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RegionDilation
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RegionDilation

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

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RegionDilation[reg1,reg2]

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

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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 allclose all

Basic Examples  (4)Summary of the most common use cases

Dilation of a disk:

In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-7470ib
Out[2]=2

Dilation of Triangle by Disk:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-5sofxx
Out[1]=1

Dilation of Rectangle by Triangle:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-zfeyxd
Out[1]=1

Visualize it:

In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-fuvyxr
Out[2]=2

Dilation of two regions:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-i47t61
Out[1]=1

Visualize it:

In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-9btejp
Out[2]=2

Scope  (8)Survey of the scope of standard use cases

Special Regions  (4)

Dilation of two lines:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-1opzvd
Out[1]=1

Visualize it:

In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-igwg55
Out[2]=2

Dilation of two disks:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-hl5gpm
In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-tsw7u5
Out[2]=2

Dilation of a rectangle and a disk:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-dpwalb
Out[1]=1

A disk and a circle:

In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-dq7yx6

Visualize it:

In[3]:=3
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https://wolfram.com/xid/0g7ijzsssey-of250
Out[3]=3

Dilation of two polygons:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-h9631g
In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-b76l3e

Visualize it:

In[3]:=3
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https://wolfram.com/xid/0g7ijzsssey-unxw
Out[3]=3

Formula Regions  (1)

RegionDilation of ImplicitRegion objects:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-crtuk1
In[3]:=3
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https://wolfram.com/xid/0g7ijzsssey-p5g6yh
Out[3]=3

2D:

In[4]:=4
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https://wolfram.com/xid/0g7ijzsssey-gc5987
In[6]:=6
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https://wolfram.com/xid/0g7ijzsssey-fbwnu3
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0g7ijzsssey-3s8ugb
Out[7]=7

Mesh Regions  (1)

Dilation of two BoundaryMeshRegion objects:

In[10]:=10
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https://wolfram.com/xid/0g7ijzsssey-bep0zh
In[11]:=11
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https://wolfram.com/xid/0g7ijzsssey-evjawf
Out[11]=11

In 3D:

In[12]:=12
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https://wolfram.com/xid/0g7ijzsssey-3h9218
In[13]:=13
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https://wolfram.com/xid/0g7ijzsssey-5r3i0z
Out[13]=13

Derived Regions  (2)

Dilation of two BooleanRegion objects:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-fzcaz8
In[3]:=3
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https://wolfram.com/xid/0g7ijzsssey-d19z6l

Visualize it:

In[4]:=4
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https://wolfram.com/xid/0g7ijzsssey-b3gvca
Out[4]=4

Dilation of two TransformedRegion objects:

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-fdpyhr
In[2]:=2
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https://wolfram.com/xid/0g7ijzsssey-hj2g71

Visualize it:

In[3]:=3
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https://wolfram.com/xid/0g7ijzsssey-c2ee5
Out[3]=3

Possible Issues  (1)Common pitfalls and unexpected behavior

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

In[1]:=1
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https://wolfram.com/xid/0g7ijzsssey-qh1w39
Out[1]=1

Turn on the message:

In[5]:=5
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https://wolfram.com/xid/0g7ijzsssey-w3t8yg
In[3]:=3
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https://wolfram.com/xid/0g7ijzsssey-l6k8sb
Out[3]=3
Wolfram Research (2021), RegionDilation, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDilation.html.
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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.

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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.

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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

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

BibTeX

@misc{reference.wolfram_2025_regiondilation, author="Wolfram Research", title="{RegionDilation}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDilation.html}", note=[Accessed: 06-June-2025 ]}

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@misc{reference.wolfram_2025_regiondilation, author="Wolfram Research", title="{RegionDilation}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDilation.html}", note=[Accessed: 06-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regiondilation, organization={Wolfram Research}, title={RegionDilation}, year={2021}, url={https://reference.wolfram.com/language/ref/RegionDilation.html}, note=[Accessed: 06-June-2025 ]}

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@online{reference.wolfram_2025_regiondilation, organization={Wolfram Research}, title={RegionDilation}, year={2021}, url={https://reference.wolfram.com/language/ref/RegionDilation.html}, note=[Accessed: 06-June-2025 ]}