On joints in arrangements of lines in space and related problems (Q1331145)

Let \(L=\{l_ 1,\dots, l_ n\}\) be an arrangement of \(n\) lines in \(\mathbb{R}^ 3\), and \(x\in\mathbb{R}^ 3\) is said to be a joint of \(L\) if this point belongs to three non-coplanar lines of \(L\). In earlier contributions it was shown that \(O(n^{7\over 4})\) is an upper bound on the number of joints in \(L\), and a construction was given which shows that this number can be \(\Omega (n^{3\over 2})\). Here it is shown that the number of joints in \(L\) is \(O(n^{{23}\over {14}} \log^{{31} \over {14}} n)\), which is \(O(n^{1.643})\). The proof is based on recent range searching techniques and structural pattern analysis with respect to the intersections occurring in \(L\).