ParametricRegion[{f1,…,fn},{u1,…,um}]
represents a region in 
given by the points {f1,…,fn} for parameters ui∈.
ParametricRegion
ParametricRegion[{f1,…,fn},{u1,…,um}]
represents a region in 
given by the points {f1,…,fn} for parameters ui∈.
ParametricRegion[{f1,…,fn},{{u1,a1,b1},…}]
constrains parameters to an interval a1≤u1≤b1 etc.
ParametricRegion[{{f1,…,fn},cond},…]
constrains parameters to satisfy the condition cond.
Details
- ParametricRegion is also known as parametric curve, surface, volume, etc.
- The value of ui in ParametricRegion[{{f1,…,fn},cond},{u1,…,um}] is taken to be localized, as in Block.
- ParametricRegion[{funs,cond},{{u1,a1,b1},…}] is equivalent to ParametricRegion[{funs,cond∧a1≤u1≤b1∧⋯},{u1,…}].
- ParametricRegion can be used in functions such as RegionDistance, Reduce, and Integrate.
Examples
open all close allBasic Examples (2)
Specify a circle as a ParametricRegion:
ℛ = ParametricRegion[{Cos[θ], Sin[θ]}, {{θ, 0, 2π}}]Region[ℛ]Region[ParametricRegion[{r Cos[θ], r Sin[θ]}, {{θ, 0, 2π}, {r, 0, 1}}]]The region between two parabolas:
ℛ = ParametricRegion[{{s, (1 + t) s ^ 2 - t}, -1 ≤ s ≤ 1 && 0 ≤ t ≤ 1}, {s, t}];Region[ℛ]Scope (20)
Purely Parametric Description (10)
Parametrically described region:
ℛ = ParametricRegion[{{s ^ 2t ^ 2, s t ^ 3}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];Region[ℛ]RegionEmbeddingDimension[ℛ]RegionDimension[ℛ]ℛ = ParametricRegion[{{s ^ 2, s ^ 3}, -1 ≤ s ≤ 1}, {s}];RegionMember[ℛ, {1, 1}]RegionMember[ℛ, {2, 2}]Get conditions for point membership:
RegionMember[ℛ, {x, y}]ℛ = ParametricRegion[{{s ^ 2t ^ 2, s t ^ 3}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];RegionMeasure[ℛ]RegionCentroid[ℛ]ℛ = ParametricRegion[{{t, s t ^ 2}, 0 ≤ s ≤ 1}, {s, t}];RegionDistance[ℛ, {1, 2}]RegionDistance[ℛ, {2, 1}]ℛ = ParametricRegion[{{t, s t ^ 2}, 0 ≤ s ≤ 1}, {s, t}];SignedRegionDistance[ℛ, {1, 2}]SignedRegionDistance[ℛ, {2, 1}]ℛ = ParametricRegion[{{t, s (1 - t ^ 2)}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];RegionNearest[ℛ, {2, 1}]N[%]RegionNearest[ℛ, {1, 0}]pts = Table[2{Cos[k 2 π / 16], Sin[k 2π / 16]}, {k, 0., 15}];npts = RegionNearest[ℛ, pts];Show[{BoundaryDiscretizeRegion[ℛ], Graphics[{{Dashed, Line[Transpose[{pts, npts}]]}, {Red, Point[npts]}, {Blue, Point[pts]}}]}, PlotRange -> All]The set of points above a parabola is not bounded:
ℛ = ParametricRegion[{{t, s + t ^ 2}, s ≥ 0}, {s, t}];BoundedRegionQ[ℛ]Integrate over an implicitly defined region:
ℛ = ParametricRegion[{{t, s (1 - t ^ 2)}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];Integrate[x ^ 2 + y ^ 2, {x, y}∈ℛ]Optimize over an implicitly defined region:
ℛ = ParametricRegion[{{t, s (1 - t ^ 2)}, -1 ≤ s ≤ 1 && -1 ≤ t ≤ 1}, {s, t}];MinValue[{x y - x, {x, y}∈ℛ}, {x, y}]Solve equations in an implicitly defined region:
ℛ = ParametricRegion[{{s ^ 2, s ^ 3}, -1 ≤ s ≤ 1}, {s}];Reduce[x^2 + y^2 == 1 && {x, y}∈ℛ, {x, y}]Mixed Implicit and Parametric Description (10)
A region given in a mixed implicit and parametric description:
ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];Region[ℛ]RegionEmbeddingDimension[ℛ]RegionDimension[ℛ]ℛ = ParametricRegion[{{s ^ 2, t ^ 3}, s t == 1}, {s, t}];{RegionMember[ℛ, {1, 1}], RegionMember[ℛ, {2, 2}]}Get conditions for point membership:
RegionMember[ℛ, {x, y}]ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];RegionMeasure[ℛ]RegionCentroid[ℛ]ℛ = ParametricRegion[{{s + t, s - t}, s ≤ t ^ 2}, {s, t}];RegionDistance[ℛ, {1, 1}]RegionDistance[ℛ, {-1, 0}]ℛ = ParametricRegion[{{s + t, s - t}, s ≤ t ^ 2}, {s, t}];SignedRegionDistance[ℛ, {1, 1}]SignedRegionDistance[ℛ, {-1, 0}]ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];RegionNearest[ℛ, {0, 1}]//RootReduceN[%]RegionNearest[ℛ, {0, 0}]pts = Table[2{Cos[k 2 π / 16], Sin[k 2π / 16]}, {k, 0., 15}];npts = RegionNearest[ℛ, pts];Show[{BoundaryDiscretizeRegion[ℛ], Graphics[{{Dashed, Line[Transpose[{pts, npts}]]}, {Red, Point[npts]}, {Blue, Point[pts]}}]}, PlotRange -> All]ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];BoundedRegionQ[ℛ]ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];Integrate[x ^ 2 + y ^ 2, {x, y}∈ℛ]ℛ = ParametricRegion[{{s, s t}, s ^ 2 + t ^ 2 ≤ 1}, {s, t}];MinValue[{x y - x, {x, y}∈ℛ}, {x, y}]//RootReduceSolve equations and inequalities in ℛ:
ℛ = ParametricRegion[{{s + t, s - t}, s ≤ t ^ 2}, {s, t}];Reduce[x^2 + y^2 == 1 && x ≥ y && {x, y}∈ℛ, {x, y}]Text
Wolfram Research (2014), ParametricRegion, Wolfram Language function, https://reference.wolfram.com/language/ref/ParametricRegion.html.
CMS
Wolfram Language. 2014. "ParametricRegion." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ParametricRegion.html.
APA
Wolfram Language. (2014). ParametricRegion. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ParametricRegion.html