# RegionDistance

RegionDistance[reg,p]

gives the minimum distance from the point p to the region reg.

RegionDistance[reg1,reg2]

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

RegionDistance[reg]

gives a that can be applied repeatedly to different points.

# Details and Options

• RegionDistance is also known as point-to-region distance, distance transform, distance field and minimum region distance.
• RegionDistance measures how close two regions are to each other.
• The distance between points q and p is taken to be Norm[p-q].
• RegionDistance[reg,p] is effectively given by MinValue[{Norm[p-q],qreg},q].
• RegionDistance[reg1,reg2] is effectively given by MinValue[Norm[p-q],{preg1,qreg2}].
• Unless the regions are closed, the nearest distance may not be attained by points in the region but in the closure of the regions.

# Examples

open allclose all

## Basic Examples(2)

Find the distance from a point to the unit disk:

Plot the distance as a function of position:

Find the distance from a point to a MeshRegion:

With one argument, you get a RegionDistanceFunction:

Apply the distance function lists of points to compute many distances:

## Scope(15)

### Special Regions(9)

Points:

Plot the distance from a three-point set:

Lines:

Plot it:

Boxes:

Plot the distance to a rectangle:

Simplices:

Plot the distance to a 2D simplex:

Balls:

Plot the distance to a disk:

Ellipsoids:

Plot the distance to a 2D ellipsoid:

Circles and ellipses:

Plot the distance to a circle:

Cylinders and cones:

Equidistance contours for a cylinder:

Regions in :

### 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 distance to a 1D BoundaryMeshRegion:

2D:

3D:

### Derived Regions(3)

The distance to a RegionIntersection:

The distance to a TransformedRegion:

The distance to a RegionBoundary:

## Applications(5)

Compute the height of a triangle:

Visualize the triangle:

Create a region that is a distance from a circle:

Visualize the region and circle:

Compute the dilation of a region:

The directed Hausdorff distance from region to is defined as . Use RegionDistance to compute the directed Hausdorff distance from =Triangle[{{0,0},{2,0},{0,1}}] to =Triangle[{{0,0},{1,0},{0,3/2}}]:

Find the nearest distance to any point in :

As expected, the distance is zero for points in that overlap with :

Find the directed Hausdorff distance by maximizing over :

If , you can conclude that where is the closure of and . Show that for the regions in this example:

The Hausdorff distance between region and is defined as where is the directed Hausdorff distance in the previous example. Use RegionDistance to compute the Hausdorff distance between =Triangle[{{0,0},{2,0},{0,1}}] and =Triangle[{{0,0},{1,0},{0,3/2}}]:

Find the directed Hausdorff distances:

If , you can conclude that and , as in the previous example. Show that and for the regions in this example:

## Properties & Relations(4)

A point is a RegionMember if the distance to the region is 0:

RegionDistance is the MinValue of the distance to any point in the region:

RegionNearest gives a point that is of minimal distance from the input:

For a point outside the region, RegionDistance and SignedRegionDistance are the same:

Wolfram Research (2014), RegionDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDistance.html (updated 2023).

#### Text

Wolfram Research (2014), RegionDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionDistance.html (updated 2023).

#### CMS

Wolfram Language. 2014. "RegionDistance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2023. https://reference.wolfram.com/language/ref/RegionDistance.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_regiondistance, author="Wolfram Research", title="{RegionDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RegionDistance.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_regiondistance, organization={Wolfram Research}, title={RegionDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/RegionDistance.html}, note=[Accessed: 14-September-2024 ]}