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RegionFarthestDistance
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RegionFarthestDistance

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gives the farthest distance between points in the regions reg1 and reg2.

Details and Options

  • RegionFarthestDistance is also known as maximum region distance and spanning distance.
  • The farthest distance measures how far apart points of two regions are.
  • RegionFarthestDistance is the greatest of all distances from a point in one region to all points in the other region.
  • The distance between points p and q is taken to be Norm[p-q].
  • RegionFarthestDistance is effectively given by MaxValue[Norm[p-q],{p∈reg1,q∈reg2}].
  • Unless the regions are closed, the farthest distance may not be attained by points in the region but in the closure of the regions.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Find the farthest distance from a point to the unit disk:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cws0re
Out[1]=1

Plot the distance as a function of position:

In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-tikzl5
Out[2]=2

Find the farthest distance between a disk and rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-lsllwt
Out[1]=1

Show the regions:

In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-y9vdkh
Out[2]=2

Find the farthest distance between two meshes:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jjktji
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-rp1s0p
Out[2]=2

Show the meshes:

In[3]:=3
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-2c2ty6
Out[3]=3

Scope  (16)Survey of the scope of standard use cases

Special Regions  (8)

Points:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-6o0jua
Out[1]=1

RegionFarthestDistance accepts coordinate lists:

In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-0ws9nd
Out[2]=2

Lines:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-7mi415
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jz43r0
Out[2]=2

Polygons:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-00041e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jdusf5
Out[2]=2

Simplices:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-esyw4l
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-0vnjs9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-flk5m8
Out[3]=3

Boxes:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-xmflef
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-0e5fc5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-37qor7
Out[3]=3

Balls:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-igxar6
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cfib7z
Out[2]=2

Spheres:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-1kaba8
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jnsvsc
Out[2]=2

Regions in n:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-5kmar3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-42668q
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-etilu9
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bfevl7
Out[4]=4

Formula Regions  (2)

The distance to a disk represented as an ImplicitRegion:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cit68
In[14]:=14
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-oe51yh
Out[14]=14
In[15]:=15
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-no49ng
Out[15]=15

A cylinder:

In[18]:=18
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https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bx9cs6
In[19]:=19
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-e766wq
Out[19]=19

The distance to a disk represented as a ParametricRegion:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-movt8j
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-echu7c
Out[2]=2

Using a rational parametrization of the disk:

In[3]:=3
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bf6y3z
In[4]:=4
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-hs0nt5
Out[4]=4

A cylinder:

In[5]:=5
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-oiyrvh
In[6]:=6
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-mfnbaw
Out[6]=6

Mesh Regions  (1)

The farthest distance between two 1D meshes:

In[134]:=134
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-xgplru
In[136]:=136
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-u97jzi
Out[136]=136
In[137]:=137
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-73wvv8
Out[137]=137

2D:

In[153]:=153
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-868gxk
In[155]:=155
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cqafsq
Out[155]=155
In[156]:=156
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-vsr1m2
Out[156]=156

3D:

In[199]:=199
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-zzum8l
In[201]:=201
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-sjlvdn
Out[201]=201
In[202]:=202
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-qtc04f
Out[202]=202

Derived Regions  (2)

The distance to a RegionIntersection:

In[13]:=13
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-iri6pf
In[18]:=18
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-9e68l
Out[18]=18
In[19]:=19
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bvehq
Out[19]=19

The distance to a TransformedRegion:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-c2wj44
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-wbcwf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-k1y8k2
Out[3]=3

Geographic Regions  (3)

The distance between polygons with GeoPosition:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-6s6gsi
In[15]:=15
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-e2z3rb
Out[15]=15

The distance between polygons with GeoGridPosition:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-rxegj0
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-yaai8f
Out[2]=2

RegionFarthestDistance works on polygons with geographic entities:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-3xgy6m
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-pqo3bi
Out[2]=2

Options  (2)Common values & functionality for each option

WorkingPrecision  (2)

RegionFarthestDistance will try to compute the distance using the same precision as its inputs:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-c9oegt
Out[1]=1

Compute the distance using machine arithmetic:

In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cuwkuv
Out[2]=2

In some cases, the exact answer cannot be computed:

In[4]:=4
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-c62f2s
Out[4]=4

Find the RegionFarthestDistance using 30 digits of precision:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-tbnvx3
Out[1]=1

Properties & Relations  (4)Properties of the function, and connections to other functions

The farthest distance between a FullRegion and a non-empty region is Infinity:

In[10]:=10
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-hwg5qt
Out[10]=10

The farthest distance to an EmptyRegion is negative Infinity:

In[9]:=9
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-4dx4hs
Out[9]=9

The farthest distance between two points is equivalent to the EuclideanDistance:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-r36bz3
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bjlsyt
Out[2]=2

The farthest distance between two point sets is the maximum EuclideanDistance between any two points:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-pkhc76
In[2]:=2
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-6ymd5k
Out[2]=2

The farthest distance between a region and a single point is equivalent to the RegionHausdorffDistance:

In[1]:=1
✖
https://wolfram.com/xid/0foj2gl4cyzusnatpm6-uq0mk1
Out[1]=1
Wolfram Research (2023), RegionFarthestDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionFarthestDistance.html.
✖
Wolfram Research (2023), RegionFarthestDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionFarthestDistance.html.

Text

Wolfram Research (2023), RegionFarthestDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionFarthestDistance.html.

✖
Wolfram Research (2023), RegionFarthestDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionFarthestDistance.html.

CMS

Wolfram Language. 2023. "RegionFarthestDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionFarthestDistance.html.

✖
Wolfram Language. 2023. "RegionFarthestDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionFarthestDistance.html.

APA

Wolfram Language. (2023). RegionFarthestDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionFarthestDistance.html

✖
Wolfram Language. (2023). RegionFarthestDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionFarthestDistance.html

BibTeX

@misc{reference.wolfram_2025_regionfarthestdistance, author="Wolfram Research", title="{RegionFarthestDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RegionFarthestDistance.html}", note=[Accessed: 10-July-2025 ]}

✖
@misc{reference.wolfram_2025_regionfarthestdistance, author="Wolfram Research", title="{RegionFarthestDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RegionFarthestDistance.html}", note=[Accessed: 10-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regionfarthestdistance, organization={Wolfram Research}, title={RegionFarthestDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/RegionFarthestDistance.html}, note=[Accessed: 10-July-2025 ]}

✖
@online{reference.wolfram_2025_regionfarthestdistance, organization={Wolfram Research}, title={RegionFarthestDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/RegionFarthestDistance.html}, note=[Accessed: 10-July-2025 ]}