rli > r2i, the global maximum is rli, and there are two
equal global minima, namely r3i and r4i. The external and
internal cylinders of the subworkspace are then the
cylinders of radii of rli and r3i, respectively.
Figure 3.23 shows the relationship between r and 8bi in
a Cartesian coordinate system in this case. It is seen that
in the portion of the subworkspace where r2i < r < rli, for
a given position of point Ci, which implies for a given
value of r, there are two corresponding solutions of 9bi"
Thus this is the two-root region of the subworkspace. The
other portion of the subworkspace is the four-root region of
the subworkspace since four solutions of 9bi can be found in
that region for a given translation di (see Fig. 3.19) and a
given position of Ci. The cylinder of radius r = r2i inside
the subworkspace divides the subworkspace into a two-root
region and a four-root region, and the surface of the
cylinder of radius r2i itself is a three-root region. It
can also be seen that the external boundary (the surface of
the cylinder of radius r = r1i) is a one-root region, and
the internal boundary (the surface of the cylinder of radius
r = r3i) is a two-root region. Since the offset is zero and
the twist angle is not 90*, without the translation of Ai,
the locus of the positions of point Ci is the surface of a
flattened form of torus. The diametral section of this kind
of torus is shown in Fig. 3.24 and the root regions can also
be visualized easily from the figure.