C= 0
ai bi(l Cabi)
D= 4
SbiS2abi
E = 1
Solving the quartic equation Eq. (3.21), we can obtain
up to four real roots for X, and then of 8bi. Back
substituting the roots of Gbi into Eq. (3.8), up to four
limiting values of the radius r of the torus are obtained.
In the case where only two limiting values of r exist,
one is the maximum rmax, and the other is the minimum rmin
in the range of 0 9bi < 2n. Hence the subworkspace can
be expressed as
xi2 i2 < rmax (3.22)
and
xi2 + yi2 r2min (3.23)
In the case where four limiting values of r exist, we
need to compare the four values and find the maximum and
the minimum which are respectively the radii of the
external and internal cylinders of the boundaries of the
subworkspace.
Furthermore, the relationship between r and 8bi, in
the two cases, are shown schematically in Figs. 3.32 and
3.33 in Cartesian coordinate systems, respectively. In the
first case (Fig. 3.32) for a given value of r in the region