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user64494
user64494
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Intersection between a line and aan n dimensional-dimensional parallelotope

Suppose that I have a line in an $n$ dimensional-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 $B$?

Intersection between a line and a n dimensional parallelotope

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 $B$?

Intersection between a line and an n-dimensional parallelotope

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 $B$?

Bumped by Community user
Bumped by Community user
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Leonardo
Leonardo
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Intersection between a line and a n dimensional parallelotope

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 $B$?