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RegionHausdorffDistance
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RegionHausdorffDistance

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gives the Hausdorff distance between the regions reg1 and reg2.

Details and Options

  • RegionHausdorffDistance is also known as Hausdorff metric and Pompeiu–Hausdorff 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],q∈reg2],p∈reg1] and MaxValue[MinValue[Norm[p-q],q∈reg1],p∈reg2].
  • 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 reg1⊆RegionDilation[reg2,ϵ] and reg2⊆RegionDilation[reg1,ϵ].

Examples

open allclose all

Basic Examples  (4)Summary of the most common use cases

Find the Hausdorff distance between two disks:

In[20]:=20
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-cws0re
Out[20]=20

Show the disks:

In[21]:=21
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-tikzl5
Out[21]=21

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

In[1]:=1
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-gjir76
Out[1]=1

Show the regions:

In[2]:=2
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-1czcc9
Out[2]=2

Plot the Hausdorff distance as the number of sides increases:

In[3]:=3
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-r89lnj
Out[3]=3

Find the Hausdorff distance between two point clouds:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-7eyj5s
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-8xo4vo
In[2]:=2
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-797pmr
Out[2]=2

Show the hull:

In[3]:=3
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-5oncx0
Out[3]=3

Scope  (9)Survey of the scope of standard use cases

Special Regions  (8)

Points:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-6o0jua
Out[1]=1

RegionHausdorffDistance accepts coordinate lists:

In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-0ws9nd
Out[2]=2

Lines:

In[1]:=1
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-7mi415
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-jz43r0
Out[2]=2

Polygons:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-00041e
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-jdusf5
Out[2]=2

Simplices:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-esyw4l
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-0vnjs9
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-flk5m8
Out[3]=3

Boxes:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-xmflef
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-0e5fc5
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-37qor7
Out[3]=3

Balls:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-igxar6
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-cfib7z
Out[2]=2

Spheres:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-1kaba8
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-jnsvsc
Out[2]=2

Regions in n:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-5kmar3
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-42668q
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-etilu9
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-bfevl7
Out[4]=4

Mesh Regions  (1)

The Hausdorff distance between two 1D meshes:

In[20]:=20
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-xgplru
In[10]:=10
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-73wvv8
Out[10]=10

2D:

In[190]:=190
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-868gxk
In[192]:=192
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-cqafsq
Out[192]=192
In[193]:=193
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-vsr1m2
Out[193]=193

3D:

In[195]:=195
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-zzum8l
In[207]:=207
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-sjlvdn
Out[207]=207
In[147]:=147
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-qtc04f
Out[147]=147

Options  (2)Common values & functionality for each option

WorkingPrecision  (2)

RegionHausdorffDistance will try to compute the distance using the same precision as its inputs:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-c9oegt
Out[1]=1

Compute the distance using machine arithmetic:

In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-cuwkuv
Out[2]=2

In some cases, the exact answer cannot be computed:

In[3]:=3
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-c62f2s
Out[3]=3

Find the RegionHausdorffDistance using 30 digits of precision:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-tbnvx3
Out[1]=1

Applications  (1)Sample problems that can be solved with this function

Region Reconstruction  (1)

Reconstruct a mesh from random point samples:

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-bu39zz

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

In[2]:=2
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-p996eq
Out[2]=2

Plot how the Hausdorff distance decreases with larger sample counts:

In[3]:=3
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-pyrh91
Out[3]=3

Properties & Relations  (4)Properties of the function, and connections to other functions

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

In[3]:=3
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-r36bz3
Out[3]=3
In[2]:=2
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-bjlsyt
Out[2]=2

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

In[1]:=1
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https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-pkhc76
In[2]:=2
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-6ymd5k
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-uq0mk1
Out[1]=1

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

In[1]:=1
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-2f75l8
In[2]:=2
✖
https://wolfram.com/xid/0e2yzb16oninfyi0ac5e-u671gz
Out[2]=2
Wolfram Research (2023), RegionHausdorffDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html.
✖
Wolfram Research (2023), RegionHausdorffDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html.

Text

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

✖
Wolfram Research (2023), RegionHausdorffDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html.

CMS

Wolfram Language. 2023. "RegionHausdorffDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html.

✖
Wolfram Language. 2023. "RegionHausdorffDistance." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html.

APA

Wolfram Language. (2023). RegionHausdorffDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html

✖
Wolfram Language. (2023). RegionHausdorffDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html

BibTeX

@misc{reference.wolfram_2025_regionhausdorffdistance, author="Wolfram Research", title="{RegionHausdorffDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html}", note=[Accessed: 24-April-2025 ]}

✖
@misc{reference.wolfram_2025_regionhausdorffdistance, author="Wolfram Research", title="{RegionHausdorffDistance}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html}", note=[Accessed: 24-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regionhausdorffdistance, organization={Wolfram Research}, title={RegionHausdorffDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html}, note=[Accessed: 24-April-2025 ]}

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@online{reference.wolfram_2025_regionhausdorffdistance, organization={Wolfram Research}, title={RegionHausdorffDistance}, year={2023}, url={https://reference.wolfram.com/language/ref/RegionHausdorffDistance.html}, note=[Accessed: 24-April-2025 ]}