SplitIntoSimplices - Maple Help

PolyhedralSets

 SplitIntoSimplices
 split a polyhedral set into simplices

 Calling Sequence SplitIntoSimplices(polyset) SplitIntoSimplices(polyset, point)

Parameters

 polyset - polyhedral set point - polyhedral set representing a vertex, list of rationals, Vector or list/set of equations of the form coordinate = value; point where polyset will be split

Description

 • 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.

Examples

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

Splitting the cube into simplices gives a list of tetrahedrons.

 > $c≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right):$
 > $\mathrm{c_pieces}≔\mathrm{SplitIntoSimplices}\left(c\right)$
 ${\mathrm{c_pieces}}{≔}\left[{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{3}}{\le }{1}{,}{-}{{x}}_{{2}}{+}{{x}}_{{3}}{\le }{0}{,}{-}{{x}}_{{1}}{+}{{x}}_{{2}}{\le }{0}{,}{{x}}_{{1}}{\le }{1}\right]\end{array}{,}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{2}}{\le }{1}{,}{{x}}_{{2}}{-}{{x}}_{{3}}{\le }{0}{,}{-}{{x}}_{{1}}{+}{{x}}_{{3}}{\le }{0}{,}{{x}}_{{1}}{\le }{1}\right]\end{array}{,}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{3}}{\le }{1}{,}{{x}}_{{2}}{\le }{1}{,}{-}{{x}}_{{1}}{+}{{x}}_{{3}}{\le }{0}{,}{{x}}_{{1}}{-}{{x}}_{{2}}{\le }{0}\right]\end{array}{,}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{{x}}_{{2}}{+}{{x}}_{{3}}{\le }{0}{,}{{x}}_{{2}}{\le }{1}{,}{-}{{x}}_{{1}}{\le }{1}{,}{{x}}_{{1}}{-}{{x}}_{{3}}{\le }{0}\right]\end{array}{,}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{{x}}_{{3}}{\le }{1}{,}{-}{{x}}_{{2}}{\le }{1}{,}{-}{{x}}_{{1}}{+}{{x}}_{{2}}{\le }{0}{,}{{x}}_{{1}}{-}{{x}}_{{3}}{\le }{0}\right]\end{array}{,}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{{x}}_{{1}}{,}{{x}}_{{2}}{,}{{x}}_{{3}}\right]\\ {\mathrm{Relations}}& {:}& \left[{{x}}_{{3}}{\le }{1}{,}{{x}}_{{2}}{-}{{x}}_{{3}}{\le }{0}{,}{-}{{x}}_{{1}}{\le }{1}{,}{{x}}_{{1}}{-}{{x}}_{{2}}{\le }{0}\right]\end{array}\right]$ (1)

The volume of the cube is equal to the sum of the volume of the simplices.

 > $\mathrm{Volume}\left(c\right)$
 ${8}$ (2)
 > $\mathrm{volumes}≔\mathrm{map}\left(\mathrm{Volume},\mathrm{c_pieces}\right)$
 ${\mathrm{volumes}}{≔}\left[\frac{{4}}{{3}}{,}\frac{{4}}{{3}}{,}\frac{{4}}{{3}}{,}\frac{{4}}{{3}}{,}\frac{{4}}{{3}}{,}\frac{{4}}{{3}}\right]$ (3)
 > $\mathrm{+}\left(\mathrm{volumes}\left[\right]\right)$
 ${8}$ (4)

Plot the simplices with transparency to see the internal structure of the sets.

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

Compatibility

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