RegionFarthestDistance
✖
RegionFarthestDistance
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 allBasic Examples (3)Summary of the most common use cases
Find the farthest distance from a point to the unit disk:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cws0re

Plot the distance as a function of position:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-tikzl5

Find the farthest distance between a disk and rectangle:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-lsllwt


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-y9vdkh

Find the farthest distance between two meshes:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jjktji

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-rp1s0p


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-2c2ty6

Scope (16)Survey of the scope of standard use cases
Special Regions (8)

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-6o0jua

RegionFarthestDistance accepts coordinate lists:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-0ws9nd


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-7mi415


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jz43r0


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-00041e


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jdusf5


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-esyw4l


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-0vnjs9


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-flk5m8


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-xmflef


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-0e5fc5


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-37qor7


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-igxar6


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cfib7z


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-1kaba8


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-jnsvsc


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-5kmar3


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-42668q


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-etilu9


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bfevl7

Formula Regions (2)
The distance to a disk represented as an ImplicitRegion:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cit68

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-oe51yh


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-no49ng


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bx9cs6

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-e766wq

The distance to a disk represented as a ParametricRegion:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-movt8j

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-echu7c

Using a rational parametrization of the disk:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bf6y3z

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-hs0nt5


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-oiyrvh

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-mfnbaw

Mesh Regions (1)
The farthest distance between two 1D meshes:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-xgplru

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-u97jzi


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-73wvv8


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-868gxk

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cqafsq


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-vsr1m2


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-zzum8l

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-sjlvdn


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-qtc04f

Derived Regions (2)
The distance to a RegionIntersection:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-iri6pf

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-9e68l


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bvehq

The distance to a TransformedRegion:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-c2wj44

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-wbcwf


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-k1y8k2

Geographic Regions (3)
The distance between polygons with GeoPosition:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-6s6gsi

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-e2z3rb

The distance between polygons with GeoGridPosition:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-rxegj0

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-yaai8f

RegionFarthestDistance works on polygons with geographic entities:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-3xgy6m

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-pqo3bi

Options (2)Common values & functionality for each option
WorkingPrecision (2)
RegionFarthestDistance will try to compute the distance using the same precision as its inputs:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-c9oegt

Compute the distance using machine arithmetic:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-cuwkuv

In some cases, the exact answer cannot be computed:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-c62f2s

Find the RegionFarthestDistance using 30 digits of precision:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-tbnvx3

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:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-hwg5qt

The farthest distance to an EmptyRegion is negative Infinity:

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-4dx4hs

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

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-r36bz3


https://wolfram.com/xid/0foj2gl4cyzusnatpm6-bjlsyt

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

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-pkhc76

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-6ymd5k

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

https://wolfram.com/xid/0foj2gl4cyzusnatpm6-uq0mk1

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
]}
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
]}