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RegionNearest
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RegionNearest

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RegionNearest[reg,p]

gives a point in the region reg that is nearest the point p.

✖

gives a RegionNearestFunction[…] that can be repeatedly applied to points.

Details

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the nearest point in the unit disk:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-h4c3pr
Out[3]=3

Plot it:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-jju9cc
Out[2]=2

Find the nearest point in a MeshRegion:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-b5gwzp
Out[1]=1

With one argument you get a RegionNearestFunction:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-e7jq48
Out[2]=2

Apply the function to lists of points to compute many nearest points:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-bi09wv
Out[3]=3

Scope  (16)Survey of the scope of standard use cases

Special Regions  (7)

Point sets:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-bij3y1
Out[1]=1

They work in any dimension:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-cdjz4a
Out[2]=2

Visualize the nearest points:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-k0p5c3
In[6]:=6
✖
https://wolfram.com/xid/09b0ef5bm-b5o963
Out[6]=6

Line:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-ntr1p
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-c2h6od
Out[2]=2

Visualize it:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-crz78j
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-oahg3
Out[4]=4

Rectangle can be used in 2D:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-jv162
Out[1]=1

Cuboid generalizes it to any dimension:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-elo97m
Out[2]=2

Visualize the nearest points on a rectangle:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-cb5t0s
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-dl95d9
Out[4]=4

Simplex:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-khpg86
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-s9hi8
Out[2]=2

Visualize the nearest points on a 2D simplex:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-q9huz
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-bxxjv4
Out[4]=4

Polygon:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-n06s0x
Out[1]=1

Visualize the nearest points in a hexagon:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-ifc9dj
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-hulmg
Out[3]=3

Disk can be used in 2D:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-c9koyz
Out[1]=1

Ball generalizes a disk in any dimension:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-h10zot
Out[2]=2

Visualize the nearest points on a 3D ball:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-kmzxz
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-iay2op
Out[4]=4

Disk can be an ellipse in 2D:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-2vzam
Out[1]=1

Ellipsoid generalizes ellipses to any dimension:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-iwctvx
Out[2]=2

Plot the nearest points on an ellipse:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-my8lut
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-l4jbke
Out[4]=4

Formula Regions  (2)

The nearest point on a disk represented as an ImplicitRegion:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-e3pxnh
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-bbebm2
Out[2]=2
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-m8ukuf
Out[4]=4

A cylinder:

In[5]:=5
✖
https://wolfram.com/xid/09b0ef5bm-jirj3
In[6]:=6
✖
https://wolfram.com/xid/09b0ef5bm-f09o8a
Out[6]=6

The nearest point on a disk represented as a ParametricRegion:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-el6xm7
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-ffcksa
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-g849lt
Out[3]=3

Using a rational parametrization of the disk:

In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-ky4tf
In[5]:=5
✖
https://wolfram.com/xid/09b0ef5bm-by6xm0
Out[5]=5

A cylinder:

In[6]:=6
✖
https://wolfram.com/xid/09b0ef5bm-hv7e8z
In[7]:=7
✖
https://wolfram.com/xid/09b0ef5bm-gj39yx
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/09b0ef5bm-gibioi
Out[8]=8

Mesh Regions  (4)

The nearest point in a 1D BoundaryMeshRegion:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-nejktm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-ekay2x
Out[2]=2

2D:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-f6bccz
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-ceuwni
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/09b0ef5bm-di3wdk
Out[5]=5

3D:

In[6]:=6
✖
https://wolfram.com/xid/09b0ef5bm-fqmu4o
In[7]:=7
✖
https://wolfram.com/xid/09b0ef5bm-idlvw
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/09b0ef5bm-kh906
Out[8]=8

The nearest point in a 0D MeshRegion in 1D:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-btvxtc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-ik7sz
Out[2]=2

A 1D region:

In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-citaof
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-fsnj3j
Out[4]=4

The nearest point in a 0D MeshRegion in 2D:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-8ds1n
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-je81r2
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-s2678
Out[3]=3

A 1D region:

In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-i80ee2
In[5]:=5
✖
https://wolfram.com/xid/09b0ef5bm-d60kip
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/09b0ef5bm-b3zcwl
Out[6]=6

A 2D region:

In[7]:=7
✖
https://wolfram.com/xid/09b0ef5bm-b8poha
In[8]:=8
✖
https://wolfram.com/xid/09b0ef5bm-chjbp1
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/09b0ef5bm-j04042
Out[9]=9

The nearest point in a 0D MeshRegion in 3D:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-kp5ocx
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-d417z5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-bmbb55
Out[3]=3

A 1D region:

In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-d6wqd3
In[5]:=5
✖
https://wolfram.com/xid/09b0ef5bm-efldpv
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/09b0ef5bm-wmdcy
Out[6]=6

A 2D region:

In[7]:=7
✖
https://wolfram.com/xid/09b0ef5bm-crt8vj
In[8]:=8
✖
https://wolfram.com/xid/09b0ef5bm-bf8122
Out[8]=8
In[9]:=9
✖
https://wolfram.com/xid/09b0ef5bm-chm3ht
Out[9]=9

A 3D region:

In[10]:=10
✖
https://wolfram.com/xid/09b0ef5bm-cc4as
In[11]:=11
✖
https://wolfram.com/xid/09b0ef5bm-b3r9b
Out[11]=11
In[12]:=12
✖
https://wolfram.com/xid/09b0ef5bm-c72n9z
Out[12]=12

Derived Regions  (3)

The distance to a RegionIntersection:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-iri6pf
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-9e68l
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-bvehq
Out[3]=3

The distance to a TransformedRegion:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-c2wj44
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-wbcwf
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-k1y8k2
Out[3]=3

The distance to a RegionBoundary:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-lstt5
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-bjh3h5
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-dao2kz
Out[3]=3

Applications  (3)Sample problems that can be solved with this function

Visualize the nearest points in a region:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-mgv8i7

The nearest points along two circles:

In[139]:=139
✖
https://wolfram.com/xid/09b0ef5bm-c8zs88
Out[139]=139

Visualize a region by mapping uniform points to the region using RegionNearest:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-ks1xh9
Out[1]=1

Find the altitude of a triangle:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-4h9btt
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-e5zzur
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-3csd0h
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-u7fgv7

Visualize altitude in red:

In[5]:=5
✖
https://wolfram.com/xid/09b0ef5bm-kxq96p
Out[5]=5

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

Only one point is returned in case there is a tie:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-6nqoqt
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-jfspe3
Out[3]=3

Visualize the nearest point chosen in red:

In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-5c9ucm
Out[4]=4

A point is a RegionMember if the nearest point to the region is itself:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-qdojf5
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-mbbak9
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-h4jhtl
Out[3]=3

RegionNearest is closely related to RegionDistance:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-itfn4t

The distance to the nearest point is minimal:

In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-bglbo1
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-dux0f3
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/09b0ef5bm-elibhs
Out[4]=4

Nearest gives nearest points for finite point regions:

In[1]:=1
✖
https://wolfram.com/xid/09b0ef5bm-fu5j25
In[2]:=2
✖
https://wolfram.com/xid/09b0ef5bm-iqtmrh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/09b0ef5bm-4fjj2
Out[3]=3
Wolfram Research (2014), RegionNearest, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionNearest.html.
✖
Wolfram Research (2014), RegionNearest, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionNearest.html.

Text

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

✖
Wolfram Research (2014), RegionNearest, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionNearest.html.

CMS

Wolfram Language. 2014. "RegionNearest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionNearest.html.

✖
Wolfram Language. 2014. "RegionNearest." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionNearest.html.

APA

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

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

BibTeX

@misc{reference.wolfram_2025_regionnearest, author="Wolfram Research", title="{RegionNearest}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RegionNearest.html}", note=[Accessed: 25-March-2025 ]}

✖
@misc{reference.wolfram_2025_regionnearest, author="Wolfram Research", title="{RegionNearest}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RegionNearest.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_regionnearest, organization={Wolfram Research}, title={RegionNearest}, year={2014}, url={https://reference.wolfram.com/language/ref/RegionNearest.html}, note=[Accessed: 25-March-2025 ]}

✖
@online{reference.wolfram_2025_regionnearest, organization={Wolfram Research}, title={RegionNearest}, year={2014}, url={https://reference.wolfram.com/language/ref/RegionNearest.html}, note=[Accessed: 25-March-2025 ]}