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RegionSymmetricDifference
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RegionSymmetricDifference

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RegionSymmetricDifference[reg1,reg2,…]

represents the symmetric difference of the regions reg1, reg2, ….

Details and Options

Examples

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Basic Examples  (2)Summary of the most common use cases

Symmetric difference of two disks:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-bwhjpw

Visualize it:

In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-je56mi
Out[2]=2

Symmetric difference of two MeshRegion objects:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-hy7mhg
Out[1]=1

Scope  (10)Survey of the scope of standard use cases

Special Regions  (4)

A symmetric difference of Line regions:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-cl5o55

Visualize it:

In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-kq6iqq
Out[2]=2

A symmetric difference of Polygon regions:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-h9631g
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-b7dsmw

Visualize it:

In[3]:=3
✖
https://wolfram.com/xid/0ywfor7dx9zm-unxw
Out[3]=3

A symmetric difference of two Cuboid regions:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-htpsnj
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-6e99sp
Out[2]=2

A symmetric difference of regions with different RegionDimension:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-dq7yx6

Visualize it:

In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-of250
Out[2]=2

Formula Regions  (2)

A symmetric difference of ImplicitRegion objects is an ImplicitRegion:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-crtuk1
In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-dx24rd
Out[3]=3

2D:

In[4]:=4
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https://wolfram.com/xid/0ywfor7dx9zm-gc5987
Out[6]=6

3D:

In[7]:=7
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https://wolfram.com/xid/0ywfor7dx9zm-daco4v
In[9]:=9
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https://wolfram.com/xid/0ywfor7dx9zm-bqtowt
Out[9]=9

nD:

In[10]:=10
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https://wolfram.com/xid/0ywfor7dx9zm-bdn04e
In[12]:=12
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https://wolfram.com/xid/0ywfor7dx9zm-cjj2u4
Out[12]=12

A symmetric difference of ParametricRegion objects:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-dgeugm
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-2azv

Visualize it:

In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-drxtd
Out[3]=3

Mesh Regions  (2)

A symmetric difference of BoundaryMeshRegion objects is a BoundaryMeshRegion:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-bep0zh
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-hel92s
Out[2]=2

2D:

In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-ndd1y3
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ywfor7dx9zm-dque2
Out[4]=4

3D:

In[5]:=5
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https://wolfram.com/xid/0ywfor7dx9zm-e0i5c3
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0ywfor7dx9zm-egkwa
Out[6]=6

A symmetric difference of full-dimensional MeshRegion objects is a MeshRegion:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-bnl5n5
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-d9p223
Out[2]=2

2D:

In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-b2vn5v
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ywfor7dx9zm-btndg3
Out[4]=4

3D:

In[5]:=5
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https://wolfram.com/xid/0ywfor7dx9zm-jhjsx
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0ywfor7dx9zm-qpxfs
Out[6]=6

Derived Regions  (2)

A symmetric difference of BooleanRegion objects:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-fzcaz8
In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-zkd1a

Visualize it:

In[4]:=4
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https://wolfram.com/xid/0ywfor7dx9zm-b3gvca
Out[4]=4

A symmetric difference of TransformedRegion objects:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-fdpyhr
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-c7maz6

Visualize it:

In[3]:=3
✖
https://wolfram.com/xid/0ywfor7dx9zm-c2ee5
Out[3]=3

Applications  (1)Sample problems that can be solved with this function

Symmetric difference of regions:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-h1ry56
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-7tcgzk
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

A point p belongs to RegionSymmetricDifference[reg1,reg2] if it belongs to an odd number of regi:

In[9]:=9
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https://wolfram.com/xid/0ywfor7dx9zm-lefxss

Use RegionMember to test membership:

In[12]:=12
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https://wolfram.com/xid/0ywfor7dx9zm-i0bat4
In[13]:=13
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https://wolfram.com/xid/0ywfor7dx9zm-s0g66
Out[13]=13

RegionSymmetricDifference is a Boolean combination Xor of regions:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-iwlkw2
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-d9lq0
Out[2]=2

RegionSymmetricDifference can be found using RegionUnion and RegionDifference:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-g7if4i
In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-k1u6e
Out[2]=2

The RegionDimension of a symmetric difference is at most the max of the input dimensions:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-i23p5l
In[2]:=2
✖
https://wolfram.com/xid/0ywfor7dx9zm-b5bikm
Out[2]=2

It can be less, however:

In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-f2sfd

This symmetric difference is two lines, and thus has dimension 1:

In[4]:=4
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https://wolfram.com/xid/0ywfor7dx9zm-ingjhp
Out[4]=4

The RegionMeasure of a symmetric difference obeys a simple formula:

In[1]:=1
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https://wolfram.com/xid/0ywfor7dx9zm-cdmjc5

Simply subtract twice the measure of the intersection from the sum of the input measures:

In[2]:=2
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https://wolfram.com/xid/0ywfor7dx9zm-ntujg
Out[2]=2

Possible Issues  (1)Common pitfalls and unexpected behavior

Symmetric difference is defined only for regions with the same RegionEmbeddingDimension:

In[1]:=1
✖
https://wolfram.com/xid/0ywfor7dx9zm-gjw969
In[3]:=3
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https://wolfram.com/xid/0ywfor7dx9zm-6k4h4
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

The symmetric difference of two spiral polygons:

In[1]:=1
✖
https://wolfram.com/xid/0ywfor7dx9zm-bv5wci
Out[1]=1
Wolfram Research (2014), RegionSymmetricDifference, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html (updated 2017).
✖
Wolfram Research (2014), RegionSymmetricDifference, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html (updated 2017).

Text

Wolfram Research (2014), RegionSymmetricDifference, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html (updated 2017).

✖
Wolfram Research (2014), RegionSymmetricDifference, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html (updated 2017).

CMS

Wolfram Language. 2014. "RegionSymmetricDifference." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html.

✖
Wolfram Language. 2014. "RegionSymmetricDifference." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html.

APA

Wolfram Language. (2014). RegionSymmetricDifference. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html

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Wolfram Language. (2014). RegionSymmetricDifference. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html

BibTeX

@misc{reference.wolfram_2025_regionsymmetricdifference, author="Wolfram Research", title="{RegionSymmetricDifference}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html}", note=[Accessed: 20-May-2025 ]}

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@misc{reference.wolfram_2025_regionsymmetricdifference, author="Wolfram Research", title="{RegionSymmetricDifference}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html}", note=[Accessed: 20-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regionsymmetricdifference, organization={Wolfram Research}, title={RegionSymmetricDifference}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html}, note=[Accessed: 20-May-2025 ]}

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@online{reference.wolfram_2025_regionsymmetricdifference, organization={Wolfram Research}, title={RegionSymmetricDifference}, year={2017}, url={https://reference.wolfram.com/language/ref/RegionSymmetricDifference.html}, note=[Accessed: 20-May-2025 ]}