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 RegionDistanceFunction[…] 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],q∈reg},q].
RegionDistance[reg₁,reg₂] is effectively given by MinValue[Norm[p-q],{p∈reg₁,q∈reg₂}].
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

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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 ℬ=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 ℬ=Triangle[{{0,0},{1,0},{0,3/2}}]: