Furthermore, in the case when ai = biS2abi, the roots
of ebi of Eq. (3.12) are coincident and equal, i.e., i =
Obi = 8- i = n. Thus, r3i = r4i = r2i, and their value is
the minimum of r, and is the radius of the internal cylinder
of the subworkspace.
The results derived are illustrated in Figs. 3.25 -
3.31, where the projections of the configurations of the
subchain are presented.
Figures 3.25 and 3.26 are shown for the case of abi = 0
or n, i.e., Sabi = 0. The projection of the circle
generated by point Ci on the AixiYi plane is also a circle.
The radii of the external and internal cylinders of the
boundary are the maximum and minimum radii r of the torus.
It is also shown that the subchain can take two branches
for point Ci to reach the same position in the
subworkspace, illustrating a two-root region.
Figures 3.27 and 3.28 are shown as the case of abi 0
or n and ai > biS2abi. The results are the same as in the
case shown in Figs. 3.25 and 3.26, except that the
projection of the circle of point Ci on the AixiYi plane is
an ellipse now.
Figures 3.29 and 3.30 are shown as the case of abi 5 0
or n and ai < biS2abi. The internal boundary is the
cylinder of radius r3i. The circle of radius r3i on the
plane oixiYi is tangent to the ellipse at two points showing
the values of r3i and r4i (at the points of tangency), which
are both the equal limiting values of the radius r of the