Intersection of a double cone and a line in the split-quaternions context
The real split-quaternion algebra \(\hat{H}\) is the \(\mathbb{R}\)-algebra with basis \(1,i,j,k:=ij=-ji\) such that \(i^{2}=-1\) and \(j^{2}=1\). The subalgebra \(\hat{H}_{0}\) of \(\hat{H}\) consisting of all elements of the form \(p=ti+xj+yk\) can be identified with the usual \((2+1)\)-Minkowski space with \(t\) as the time coordinate. The norm of \(p\) is defined by \(n(p):=t^{2}-x^{2}-y^{2}\) and \(a\wedge b\) denotes the (exterior) cross product of two elements of \(\hat {H}_{0}\). A line in \(\hat{H}_{0}\) is a set of the form \(\left\{ a+\lambda b~|~\lambda\in\mathbb{R}\right\} \) for \(a,b\in\) \(\hat{H}_{0}\) with \(b\neq0\) and the double cone is the set \(\left\{ ti+xj+yk~|~x^{2}+y^{2}=t^{2}\text{ with }x,y,t\in\mathbb{R}\right\} \). The main theorem describes the number \(N\) of points of intersection of the line with double cone: (1) if \(n(b)\neq0\), then \(N=0,1\) or \(2\) depending on whether \(n(a\wedge b)\) is positive, zero or negative; (2) (i) if \(n(b)=0\) and \(a\) belongs to a separation plane of \(b\), then \(N=\infty\) or \(0\) depending on whether \(n(a)=0\) or \(n(a)\neq0\); (2) (ii) if \(n(b)=0\) and \(a\) does not belong to a separation plane of \(b\), then \(N=1\).
scientific article; zbMATH DE number 6835303