A one-dimensional variety

The most interesting and complicated situations occur when there is a continuum of solutions. For example, if one of the constraints is removed, then a one-dimensional set of solutions can be obtained. Suppose only one variable is constrained for the example in Figure 4.22. Intuitively, this should yield a one-dimensional variety. Set the $x$ coordinate to 0, which yields

$$a_1 a_2 - b_1 b_2 + a_1 = 0 ,$$ (4.65)

and allow any possible value for $y$. As shown in Figure 4.23a, the point $p$ must follow the $y$-axis. (This is equivalent to a three-bar linkage that can be constructed by making a third joint that is prismatic and forced to stay along the $y$-axis.) Figure 4.23b shows the resulting variety $V(a_1 a_2 - b_1 b_2 + a_1)$ but plotted in $\theta_1-\theta_2$ coordinates to reduce the dimension from $4$ to $2$ for visualization purposes. To correctly interpret the figures in Figure 4.23, recall that the topology is ${\mathbb{S}}^1 \times {\mathbb{S}}^1$, which means that the top and bottom are identified, and also the sides are identified. The center of Figure 4.23b, which corresponds to $(\theta_1,\theta_2) = (\pi,\pi)$, prevents the variety from being a manifold. The resulting space is actually homeomorphic to two circles that touch at a point. Thus, even with such a simple example, the nice manifold structure may disappear. Observe that at $(\pi,\pi)$ the links are completely overlapped, and the point $p$ of ${\cal A}_2$ is placed at $(0,0)$ in ${\cal W}$. The horizontal line in Figure 4.23b corresponds to keeping the two links overlapping and swinging them around together by varying $\theta_1$. The diagonal lines correspond to moving along configurations such as the one shown in Figure 4.23a. Note that the links and the $y$-axis always form an isosceles triangle, which can be used to show that the solution set is any pair of angles, $\theta_1$, $\theta_2$ for which $\theta_2 = \pi - \theta_1$. This is the reason why the diagonal curves in Figure 4.23b are linear. Figures 4.23c and 4.23d show the varieties for the constraints

$$a_1 a_2 - b_1 b_2 + a_1 = \begin{matrix}\frac{1}{8} \end{matrix},$$ (4.66)

and

$$a_1 a_2 - b_1 b_2 + a_1 = 1 ,$$ (4.67)

respectively. In these cases, the point $(0,1)$ in ${\cal A}_2$ must follow the $x=1/8$ and $x=1$ axes, respectively. The varieties are manifolds, which are homeomorphic to ${\mathbb{S}}^1$. The sequence from Figure 4.23b to 4.23d can be imagined as part of an animation in which the variety shrinks into a small circle. Eventually, it shrinks to a point for the case $a_1 a_2 - b_1 b_2 + a_1 = 2$, because the only solution is when $\theta_1 = \theta_2 = 0$. Beyond this, the variety is the empty set because there are no solutions. Thus, by allowing one constraint to vary, four different topologies are obtained: 1) two circles joined at a point, 2) a circle, 3) a point, and 4) the empty set.

Figure 4.23: A single constraint was added to the point $p$ on ${\cal A}_2$, as shown in (a). The curves in (b), (c), and (d) depict the variety for the cases of $f_1=0$, $f_1=1/8$, and $f_1=1$, respectively.