# RegionHausdorffDistance

RegionHausdorffDistance[reg1,reg2]

gives the Hausdorff distance between the regions reg1 and reg2.

# Details and Options • RegionHausdorffDistance is also known as Hausdorff metric and PompeiuHausdorff distance.
• The Hausdorff distance measures how different two regions are from each other.
• RegionHausdorffDistance is the greatest of all distances from a point in one region to the closest point in the other region.
• • The distance between points p and q is taken to be Norm[p-q].
• RegionHausdorffDistance is effectively given by the maximum of MaxValue[MinValue[Norm[p-q],qreg2],preg1] and MaxValue[MinValue[Norm[p-q],qreg1],preg2].
• Unless the regions are closed, the Hausdorff distance may not be attained by points in the region but in the closure of the regions.
• The Hausdorff distance between two regions reg1 and reg2 is ϵ if reg1RegionDilation[reg2,ϵ] and reg2RegionDilation[reg1,ϵ].

# Examples

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## Basic Examples(4)

Find the Hausdorff distance between two disks:

Show the disks:

Find the Hausdorff distance between the unit disk and a regular polygon:

Show the regions:

Plot the Hausdorff distance as the number of sides increases:

Find the Hausdorff distance between two point clouds:

Find the Hausdorff distance between a MeshRegion and its convex hull:

Show the hull:

## Scope(9)

### Special Regions(8)

Points:

RegionHausdorffDistance accepts coordinate lists:

Lines:

Polygons:

Simplices:

Boxes:

Balls:

Spheres:

Regions in n:

### Mesh Regions(1)

The Hausdorff distance between two 1D meshes:

2D:

3D:

## Options(2)

### WorkingPrecision(2)

RegionHausdorffDistance 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 RegionHausdorffDistance using 30 digits of precision:

## Applications(1)

### Region Reconstruction(1)

Reconstruct a mesh from random point samples:

Determine the Hausdorff distance between the ground truth and reconstructed mesh:

Plot how the Hausdorff distance decreases with larger sample counts:

## Properties & Relations(4)

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

The Hausdorff distance between two point sets is the maximum EuclideanDistance from any point to the other set:

The Hausdorff distance between a region and a single point is the furthest distance of the point to any point in the region:

RegionDistance can be used to find the nearest distance from a point to a region: