Suppose that we fix points $a,b,c,d \in \mathbb{R}^3$ which do not belong to the same plane.
How can we check algebraically whether $x \in \mathbb{R}^3$ to their convex hull?
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I suppose one option is to use an affine transformation $T$ to map $a, b, c, d$ respectively to $(0, 0, 0, 0)$, $(0, 1, 0, 0)$, $(0, 0, 1, 0)$, $(0, 0, 0, 1)$. Then, $x$ is in the convex hull iff $Tx$ is in the transformed pyramid, that is, if all of the coordinates of $Tx$ are nonnegative and the sum of its coordinates is $\leq 1$. – Travis Willse Mar 12 '16 at 21:43
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Im sorry Dietrich, I wasn't able to find that question. Close it if necessary.. (and thanks for the comment!) – Mar 12 '16 at 21:45
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One approach is to compute the plane containing $a,b,c$ and check that $x$ and $d$ are on the same side. Repeat this with all other sets of three points from $a,b,c,d$. If $x$ is always on the same side of the plane as the odd point out, it is in the convex hull.
Ross Millikan
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