A numerical scheme is presented for computing the intersection of two \(m\)-dimensional algebraic subsets \(Y,Z\) of a \(2m\)-dimensional quadric hypersurface \(X\) in a complex projective space. The scheme is obtained by defining an appropriate action of the nonzero complex numbers \({\mathbb C}^\ast\) on \(X\), which then leads to two Bialynicki-Birula style cell decompositions: the positive cells, which consist of points \(x\in X\) that share the same limit point \(\lim_{t\to 0}tx\), and the negative cells, which consist of points that share the same limit \(\lim_{t\to\infty}tx\). The authors then use the \({\mathbb C}^\ast\) action to construct a homotopy \(H(t)\) on \(Y\times Z\) whose zero set \(S_0\) at \(t=0\) consists of the intersection of the positive \(m\)-dimensional cells with \(Y\), as well as the negative \(m\)-dimensional cells with \(Z\); and at \(t=1\) consists of points in \(Y\cap Z\). It is shown that, under an assumption of general position, homotopy continuation methods can be used to obtain all points in \(Y\cap Z\) by tracking points from \(S_0\) via \(H\). Versions of this result are provided in the cases when the sets \(Y,Z\) are defined implicitly, and when they are defined parametrically.
The authors then apply the above scheme to the inverse kinematical problem of determining the possible rotation angles of a robot arm, formed from six revolute joints connected by rigid links (6R chain), that will result in a desired arm configuration. In this case, the hypersurface \(X\) corresponds to the configuration space of rigid Euclidean motions in three-dimensions, which can be identified with the Study quadric – a six-dimensional hypersurface in seven-dimensional projective space. The three-dimensional subsets \(Y,Z\) are obtained by considering the first three joints and the last three joints of the arm separately; i.e., breaking the arm into two 3R chains.
Readers are assumed to be familiar with concepts from algebraic geometry; a knowledge of numerical homotopy continuation methods and robotics is useful. The article is organized and readable.