TransformedRegion
TransformedRegion
TransformedRegion[reg,f]
代表变换区域 
，其中 reg 是一个区域，f 是一个函数.
更多信息和选项
- TransformedRegion 又被称为区域的图像.
范例
打开所有单元 关闭所有单元基本范例 (2)
ℛ = TransformedRegion[Rectangle[], RotationTransform[π / 4]]Region[ℛ]Area[ℛ]
ℛ = TransformedRegion[Disk[{1, 1}, 4], {Indexed[#, 1] Indexed[#, 2], Indexed[#, 1] + Indexed[#, 2]}&];Region[ℛ]RegionMeasure[ℛ]范围 (26)
特殊区域 (10)
Subscript[ℛ, 1] = Rectangle[];
Subscript[ℛ, 2] = TransformedRegion[Subscript[ℛ, 1], ShearingTransform[Pi / 4, {1, 0}, {0, 1}]]Graphics[{Opacity[0.5], {Red, Subscript[ℛ, 1]}, {Green, Subscript[ℛ, 2]}}]Subscript[ℛ, 1] = Ball[];
𝒯 = TransformationFunction[{{7, 5, -2, -4}, {3, -8, 9, 4}, {-8, 3, 1, -6}, {1, 0, 0, 2}}];
Subscript[ℛ, 2] = TransformedRegion[Subscript[ℛ, 1], 𝒯]Graphics3D[{Opacity[0.5], {Red, Subscript[ℛ, 1]}, {Green, Subscript[ℛ, 2]}}]平移一个单位 Disk：
ℛ = TransformedRegion[Disk[], TranslationTransform[{2, 3}]];{RegionMember[ℛ, {0, 0}], RegionMember[ℛ, {2, 3}]}RegionMember[ℛ, {x, y}]Region[ℛ]错切一个单位 Rectangle：
ℛ = TransformedRegion[Rectangle[], ShearingTransform[30Degree, {1, 0}, {0, 1}]];计算 RegionBounds：
RegionBounds[ℛ]Region[ℛ]旋转一个标准 Triangle：
r = Triangle[];
Region[r]ℛ = TransformedRegion[Triangle[], RotationTransform[π / 3]];Region[ℛ]{ConstantRegionQ[r], ConstantRegionQ[ℛ]}{BoundedRegionQ[r], BoundedRegionQ[ℛ]}缩放一个 Circle:
r = Circle[];
Region[r]ℛ = TransformedRegion[r, ScalingTransform[2., {1, 1}]];Region[ℛ]计算 ArcLength：
{ArcLength[r], ArcLength[ℛ]}
r1 = TransformedRegion[Disk[{1, 1}, 4], Function[p, {p[[1]] p[[2]], p[[1]] + p[[2]]}]];Region[r1]用 Indexed 来表达同样的点映射：
r2 = TransformedRegion[Disk[{1, 1}, 4], {Indexed[#, 1] Indexed[#, 2], Indexed[#, 1] + Indexed[#, 2]}&];Region[r2]计算 RegionMeasure：
RegionMeasure[r2]通过非线性变换 
，将三维 Cuboid 的点映射至二维：
ℛ = TransformedRegion[Cuboid[{0, 0, 0}, {1, 1, 1}], {Indexed[#, 1] + Indexed[#, 2], Indexed[#, 1] + Indexed[#, 3]}&];{RegionDimension[ℛ], RegionEmbeddingDimension[ℛ]}计算 RegionCentroid：
c = RegionCentroid[ℛ]RegionPlot[ℛ, Epilog -> {Red, Point[c]}]SignedRegionDistance[ℛ, {1, 1}]SignedRegionDistance[ℛ, {0, 2}]旋转一个三维 Cuboid：
ℛ = TransformedRegion[Cuboid[{0, 0, 0}, {1, 1, 1}], RotationTransform[Pi / 3, {1, 1, 1}]];RegionDistance[ℛ, {0, 0, 0}]RegionDistance[ℛ, {2, 1, 0}]RegionNearest[ℛ, {1, 1, 1}]RegionNearest[ℛ, {2, 1, 0}]pts = Flatten[Table[{1 / 2, 1 / 2, 1 / 2} + {Cos[k 2π / 10]Sin[l π / 5], Sin[k 2π / 10]Sin[l π / 5], Cos[l π / 5]}, {k, 0., 9}, {l, 0., 4}], 1];
nst = RegionNearest[ℛ, #]& /@ pts;Graphics3D[{GeometricTransformation[Cuboid[], RotationTransform[Pi / 3, {1, 1, 1}]], {Dashed, Line[Transpose[{pts, nst}]]}, {Red, Point[pts]}, {Blue, Point[nst]}}]ℛ = TransformedRegion[Cuboid[{0, 0, 0}, {1, 1, 1}], RotationTransform[Pi / 3, {1, 1, 1}]];Integrate[x y z, {x, y, z}∈ℛ]MinValue[{x y z - x y, {x, y, z}∈ℛ}, {x, y, z}]Reduce[x^2 + y^2 + z^2 == 3 / 2 && z^2 == x y && {x, y, z}∈ℛ, {x, y, z}]公式区域 (6)
错切一个 ParametricRegion:
pr = ParametricRegion[{Sin[θ], Sin[2θ]}, {{θ, 0, 2π}}];Region[pr]ℛ = TransformedRegion[pr, ShearingTransform[30Degree, {1., 0}, {0, 1}]];计算 ArcLength：
ArcLength[ℛ]pt = {1.2, -0.75};
npt = RegionNearest[ℛ, pt]RegionPlot[ℛ, Epilog -> {PointSize[Large], {Red, Point[pt]}, {Blue, Point[npt]}, {Dashed, Line[{pt, npt}]}}]一个旋转的 ParametricRegion：
pr = ParametricRegion[{{s, (1 + t) s ^ 2 - t}, -1 ≤ s ≤ 1 && 0 ≤ t ≤ 1}, {s, t}];Region[pr]ℛ = TransformedRegion[pr, RotationTransform[Pi / 4, {1, 1}]];Region[ℛ]计算 Area：
Area[ℛ]计算 RegionBounds：
RegionBounds[ℛ]RegionDistance[ℛ, {1, -1}]RegionDistance[ℛ, {2., 0}]错切一个 ImplicitRegion：
ir = ImplicitRegion[x ^ 2 - y ^ 2 ≤ 1, {x, y}];DiscretizeRegion[ir, {{-2, 2}, {-2, 2}}]ℛ = TransformedRegion[ir, ShearingTransform[30Degree, {1, 0}, {0, 1}]];DiscretizeRegion[ℛ, {{-3, 3}, {-2, 2}}]{BoundedRegionQ[ir], BoundedRegionQ[ℛ]}{RegionMember[ℛ, {0, 0}], RegionMember[ℛ, {2, 0}]}RegionMember[ℛ, {x, y}]缩放一个 ImplicitRegion：
ir = ImplicitRegion[x^2 + y^2 + z^2 < 1 || (x + 1)^2 + y^2 + z^2 < 1, {x, y, z}];Region[ir]ℛ = TransformedRegion[ir, ScalingTransform[{1, 2, 2}]];Region[ℛ]计算 Volume：
Volume[ℛ]在区域 Integrate：
Integrate[x + y z, {x, y, z}∈ℛ]MinValue[{x + y z, {x, y, z}∈ℛ}, {x, y, z}]Reduce[x + y z == 3 / 2 && y == x z && {x, y, z}∈ℛ, {x, y, z}]一个 ImplicitRegion 经过非线性变换 
：
ir = ImplicitRegion[0 ≤ x ≤ 2 && 0 ≤ y ≤ 2 && x y ≤ 1, {x, y}];RegionPlot[ir]r1 = TransformedRegion[ir, Function[p, {p[[1]] p[[2]], p[[1]] + p[[2]]}]];RegionPlot[r1]用 Indexed 表达同样的点映射：
r2 = TransformedRegion[ir, {Indexed[#, 1] Indexed[#, 2], Indexed[#, 1] + Indexed[#, 2]}&];RegionPlot[r2]计算 RegionMeasure：
RegionMeasure[r2]
ir = ImplicitRegion[x^2 + y^2 + z^2 < 1, {x, y, z}];Region[ir]ℛ = TransformedRegion[ir, {Indexed[#, 1] + Indexed[#, 2], Indexed[#, 1] + Indexed[#, 3]}&];{RegionDimension[ℛ], RegionEmbeddingDimension[ℛ]}计算 RegionCentroid：
c = RegionCentroid[ℛ]RegionPlot[ℛ, Epilog -> {Red, Point[c]}]网格区域 (3)
旋转一个 BoundaryMeshRegion：
br = BoundaryMeshRegion[{{2, 1}, {6, -1}, {4, 1}, {6, 3}}, Line[{1, 2, 3, 4, 1}]]ℛ = TransformedRegion[br, RotationTransform[Pi / 3, {1, 1}]];经过变换的区域仍然是 BoundaryMeshRegionQ：
{BoundaryMeshRegionQ[br], BoundaryMeshRegionQ[ℛ]}{RegionMember[ℛ, {2, 3}], RegionMember[ℛ, {1, 3}]}计算 RegionCentroid：
c = RegionCentroid[ℛ]RegionPlot[ℛ, Epilog -> {Red, Point[c]}]计算 Area：
Area[ℛ]Integrate[x y, {x, y}∈ℛ]错切一个 MeshRegion：
mr = MeshRegion[{{0, 0, 0}, {1, 0, 0}, {1, 1, 0}, {0, 1, 0}, {0, 0, 1}, {1, 0, 1}, {1, 1, 1}, {0, 1, 1}, {0.5, 0.5, 1.5}, {1.5, 0, 0}, {1.5, 1, 0}, {1, -0.5, 0}}, Hexahedron[{1, 2, 3, 4, 5, 6, 7, 8}]]ℛ = TransformedRegion[mr, ShearingTransform[30Degree, {1, 0, 0}, {0, 1, 0}]]经过变换的区域仍然是 MeshRegionQ：
{MeshRegionQ[mr], MeshRegionQ[ℛ]}计算 Volume：
Volume[ℛ]计算 RegionBounds：
RegionBounds[ℛ]到一个点的 RegionDistance：
RegionDistance[ℛ, {1, -1, 1}]RegionDistance[ℛ, {1, 1, 1}]在区域上 Integrate：
Integrate[x y z, {x, y, z}∈ℛ]缩放一个低维 MeshRegion:
mr = MeshRegion[{{0, 0}, {1, 0}, {2, 1 / 2}, {2, -1 / 2}}, {Line[{1, 2}], Line[{2, 3}], Line[{3, 4}], Line[{4, 2}]}]ℛ = TransformedRegion[mr, ScalingTransform[2, {1, 1}]]计算 ArcLength：
{ArcLength[mr], ArcLength[ℛ]}pt = {1.5, 1.5};
npt = RegionNearest[ℛ, pt]RegionPlot[ℛ, Epilog -> {PointSize[Large], {Red, Point[pt]}, {Blue, Point[npt]}, {Dashed, Line[{pt, npt}]}}]衍生区域 (5)
变换一个 TransformedRegion：
tr = TransformedRegion[Cuboid[], ShearingTransform[30Degree, {1, 0, 0}, {0, 1, 1}]];Region[tr]ℛ = TransformedRegion[tr, ReflectionTransform[{0, 1, 1}]];Region[ℛ]计算 Volume：
Volume[ℛ]在区域上 Integrate：
Integrate[x + y z, {x, y, z}∈ℛ]MinValue[{x + y z, {x, y, z}∈ℛ}, {x, y, z}]Reduce[x + y z == 3 / 2 && y == x z && {x, y, z}∈ℛ, {x, y, z}]变换一个 RegionDifference：
rd = RegionDifference[Rectangle[{0, 0}], Disk[{1, 0}, 1 / 2]];Region[rd]ℛ = TransformedRegion[rd, RotationTransform[π / 4]];Region[ℛ]计算 Area：
Area[ℛ]计算 RegionBounds：
RegionBounds[ℛ]到一个点的 SignedRegionDistance：
SignedRegionDistance[ℛ, {-1 / 4, 1}]SignedRegionDistance[ℛ, {2, -3}]变换一个 RegionBoundary：
rb = RegionBoundary[Polygon[{{0, 0}, {3, -1}, {2, 0}, {3, 1}}]];Region[rb]ℛ = TransformedRegion[rb, ScalingTransform[2, {1, 1}]];计算 ArcLength：
ArcLength[ℛ]pt = {3 / 2, 2};
npt = RegionNearest[ℛ, pt]RegionPlot[ℛ, Epilog -> {PointSize[Large], {Red, Point[pt]}, {Blue, Point[npt]}, {Dashed, Line[{pt, npt}]}}]变换一个 RegionProduct:
rp = RegionProduct[Disk[{0, 0}, 1], Line[{{0}, {1}}]];Region[rp]ℛ = TransformedRegion[rp, ReflectionTransform[{1, 0, 1}]];Region[ℛ]计算 Volume：
Volume[ℛ]计算 RegionBounds：
RegionBounds[ℛ]到一个点的 RegionDistance：
RegionDistance[ℛ, {0, 0, 0}]RegionDistance[ℛ, {1, 1, 1}]通过非线性变换 
变换一个 RegionUnion：
ru = RegionUnion[Disk[{0, 0}, 2], Disk[{3, 0}, 2]];Region[ru]r1 = TransformedRegion[ru, Function[p, {p[[1]] p[[2]], p[[1]] + p[[2]]}]];Region[r1]用 Indexed 表达同样的点映射：
r2 = TransformedRegion[ru, {Indexed[#, 1] Indexed[#, 2], Indexed[#, 1] + Indexed[#, 2]}&];Region[r2]计算 RegionBounds：
RegionBounds[r2]CSG 区域 (1)
转换二维 CSGRegion：
TransformedRegion[CSGRegion["Difference", {Disk[], Disk[{0, 0}, 1 / 2]}], ScalingTransform[{3, 1}]]TransformedRegion[CSGRegion["Difference", {Ball[], Cuboid[{-2, -2, 0}, {2, 2, -2}]}], ScalingTransform[{3, 1, 1}]]细分区域 (1)
转换二维 SubdivisionRegion：
TransformedRegion[SubdivisionRegion[Rectangle[]], ScalingTransform[{3, 1}]]TransformedRegion[SubdivisionRegion[Cube[]], ScalingTransform[{3, 1, 1}]]应用 (2)
{p0, p1, p2} = {{1, 1}, {2, 2}, {3, 1}};
![A=TemplateBox[{{{, {{{p, _, 1}, -, {p, _, 0}}, ,, ..., ,, {{p, _, 3}, -, {p, _, 0}}}, }}}, Transpose]](Files/TransformedRegion.zh/2.png "A=TemplateBox[{{{, {{{p, _, 1}, -, {p, _, 0}}, ,, ..., ,, {{p, _, 3}, -, {p, _, 0}}}, }}}, Transpose]"){kind=small}
A = {p1 - p0, p2 - p0};t = AffineTransform[{A, p0}];Subscript[ℛ, 1] = Triangle[{p0, p1, p2}];
Subscript[ℛ, 2] = Triangle[{{0, 0}, {1, 0}, {0, 1}}];{Region[Subscript[ℛ, 1], Frame -> True],
Region[TransformedRegion[Subscript[ℛ, 2], t], Frame -> True]}找到中心为 
的单位 Cuboid 的透视变换：
tr = LinearFractionalTransform[{{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {-(1/2), 0, 0, 1}}];ℛ = TransformedRegion[Cuboid[], tr]RegionPlot3D[Evaluate@RegionMember[ℛ, {x, y, z}], {x, -2, 2}, {y, -0.5, 2.5}, {z, -0.5, 2.5}, Mesh -> None, PlotPoints -> 100, PlotStyle -> Opacity[0.3, Yellow]]计算 Volume：
Volume[ℛ]
Wolfram Research (2014)，TransformedRegion，Wolfram 语言函数，https://reference.wolfram.com/language/ref/TransformedRegion.html.
文本
Wolfram Research (2014)，TransformedRegion，Wolfram 语言函数，https://reference.wolfram.com/language/ref/TransformedRegion.html.
CMS
Wolfram 语言. 2014. "TransformedRegion." Wolfram 语言与系统参考资料中心. Wolfram Research. https://reference.wolfram.com/language/ref/TransformedRegion.html.
APA
Wolfram 语言. (2014). TransformedRegion. Wolfram 语言与系统参考资料中心. 追溯自 https://reference.wolfram.com/language/ref/TransformedRegion.html 年