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Suppose that I have a line in an $n$-dimensional space described by $$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$ here $A$ is known and I want to find all the possible vectors $B$ such that this line has one and only one intersection with the set $S$ described by: $$S=\{ X \in \mathbb{R}^n \: | \: 0 \le X \le W\} $$ where $W \in \mathbb{R}_+^n$ and the inequalities are to be intended component-wise.

Is there any closed-form formula for such vectors $B$?

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  • $\begingroup$ I would not expect a closed form formula. Are you interested in a general procedure to find this set? $\endgroup$
    – M. Winter
    Commented Mar 17, 2020 at 22:30

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Line-Plane Intersection. In 3D, a line L is either parallel to a plane P or intersects it in a single point. ... We first check if L is parallel to P by testing if , which means that the line direction vector u is perpendicular to the plane normal n.

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  • $\begingroup$ Yes but since I want to find all vectors $B$ such that there is just one intersection, that can happen if and only if the line passes through either a vertex or an edge of my parallelotope, isn't that right? $\endgroup$
    – Leonardo Massai
    Commented Jan 22, 2019 at 15:37

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