DualSet - Maple Help

PolyhedralSets

 DualSet
 polar dual of a polyhedral set

 Calling Sequence DualSet(polyset)

Parameters

 polyset - polyhedral set

Description

 • This command computes the polar dual of the polyhedral set polyset.

Examples

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

The dual of the cube is an octahedron

 > $\mathrm{cube}≔\mathrm{ExampleSets}:-\mathrm{Cube}\left(\right):$$\mathrm{cube_dual}≔\mathrm{DualSet}\left(\mathrm{cube}\right):$$\mathrm{Plot}\left(\mathrm{cube_dual}\right)$

DualSet is an involution for bounded sets that include the origin, that is taking the dual of octahedron yields the original cube

 > $\mathrm{cube_dual_dual}≔\mathrm{DualSet}\left(\mathrm{cube_dual}\right):$$\mathrm{Plot}\left(\mathrm{cube_dual_dual}\right)$

The dual of the dual of a half-space, on the other hand, does not yield the original set.

 > $p≔\mathrm{PolyhedralSet}\left(\left[x\le -1\right],\left[x,y\right]\right)$
 ${p}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}\right]\\ {\mathrm{Relations}}& {:}& \left[{x}{\le }{-1}\right]\end{array}$ (1)
 > $\mathrm{p_dual}≔\mathrm{DualSet}\left(p\right)$
 ${\mathrm{p_dual}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}\right]\\ {\mathrm{Relations}}& {:}& \left[{y}{=}{0}{,}{-}{x}{\le }{0}\right]\end{array}$ (2)
 > $\mathrm{p_dual_dual}≔\mathrm{DualSet}\left(\mathrm{p_dual}\right)$
 ${\mathrm{p_dual_dual}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}\right]\\ {\mathrm{Relations}}& {:}& \left[{x}{\le }{0}\right]\end{array}$ (3)
 > $\mathrm{Equal}\left(p,\mathrm{p_dual_dual}\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{Plot}\left(\left[p,\mathrm{p_dual_dual}\right]\right)$

Compatibility

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