RegionFarthestDistance

RegionFarthestDistance[reg1,reg2]

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

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Basic Examples  (3)

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

Plot the distance as a function of position:

Find the farthest distance between a disk and rectangle:

Show the regions:

Find the farthest distance between two meshes:

Show the meshes:

Scope  (16)

Special Regions  (8)

Points:

RegionFarthestDistance accepts coordinate lists:

Lines:

Polygons:

Simplices:

Boxes:

Balls:

Spheres:

Regions in n:

Formula Regions  (2)

The distance to a disk represented as an ImplicitRegion:

A cylinder:

The distance to a disk represented as a ParametricRegion:

Using a rational parametrization of the disk:

A cylinder:

Mesh Regions  (1)

The farthest distance between two 1D meshes:

2D:

3D:

Derived Regions  (2)

The distance to a RegionIntersection:

The distance to a TransformedRegion:

Geographic Regions  (3)

The distance between polygons with GeoPosition:

The distance between polygons with GeoGridPosition:

RegionFarthestDistance works on polygons with geographic entities:

Options  (2)

WorkingPrecision  (2)

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

Compute the distance using machine arithmetic:

In some cases, the exact answer cannot be computed:

Find the RegionFarthestDistance using 30 digits of precision:

Properties & Relations  (4)

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

The farthest distance to an EmptyRegion is negative Infinity:

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

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

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

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.

CMS

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

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: 30-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: 30-July-2025 ]}