PolyhedralSets - Maple Programming Help

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PolyhedralSets

 InteriorPoint
 get an interior point of a polyhedral set

 Calling Sequence InteriorPoint(polyset)

Parameters

 polyset -

Description

 • This command computes an interior point of the highest dimensional face for the given polyhedral set polyset.  If polyset is of lower dimension than its ambient space, then the point lies on the boundary of polyset, but the point is not an extreme point of the set.
 • The command returns NULL if there are no interior points in polyset. In other words, if polyset is a vertex or an empty set.  Otherwise, the command returns a list of rationals representing a point in polyset's coordinate space.

Examples

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

Get a point inside the cube

 > $c≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right):$$p≔\mathrm{InteriorPoint}\left(c\right)$
 ${p}{≔}\left[{0}{,}{0}{,}{0}\right]$ (1)

For a 2-D triangular set in 3-D space, the command returns a point which is interior to the triangle, but lies on the boundary of the set strictly speaking.

 > $\mathrm{triangle}≔\mathrm{PolyhedralSet}\left(\left[z=1,-1\le y,-1\le x,x\le -y+1\right],\left[x,y,z\right]\right)$
 ${\mathrm{triangle}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}{,}{z}\right]\\ {\mathrm{Relations}}& {:}& \left[{z}{=}{1}{,}{-}{y}{\le }{1}{,}{-}{x}{\le }{1}{,}{x}{+}{y}{\le }{1}\right]\end{array}$ (2)
 > $p≔\mathrm{InteriorPoint}\left(\mathrm{triangle}\right)$
 ${p}{≔}\left[{0}{,}{0}{,}{1}\right]$ (3)

The point lies on a 2 dimensional set, while the set is in a 3 dimensional space.

 > $\mathrm{face_with_p}≔\mathrm{LocatePoint}\left(p,\mathrm{triangle}\right)$
 ${\mathrm{face_with_p}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}{,}{z}\right]\\ {\mathrm{Relations}}& {:}& \left[{z}{=}{1}{,}{-}{y}{\le }{1}{,}{-}{x}{\le }{1}{,}{x}{+}{y}{\le }{1}\right]\end{array}$ (4)
 > $\mathrm{Dimension}\left(\mathrm{face_with_p}\right)$
 ${2}$ (5)
 > $\mathrm{nops}\left(\mathrm{Coordinates}\left(\mathrm{face_with_p}\right)\right)$
 ${3}$ (6)

Compatibility

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