Area - Maple Help

PolyhedralSets

 Volume
 volume of polyhedral set
 Area
 area of polyhedral set
 Length
 length of polyhedral set

 Calling Sequence Volume(polyset) Volume(polyset, d) Area(polyset) Length(polyset)

Parameters

 polyset - polyhedral set d - (optional) non-negative integer, dimensionality of the set in which the volume should be computed; default is the dimension of polyset's coordinate space

Description

 • The Volume command computes the volume of a polyhedral set in the generalized sense of a Lebesgue measure, defining a $d$-volume of a set in higher dimensions.  For sets in 3, 2, and 1 dimensional spaces, this corresponds to the standard definitions of volume, area, and length, respectively.
 • The optional parameter d specifies which $d$-volume to compute for the set, where d is less than or equal to the number of polyset's coordinates.  For sets of dimension greater than d, the sum of the volumes for the $d$-faces is computed.
 • The Area and Length commands are used to compute the 2-volume and 1-volume of polyset, respectively, and are equivalent to Volume(polyset, 2) and Volume(polyset, 1).

Examples

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

For the standard cube

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

the edges of the cube all have length 2

 > $\mathrm{c_edges}≔\mathrm{Edges}\left(c\right):$$\mathrm{map}\left(\mathrm{Length},\mathrm{c_edges}\right)$
 $\left[{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}{,}{2}\right]$ (2)

the cube has a volume of 8

 > $\mathrm{Volume}\left(c\right)$
 ${8}$ (3)

and the total surface area of the cube is

 > $\mathrm{Area}\left(c\right)$
 ${24}$ (4)

The polyhedral set ps is a three-dimensional polytope in four-dimensional space.

 > $\mathrm{ps}≔\mathrm{PolyhedralSet}\left(\left[\left[1,0,0,0\right],\left[0,1,0,0\right],\left[0,0,1,0\right],\left[0,0,0,1\right]\right],\left[x,y,z,u\right]\right)$
 ${\mathrm{ps}}{≔}{{}\begin{array}{lll}{\mathrm{Coordinates}}& {:}& \left[{x}{,}{y}{,}{z}{,}{u}\right]\\ {\mathrm{Relations}}& {:}& \left[{-}{z}{\le }{0}{,}{-}{y}{\le }{0}{,}{-}{x}{\le }{0}{,}{x}{+}{y}{+}{z}{\le }{1}{,}{x}{+}{y}{+}{z}{+}{u}{=}{1}\right]\end{array}$ (5)

Its four-dimensional volume is 0, but its three-dimensional volume is positive.

 > $\mathrm{Volume}\left(\mathrm{ps}\right)$
 ${0}$ (6)
 > $\mathrm{Volume}\left(\mathrm{ps},3\right)$
 $\frac{{1}}{{3}}$ (7)

Compatibility

 • The PolyhedralSets[Volume], PolyhedralSets[Area] and PolyhedralSets[Length] commands were introduced in Maple 2015.