# RegionDistance

RegionDistance[reg,p]

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

RegionDistance[reg]

gives a that can be applied repeatedly to different points.

# Details • RegionDistance is also known as point-to-region distance, distance transform, distance field and minimum region distance.
• • The distance between points q and p is taken to be Norm[p-q].
• Region distance is effectively given by MinValue[{Norm[p-q],qreg},q].
• RegionDistance can be used with symbolic regions and points in GeometricScene.

# Examples

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## 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:

Introduced in 2014
(10.0)