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RegionDilation[reg,r]

gives the dilation of the region reg by a disk of radius r centered at the origin.

RegionDilation[reg1,reg2]

gives the dilation of the region reg1 by the region reg2.

RegionDilation[reg1,reg2,{u,v}]

gives the dilation of reg1 scaled by a factor u and reg2 scaled by a factor v.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Special Regions  
Formula Regions  
Mesh Regions  
Derived Regions  
Possible Issues  
See Also
Related Guides
History
Cite this Page

RegionDilation

RegionDilation[reg,r]

gives the dilation of the region reg by a disk of radius r centered at the origin.

RegionDilation[reg1,reg2]

gives the dilation of the region reg1 by the region reg2.

RegionDilation[reg1,reg2,{u,v}]

gives the dilation of reg1 scaled by a factor u and reg2 scaled by a factor v.

Details and Options

  • RegionDilation is also known as Minkowski sum, geometric sum or offset.
  • Typically used to construct the union of positions of a region through translation.
  • RegionDilation[reg,r] gives the dilation of the region reg by a disk of radius r.
  • RegionDilation[reg1,reg2] returns the region reg1⊕reg2={a+b|a∈reg1∧b∈reg2}.
  • RegionDilation[reg1,reg2,{u,v}] returns the region u reg1⊕v reg2={u a+v b|a∈reg1∧b∈reg2}.
  • RegionDilation takes the same options as Region.

Examples

open all close all

Basic Examples  (4)

Dilation of a disk:

Wolfram Language code: RegionDilation[Disk[], 1 / 2]

Dilation of Triangle by Disk:

Wolfram Language code: Region[RegionDilation[Triangle[], Disk[]]]

Dilation of Rectangle by Triangle:

Wolfram Language code: RegionDilation[Rectangle[], Triangle[]]

Visualize it:

Wolfram Language code: Region[%]

Dilation of two regions:

Wolfram Language code: RegionDilation[[image], [image]]

Visualize it:

Wolfram Language code: Region[%]

Scope  (8)

Special Regions  (4)

Dilation of two lines:

Wolfram Language code: RegionDilation[Line[{{0, 0}, {1, 1}}], Line[{{1, 1}, {2, 1}}]]

Visualize it:

Wolfram Language code: Region[%]

Dilation of two disks:

Wolfram Language code: RegionDilation[Disk[{0, 0}, 1], Disk[{1, 0}, 1]];
Wolfram Language code: Region[%]

Dilation of a rectangle and a disk:

Wolfram Language code: RegionDilation[Region[Rectangle[]], Disk[]]

A disk and a circle:

Wolfram Language code: ℛ = RegionDilation[Disk[{0, 0}, 1], Circle[{1, 0}, 1]];

Visualize it:

Wolfram Language code: Region[ℛ]

Dilation of two polygons:

Wolfram Language code: Subscript[ℛ, 1] = Polygon[{{0, 0}, {3, -1}, {2, 0}, {3, 1}}]; Subscript[ℛ, 2] = Polygon[{{5, 0}, {2, 1}, {3, 0}, {2, -1}}];
Wolfram Language code: ℛ = RegionDilation[Subscript[ℛ, 1], Subscript[ℛ, 2]];

Visualize it:

Wolfram Language code: Region[ℛ]

Formula Regions  (1)

RegionDilation of ImplicitRegion objects:

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[x ≤ -1, {x}]; Subscript[ℛ, 2] = ImplicitRegion[x ≥ 1, {x}];
Wolfram Language code: RegionDilation[Subscript[ℛ, 1], Subscript[ℛ, 2]]

2D:

Wolfram Language code: Subscript[ℛ, 1] = ImplicitRegion[x^2 + y^2 ≤ 1, {x, y}]; Subscript[ℛ, 2] = ImplicitRegion[x^2 + (y - 1)^2 ≤ 1, {x, y}];
Wolfram Language code: RegionDilation[Subscript[ℛ, 1], Subscript[ℛ, 2]]
Wolfram Language code: DiscretizeRegion[%]

Mesh Regions  (1)

Dilation of two BoundaryMeshRegion objects:

Wolfram Language code: RegionDilation[[image], [image]];
Wolfram Language code: Region[%]

In 3D:

Wolfram Language code: RegionDilation[[image], [image]];
Wolfram Language code: Region[%]

Derived Regions  (2)

Dilation of two BooleanRegion objects:

Wolfram Language code: Subscript[ℛ, 1] = BooleanRegion[Or, {Triangle[{{0, 0}, {2, 3}, {-2, 3}}], Triangle[{{0, 2}, {2, -1}, {-2, -1}}]}]; Subscript[ℛ, 2] = BooleanRegion[And, {Triangle[{{0, 0}, {2, 3}, {-2, 3}}], Triangle[{{0, 2}, {2, -1}, {-2, 2}}]}];
Wolfram Language code: ℛ = RegionDilation[Subscript[ℛ, 1], Subscript[ℛ, 2]];

Visualize it:

Wolfram Language code: Region[ℛ]

Dilation of two TransformedRegion objects:

Wolfram Language code: Subscript[ℛ, 1] = TransformedRegion[Cuboid[], RotationTransform[Pi / 8, {1, 0, 0}]]; Subscript[ℛ, 2] = TransformedRegion[Cuboid[], RotationTransform[Pi / 8, {0, 1, 0}]];
Wolfram Language code: ℛ = RegionDilation[Subscript[ℛ, 1], Subscript[ℛ, 2]];

Visualize it:

Wolfram Language code: Region[ℛ]

Possible Issues  (1)

RegionDilation gives an approximate result when the exact result cannot be computed:

Wolfram Language code: RegionDilation[Polygon[{{0, 0}, {1, 0}, {1, 1}}], Disk[]]

Turn on the message:

Wolfram Language code: On[General::approx]
Wolfram Language code: RegionDilation[Polygon[{{0, 0}, {1, 0}, {1, 1}}], Disk[]]
Wolfram Research (2021), RegionDilation, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDilation.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_regiondilation, author="Wolfram Research", title="{RegionDilation}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDilation.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_regiondilation, organization={Wolfram Research}, title={RegionDilation}, year={2021}, url={https://reference.wolfram.com/language/ref/RegionDilation.html}, note=[Accessed: 12-June-2026]}