# 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],{preg1,qreg2}].
• 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)

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_2023_regionfarthestdistance, author="Wolfram Research", title="{RegionFarthestDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RegionFarthestDistance.html}", note=[Accessed: 04-October-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_regionfarthestdistance, organization={Wolfram Research}, title={RegionFarthestDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/RegionFarthestDistance.html}, note=[Accessed: 04-October-2023 ]}