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RegionSimilar[reg1,reg2]

tests whether the regions reg1 and reg2 are similar.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Special Regions  
Mesh Regions  
Derived Regions  
Applications  
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RegionSimilar

RegionSimilar[reg1,reg2]

tests whether the regions reg1 and reg2 are similar.

Details and Options

  • The regions reg1 and reg2 are similar if reg2 can be obtained from reg1 by a combination of translations, rotations, reflections and uniform scalings.
  • Typically used to test whether two regions have the same shape or whether one has the same shape as the mirror image of the other.
  • RegionSimilar[reg1,reg2] gives True if there is an affine transformation from reg1 to reg2 with matrix m satisfying TemplateBox[{m}, Transpose]=TemplateBox[{m}, Det] TemplateBox[{m}, Inverse].
  • Two regions reg1 and reg2 are similar if there is an affine transformation that maps reg1 into reg2 with a constant ratio between distance of two points and their images.

Examples

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Basic Examples  (1)

Check if two regions are similar:

Wolfram Language code: RegionSimilar[Rectangle[{1, 1}], Rectangle[{2, 2}, {4, 4}]]

Scope  (9)

Basic Uses  (2)

Test whether two regions are similar:

Wolfram Language code: Subscript[ℛ, 1] = Rectangle[{0, 0}, {2, 1}]; Subscript[ℛ, 2] = ImplicitRegion[0 ≤ x ≤ 2 && 0 ≤ y ≤ 1, {x, y}];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Show two regions are not similar:

Wolfram Language code: RegionSimilar[Disk[], Ellipsoid[{0, 0}, {3, 2}]]

Special Regions  (3)

RegionSimilar works in 1D:

Wolfram Language code: RegionSimilar[Ball[1], Interval[{-1, 1}]]

Line and Interval:

Wolfram Language code: RegionSimilar[Line[{{0}, {1}}], Interval[{0, 1}]]

RegionSimilar works in 2D:

Wolfram Language code: Subscript[ℛ, 1] = Point[Tuples[Range[3], 2]]; Subscript[ℛ, 2] = Point[Join@@Table[{i, j}, {j, 3}, {i, 3}]];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Disk and Ball:

Wolfram Language code: Subscript[ℛ, 1] = Disk[]; Subscript[ℛ, 2] = Ball[2];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Polygon:

Wolfram Language code: Subscript[ℛ, 1] = Polygon[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}]; Subscript[ℛ, 2] = Rectangle[];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

RegionSimilar works in 3D:

Wolfram Language code: Subscript[ℛ, 1] = Point[Tuples[Range[5], 3]]; Subscript[ℛ, 2] = Point[Flatten[Table[{i, j, k}, {j, 5}, {k, 5}, {i, 5}], 2]];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Mesh Regions  (3)

Test whether two mesh regions are similar:

Wolfram Language code: Subscript[ℛ, 1] = MeshRegion[{{0, 0}, {1, 0}, {1, 1}, {0, 1}}, Polygon[{1, 2, 3, 4}]]; Subscript[ℛ, 2] = TriangulateMesh[Subscript[ℛ, 1]];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Test whether two boundary mesh regions are similar:

Wolfram Language code: Subscript[ℛ, 1] = BoundaryMeshRegion[{{0}, {1}}, Point[{1, 2}]]; Subscript[ℛ, 2] = BoundaryMeshRegion[{{0}, {1}}, Point[{2, 1}]];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]
Wolfram Language code: Subscript[ℛ, 1] = BoundaryMeshRegion[{{0, 0}, {2, 0}, {2, 2}, {0, 2}}, Line[{1, 2, 3, 4, 1}]]; Subscript[ℛ, 2] = BoundaryMeshRegion[{{0, 0}, {1, 0}, {2, 0}, {2, 2}, {0, 2}}, Line[{1, 2, 3, 4, 5, 1}]];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Test whether a mesh region and a boundary mesh region are similar:

Wolfram Language code: Subscript[ℛ, 1] = DelaunayMesh[RandomReal[{-1, 1}, {25, 2}]]; Subscript[ℛ, 2] = BoundaryMesh[Subscript[ℛ, 1]];
Wolfram Language code: {Subscript[ℛ, 1], Subscript[ℛ, 2]}
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

In :

Wolfram Language code: Subscript[ℛ, 1] = DelaunayMesh[RandomReal[{-1, 1}, {25, 3}]]; Subscript[ℛ, 2] = BoundaryMesh[Subscript[ℛ, 1]];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Derived Regions  (1)

Test whether Boolean regions are similar:

Wolfram Language code: Subscript[ℛ, 1] = BooleanRegion[Xnor, {Disk[], Disk[{1, 0}]}]; Subscript[ℛ, 2] = BooleanRegion[(!#1 || #2) && (#1 || !#2)&, {Disk[], Disk[{1, 0}]}];
Wolfram Language code: RegionSimilar[Subscript[ℛ, 1], Subscript[ℛ, 2]]

Applications  (1)

Two polygons are similar if they have the same vertex angles:

Wolfram Language code: RegionSimilar[Polygon[{{0, 0}, {1, 0}, {1, 1}}], Polygon[{{0, 0}, {2, 0}, {2, 2}}]]
Wolfram Language code: PolygonAngle[Polygon[{{0, 0}, {1, 0}, {1, 1}}]] === PolygonAngle[Polygon[{{0, 0}, {2, 0}, {2, 2}}]]
Wolfram Research (2021), RegionSimilar, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionSimilar.html.

Text

Wolfram Research (2021), RegionSimilar, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionSimilar.html.

CMS

Wolfram Language. 2021. "RegionSimilar." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionSimilar.html.

APA

Wolfram Language. (2021). RegionSimilar. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionSimilar.html

BibTeX

@misc{reference.wolfram_2026_regionsimilar, author="Wolfram Research", title="{RegionSimilar}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/RegionSimilar.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_regionsimilar, organization={Wolfram Research}, title={RegionSimilar}, year={2021}, url={https://reference.wolfram.com/language/ref/RegionSimilar.html}, note=[Accessed: 12-June-2026]}