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I am trying to check if a point lies beneath vectors convex to it by trying to deduce which of the four cases below fits any given scenario:

cases

I am hoping to isolate B, in that there's some general way to deduce whether a given scenario is case B or not (without the need to compute angles between vectors) i.e. since only $\vec{a}$ and $\vec{b}$ in case B are convex to the point $p$

linker
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1 Answers1

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Just find the coordinates of $p - h$ with respect to the frame whose axes are vectors $\vec{a}, \vec{b}$ and whose origin is $h$

The associated linear system can be written as

$ [ \vec{a} , \vec{b} ] [x_1, y_1]^T = p - h $

Then, it follows that

$ [x_1, y_1]^T = [ \vec{a}, \vec{b} ]^{-1} (p - h) $

Since you're interested in Case $B$, then you want $x_1 \gt 0 , y_1 \gt 0$.

For example, if point $h = [2, 3]^T , \vec{a} = [1, -3]^T , \vec{b} = [2, 6]^T $, and $p$ is given as $ [ 5, 7 ]^T$, then

$ \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = \begin{bmatrix} 1 && 2 \\ -3 && 6 \end{bmatrix}^{-1} \begin{bmatrix} 3 \\ 4 \end{bmatrix} $

And this solves to

$ \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} = \dfrac{1}{12} \begin{bmatrix} 6 && -2 \\ 3 && 1 \end{bmatrix} \begin{bmatrix} 3 \\ 4 \end{bmatrix} =\dfrac{1}{12} \begin{bmatrix} 10 \\ 13 \end{bmatrix} $

Hence, $x_1 \gt 0 , y_1 \gt 0 $, so this is Case $B$.

  • Hi @Robin’sPremiumCoffee would this still work if $\vec{a}$ and $\vec{b}$ for example are not perpendicular to one another, also say we are dealing with a 3D frame $\vec{a} , \vec{b} ,\vec{d}$ instead would it be okay to apply the same logic asper: $ [x_1, y_1, z_1]^T = [ \vec{a} , \vec{b} ,\vec{d}]^{-1} (p - h) $ such that $x_1 \gt 0 , y_1 \gt 0, z_1 \gt 0 $ – linker May 08 '22 at 14:19
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    Yes. It also works if $\vec{a}, \vec{b}$ are not mutually perpendicular. And the equation you wrote for $3D$ is also correct, and its conclusion is correct. –  May 08 '22 at 15:05
  • Hi @Hosam Y. Hajjir, please could you help out with this https://math.stackexchange.com/q/4484719/585488 – linker Jul 03 '22 at 07:57
  • What do you mean by polytope ? What does that mean ? –  Jul 03 '22 at 07:59
  • just a polytope in general e.g. a 2-polytope (polygon), 3-polytope (polyhedron/3d mesh) .etc you may also have a look at https://en.m.wikipedia.org/wiki/Polytope – linker Jul 03 '22 at 08:06
  • Sorry. I cannot help with the indicated question. –  Jul 03 '22 at 08:13
  • no problem, I appreciate the effort my mathematical friend :) – linker Jul 03 '22 at 08:15