VerticesAndRays - Maple Help

PolyhedralSets

 VerticesAndRays
 vertices and rays of a polyhedral set

 Calling Sequence VerticesAndRays(polyset, options)

Parameters

 polyset - polyhedral set options - choice of method and type of vertices returned; see Description Section for details

Description

 • This command computes the vertices and rays of the polyhedral set polyset.  It returns two lists, the first being the vertices of the set and the second its rays.  Either list could be empty, depending on the set and the options specified.
 • The available options are:
 – The method = enumeration option (default) computes a set's V-Representation by recursively computing its facets until its minimal dimensional faces have been calculated.  The vertices and rays are then inferred and returned.  The method = duality option computes the set's V-Representation using its polar dual.  This method is applicable to bounded polyhedral sets that contain the origin in their interior.
 – The vertextype option selects whether or not the vertices returned by VerticesAndRays are strict vertices of polyset.  The default vertextype = extremepoints returns only vertices that are extreme points of polyset, while vertextype = vrepresentation return the vertices necessary to form a V-Representation of polyset.  With vertextype = vrepresentation, this command will return vertices even if the set does not have any extreme points, i.e. when Faces(polyset, dimension = 0) returns an empty list.  This option has no effect on sets that have vertices.

Examples

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

Bounded sets like the octahedron have vertices, but the list of rays is empty.

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

The rays of an unbounded set indicate the direction(s) in which the set extends infinitely.  The half-space

 > $H≔\mathrm{PolyhedralSet}\left(\left[1\le x\right],\left[x,y\right]\right):$

extends in the positive $x$, positive $y$ and negative $y$ directions, as indicated by its rays.

 > $v,r≔\mathrm{VerticesAndRays}\left(H\right)$
 ${v}{,}{r}{≔}\left[\right]{,}\left[\left[{0}{,}{1}\right]{,}\left[{0}{,}{-1}\right]{,}\left[{1}{,}{0}\right]\right]$ (3)

Since $H$ has no extreme points, the list of vertices $v$ is empty. We can query the V-Representation of $H$ using the vertextype = vrepresentation option, which returns an arbitrary point on the surface of the half-space as a vertex.

 > $v,r≔\mathrm{VerticesAndRays}\left(H,\mathrm{vertextype}=\mathrm{vrepresentation}\right)$
 ${v}{,}{r}{≔}\left[\left[{1}{,}{0}\right]\right]{,}\left[\left[{0}{,}{1}\right]{,}\left[{0}{,}{-1}\right]{,}\left[{1}{,}{0}\right]\right]$ (4)

Compatibility

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