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FindRegionTransform[reg1,reg2]

gives the transformation function that maps the region reg1 to the region reg2.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Basic Uses  
Special Regions  
Mesh Regions  
Applications  
Basic Applications  
See Also
Related Guides
History
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FindRegionTransform

FindRegionTransform[reg1,reg2]

gives the transformation function that maps the region reg1 to the region reg2.

Details and Options

  • FindRegionTransform is typically used to get transformations between regions that preserve geometric properties such as lines and parallelism.
  • FindRegionTransform[reg1,reg2] gives the affine transformation that maps reg1 to reg2.
  • The following option can be given:
  • WorkingPrecisionAutomaticthe precision to use in internal computations

Examples

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Basic Examples  (2)

Determine the transformation between two regions:

Wolfram Language code: tf = FindRegionTransform[Rectangle[], Parallelogram[]]

Apply the transformation:

Wolfram Language code: tf[Rectangle[]]

Find an affine transformation that aligns two sets of positions:

Wolfram Language code: FindRegionTransform[{{0., 0.}, {0., 1.}}, {{0, 0}, {-0.5, 0.5}}]

Scope  (11)

Basic Uses  (4)

Find an affine transformation between two regions:

Wolfram Language code: FindRegionTransform[Rectangle[], Parallelogram[]]

An affine transformation on a random sample of points:

Wolfram Language code: p = RandomReal[1, {10, 2}]; q = RandomSample[p, 10];
Wolfram Language code: FindRegionTransform[p, q]

FindRegionTransform works in:

2D:

Wolfram Language code: FindRegionTransform[Rectangle[], Parallelogram[]]

3D:

Wolfram Language code: FindRegionTransform[Cuboid[], Cube[]]

nD:

Wolfram Language code: FindRegionTransform[Ball[{0, 0, 0, 0, 0}], Ellipsoid[{0, 0, 0, 0, 0}, {1, 2, 3, 4, 5}]]

FindRegionTransform works with nonconstant regions:

Wolfram Language code: FindRegionTransform[Point[{a, 1}], Point[{2, 0}]]

Special Regions  (4)

A transformation between a rectangle and a parallelogram:

Wolfram Language code: FindRegionTransform[Rectangle[], Parallelogram[]]

Triangles and polygons:

Wolfram Language code: FindRegionTransform[Triangle[], Polygon[{{0, 0}, {0, 1}, {1 / 2, 1 / 2}}]]

Balls and ellipsoids:

Wolfram Language code: FindRegionTransform[Ball[{1, 1}], Ellipsoid[{0, 0}, {3, 2}]]

Polygons:

Wolfram Language code: FindRegionTransform[Polygon[{{-Sqrt[5/8 + Sqrt[5]/8], (-1 + Sqrt[5])/4}, {-Sqrt[5/8 - Sqrt[5]/8], (-1 - Sqrt[5])/4}, {0, 1}, {Sqrt[5/8 - Sqrt[5]/8], (-1 - Sqrt[5])/4}, {Sqrt[5/8 + Sqrt[5]/8], (-1 + Sqrt[5])/4}}, {1, 2, 4, 5, 3}], Polygon[{{1 - Sqrt[(5/8 + Sqrt[5]/8)/2] + (-1/4*1/Sqrt[2])*(-1 + Sqrt[5]), 2 - Sqrt[(5/8 + Sqrt[5]/8)/2] + (1/(4*Sqrt[2]))*(-1 + Sqrt[5])}, {1 + (-1/4*1/Sqrt[2])*(-1 - Sqrt[5]) - Sqrt[(5/8 - Sqrt[5]/8)/2], 2 + (1/(4*Sqrt[2]))*(-1 - Sqrt[ ... [(5/8 - Sqrt[5]/8)/2], 2 + (1/(4*Sqrt[2]))*(-1 - Sqrt[5]) + Sqrt[(5/8 - Sqrt[5]/8)/2]}, {1 + Sqrt[(5/8 + Sqrt[5]/8)/2] + (-1/4*1/Sqrt[2])*(-1 + Sqrt[5]), 2 + Sqrt[(5/8 + Sqrt[5]/8)/2] + (1/(4*Sqrt[2]))*(-1 + Sqrt[5])}}, {1, 2, 4, 5, 3}]]

Mesh Regions  (3)

A transformation between mesh regions:

Wolfram Language code: FindRegionTransform[MeshRegion[{{0, 0}, {0, 1}, {1, 0}, {1, 1}}, Polygon[{1, 3, 4, 2}]], MeshRegion[{{0, 0}, {1, 0}, {1, 1}, {2, 1}}, Polygon[{1, 2, 4, 3}]]]

A transformation between boundary mesh regions:

Wolfram Language code: FindRegionTransform[BoundaryMeshRegion[{{0, 0}, {0, 1}, {1, 0}, {1, 1}}, Line[{1, 3, 4, 2, 1}]], BoundaryMeshRegion[{{0, 0}, {1, 0}, {1, 1}, {2, 1}}, Line[{1, 2, 4, 3, 1}]]]

A transformation between mesh regions and boundary mesh regions:

Wolfram Language code: FindRegionTransform[MeshRegion[{{0, 0}, {0, 1}, {1, 0}, {1, 1}}, Polygon[{1, 3, 4, 2}]], BoundaryMeshRegion[{{0, 0}, {1, 0}, {1, 1}, {2, 1}}, Line[{1, 2, 4, 3, 1}]]]

Applications  (1)

Basic Applications  (1)

FindRegionTransform finds affine transformations:

Wolfram Language code: p = Rectangle[];

Translation:

Wolfram Language code: q = Translate[p, {1, 1}];
Wolfram Language code: FindRegionTransform[p, q]

Rotation:

Wolfram Language code: q = Rotate[p, Pi / 2];
Wolfram Language code: FindRegionTransform[p, q]

Scaling:

Wolfram Language code: q = Scale[p, {2, 2}];
Wolfram Language code: FindRegionTransform[p, q]
Wolfram Research (2021), FindRegionTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/FindRegionTransform.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_findregiontransform, author="Wolfram Research", title="{FindRegionTransform}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/FindRegionTransform.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_findregiontransform, organization={Wolfram Research}, title={FindRegionTransform}, year={2021}, url={https://reference.wolfram.com/language/ref/FindRegionTransform.html}, note=[Accessed: 12-June-2026]}