 1 −1 plane
z 2 − 1 = 0 0 0 1 0 −1 pair of parallel planes
x 2 + y 2 + z 2 − 1 = 0 1 1 1 0 −1 sphere
x 2 + y 2 − 1 = 0 1 1 0 0 −1 cylinder
x 2 − y 2 − 1 = 0 1 −1 0 0 −1 hyperbolic cylinder
x 2 + y 2 − z 2 = 0 1 1 −1 0 0 cone
x 2 + y 2 − z 2 − 1 = 0 1 1 −1 0 −1 one sheet hyperboloid
x 2 + y 2 − z 2 + 1 = 0 1 1 −1 0 1 two sheet hyperboloid
x 2 + y 2 − z = 0 1 1 0 −1 0 paraboloid
x 2 − z = 0 1 0 0 −1 0 parabolic cylinder
x 2 − y 2 − z = 0 1 −1 0 −1 0 hyperbolic paraboloid
. . . and permutations of x, y and z that preserve the central symmetry with respect
to the z-axis.
For instance, this transforms the reduced sphere into an ellipsoid with semiaxes
equal to the scaling factors.
(ii) A rotation, R(ω, θ, φ), defined through the Euler angles OMEGA= ω, THETA= θ and
PHI= φ. Notice that the rotation R(ω, θ, φ) transforms a plane perpendicular to
the z-axis into a plane perpendicular to the direction with polar and azimuthal
angles THETA and PHI, respectively. The first Euler angle, ω has no effect when
the initial (scaled) quadric is symmetric about the z-axis.
(iii) A translation, defined by the components of the displacement vector t (X-SHIFT=
t x , Y-SHIFT= t y , Z-SHIFT= t z ).
A quadric is completely specified by giving the set of indices (I 1 , I 2 , I 3 , I 4 , I 5 ), the scale
factors (X-SCALE, Y-SCALE, Z-SCALE), the Euler angles (OMEGA, THETA, PHI) and the
displacement vector (X-SHIFT, Y-SHIFT, Z-SHIFT). Any quadric surface can be expressed
in this way. The implicit equation of the quadric is obtained as follows. We define the
matrix
⎛
1
A xx A 2 xy 1 A ⎞
2 xz
A =
1
⎜ A 1
2 xy A yy A 2 yz ⎟
(5.15)
⎝
⎠
1
A 2 xz 1 A 2 yz A zz
and write the generic quadric equation (5.12) in matrix form
r T A r + A T r + A 0 = 0, (5.16)