The command SimplicialDecomposition(pc) returns a simplicial decomposition, as a list of polyhedral sets, of the polyhedral cone pc.

The command SimplicialDecomposition(pc,polyhedralsets) returns the same as a SimplicialDecomposition(pc).

The command SimplicialDecomposition(pc,polyhedralcones) returns  a simplicial decomposition, as a list of polyhedral cones, of the polyhedral cone pc.

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

$\mathrm{ps}≔\mathrm{PolyhedralSet}\left(\left[1\le x\,1\le y\right]\right)\;$$\mathrm{PolyhedralSets}:-\mathrm{Plot}\left(\mathrm{ps}\right)$

$\mathrm{ps}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x\,y\right]\\ \mathrm{Relations}& \:& \left[-y\le \mathrm{-1}\,-x\le \mathrm{-1}\right]\end{array}$

$\mathrm{pc}≔\mathrm{PolyhedralCone}\left(\mathrm{ps}\right)$

$\mathrm{pc}≔⟨\text{polyhedral cone with vertex}\left[1\,1\right]\text{and rays}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)⟩$

$\mathrm{SimplicialDecomposition}\left(\mathrm{pc}\right)$

$\left[⟨\text{polyhedral cone with vertex}\left[1\,1\right]\text{and rays}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)⟩\right]$

$\mathrm{SimplicialDecomposition}\left(\mathrm{pc}\,\mathrm{polyhedralsets}\right)$

$\left[\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\right]\\ \mathrm{Relations}& \:& \left[-x_2\le \mathrm{-1}\,-x_1\le \mathrm{-1}\right]\end{array}\right]$

$\mathrm{ps}≔\mathrm{PolyhedralSet}\left(\left[-x\left[1\right]-x\left[2\right]-x\left[3\right]\le 0\,-x\left[1\right]+x\left[2\right]+x\left[3\right]\le 0\,x\left[1\right]-x\left[2\right]+x\left[3\right]\le 0\right]\right)\;$$\mathrm{PolyhedralSets}:-\mathrm{Plot}\left(\mathrm{ps}\right)$

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

$\mathrm{pc}≔\mathrm{PolyhedralCone}\left(\mathrm{ps}\right)$

$\mathrm{pc}≔⟨\text{polyhedral cone with vertex}\left[0\,0\,0\right]\text{and rays}\left(\begin{array}{ccc}1& 1& 0\\ 1& 0& \mathrm{-1}\\ 0& 1& \mathrm{-1}\end{array}\right)⟩$

$\mathrm{SimplicialDecomposition}\left(\mathrm{pc}\right)$

$\left[⟨\text{polyhedral cone with vertex}\left[0\,0\,0\right]\text{and rays}\left(\begin{array}{ccc}1& 1& 0\\ 1& 0& \mathrm{-1}\\ 0& 1& \mathrm{-1}\end{array}\right)⟩\right]$

$\mathrm{SimplicialDecomposition}\left(\mathrm{pc}\,\mathrm{polyhedralsets}\right)$

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

$\mathrm{ps}≔\mathrm{PolyhedralSet}\left(\left[\left[0\,0\,0\right]\right]\,\left[\left[1\,-1\,1\right]\,\left[-1\,1\,1\right]\,\left[-1\,-1\,1\right]\,\left[1\,1\,11\right]\right]\right)\;$$\mathrm{PolyhedralSets}:-\mathrm{Plot}\left(\mathrm{ps}\right)$

$\mathrm{ps}≔\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_2-x_3\le 0\,-x_1-x_3\le 0\,x_1+\frac{5x_2}{6}-\frac{x_3}{6}\le 0\,x_1+\frac{6x_2}{5}-\frac{x_3}{5}\le 0\right]\end{array}$

$\mathrm{pc}≔\mathrm{PolyhedralCone}\left(\mathrm{ps}\right)$

$\mathrm{pc}≔⟨\text{polyhedral cone with vertex}\left[0\,0\,0\right]\text{and rays}\left(\begin{array}{ccc}1& \mathrm{-1}& 1\\ \mathrm{-1}& \mathrm{-1}& 1\\ \frac{1}{11}& \frac{1}{11}& 1\\ \mathrm{-1}& 1& 1\end{array}\right)⟩$

$\mathrm{SimplicialDecomposition}\left(\mathrm{pc}\right)$

$\left[⟨\text{polyhedral cone with vertex}\left[0\,0\,0\right]\text{and rays}\left(\begin{array}{ccc}\mathrm{-1}& \mathrm{-1}& 1\\ 1& 1& 11\\ \mathrm{-1}& 1& 1\end{array}\right)⟩\,⟨\text{polyhedral cone with vertex}\left[0\,0\,0\right]\text{and rays}\left(\begin{array}{ccc}1& \mathrm{-1}& 1\\ \mathrm{-1}& \mathrm{-1}& 1\\ 1& 1& 11\end{array}\right)⟩\right]$

$\mathrm{SimplicialDecomposition}\left(\mathrm{pc}\,\mathrm{polyhedralsets}\right)$

$\left[\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_2-x_3\le 0\,-x_1-x_3\le 0\,x_1+x_2\le 0\right]\end{array}\,\{\begin{array}{lll}\mathrm{Coordinates}& \:& \left[x_1\,x_2\,x_3\right]\\ \mathrm{Relations}& \:& \left[-x_1-x_2\le 0\,x_1+\frac{5x_2}{6}-\frac{x_3}{6}\le 0\,x_1+\frac{6x_2}{5}-\frac{x_3}{5}\le 0\right]\end{array}\right]$

The PolyhedralSets[PolyhedralCones][SimplicialDecomposition] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.