This command splits the polyhedral set polyset into simplices, returning a list of polyhedral sets whose union is polyset. Only bounded polyhedral sets, that is to say polytopes, can be divided into simplices.

If point is provided, it will be a vertex of all the simplices when polyset is split. If point is omitted, an arbitrary vertex of polyset is chosen for the splitting point.

$\mathrm{with}\left(\mathrm{PolyhedralSets}\right)\:$

$c≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right)\:$

$\mathrm{c\_pieces}≔\mathrm{SplitIntoSimplices}\left(c\right)$

$\mathrm{c\_pieces}≔\left[\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_3\le 1\,-x_2+x_3\le 0\,-x_1+x_2\le 0\,x_1\le 1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_2\le 1\,x_2-x_3\le 0\,-x_1+x_3\le 0\,x_1\le 1\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_3\le 1\,x_2\le 1\,-x_1+x_3\le 0\,x_1-x_2\le 0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_2+x_3\le 0\,x_2\le 1\,-x_1\le 1\,x_1-x_3\le 0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[x_3\le 1\,-x_2\le 1\,-x_1+x_2\le 0\,x_1-x_3\le 0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[x_3\le 1\,x_2-x_3\le 0\,-x_1\le 1\,x_1-x_2\le 0\right]\end{array}\right]$

$\mathrm{Volume}\left(c\right)$

$8$

$\mathrm{volumes}≔\mathrm{map}\left(\mathrm{Volume}\,\mathrm{c\_pieces}\right)$

$\mathrm{volumes}≔\left[\frac{4}{3}\,\frac{4}{3}\,\frac{4}{3}\,\frac{4}{3}\,\frac{4}{3}\,\frac{4}{3}\right]$

$\mathrm{`+`}\left(\mathrm{volumes}\left[\right]\right)$

$8$

$\mathrm{Plot}\left(\mathrm{c\_pieces}\,\mathrm{transparency}=0.5\,\mathrm{orientation}=\left[23\,62\,0\right]\right)$

The PolyhedralSets[SplitIntoSimplices] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.