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Insphere
✖
Insphere

✖
Insphere[{p1,…,pn+1}]

gives the sphere that can be inscribed in the simplex defined by points pi in .

✖
Insphere[poly]

gives the insphere of a polyhedron or polygon poly.

Details

  • Insphere is also known as incircle, inscribed circle, or inscribed disk.
  • Insphere gives the Sphere of largest measure (arc length, area, etc.) that can be inscribed in the simplex (triangle, tetrahedron, etc.) defined by points pi.
  • Insphere evaluates to a Sphere[c,r], where the center c is known as the incenter and radius r is known as the inradius for the related simplex.
  • Insphere is defined for and affinely independent.
  • For polyhedra, Insphere[poly] returns a sphere that is contained within the polyhedron poly and tangent to each of the polyhedron faces.
  • For polygons, Insphere[poly] returns a sphere that is contained within the polygon poly and tangent to each of the polygon edges.
  • Insphere can be used with symbolic points in GeometricScene.

Examples

open allclose all

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

An insphere in 2D:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-fz03yw
Out[1]=1

And in 3D:

In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-tld766
Out[2]=2

The insphere of the regular octahedron:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-nw12dx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-g341iw
Out[2]=2

Its surface area:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-36b300
Out[3]=3

Scope  (17)Survey of the scope of standard use cases

Graphics  (6)

Specification  (2)

Inspheres in different dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-wntg1a
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-zr7ghg
Out[2]=2

Insphere evaluates to a Sphere:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-do6gu6
Out[1]=1

Get the center and radius:

In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-e7tdeq
Out[2]=2

Styling  (4)

Colored circumspheres:

In[1]:=1
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https://wolfram.com/xid/0d6gh876r6-n62odp
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-5h7y6f
Out[2]=2

Different properties can be specified for the front and back of faces using FaceForm:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-qlfjpq
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-gokgh8
Out[2]=2

Inspheres with different specular exponents:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-21ednz
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-8n1xyu
Out[2]=2

Black circumsphere that glows red:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-tdbv9f
Out[3]=3

Opacity specifies the face opacity:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-7orfqj
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-v6r7w7
Out[2]=2

Regions  (11)

Insphere works in any number of dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-u933r
Out[1]=1

Get the circumcenter and circumradius:

In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-fekmm
Out[2]=2

Embedding dimension is the dimension of the space in which the sphere lives:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-eq5j78
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-f2f0kz
Out[2]=2

Geometric dimension is the dimension of the shape itself:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-ca28v8
Out[3]=3

Membership testing:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-uyn7nu
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-tg4hq
Out[2]=2

Get conditions for membership:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-9ge5ph
Out[3]=3

Area:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-se0twe
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-e06l44
Out[2]=2

Centroid:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-gwq4b4
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-oknxhk
Out[4]=4

Distance from a point:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-8t82nc
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-yy0poa
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-bxuxrk
Out[3]=3

Signed distance from a point:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-lpnyc3
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-q2k9sm
Out[2]=2

Plot it:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-f514j9
Out[3]=3

Nearest point in the region:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-btejnv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-et4yza
Out[2]=2

Nearest points to an enclosing sphere:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-fccj33
In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-m59d3a
In[5]:=5
✖
https://wolfram.com/xid/0d6gh876r6-g70sch
Out[5]=5

A sphere is bounded:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-kpktle
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-wy4vq0
Out[2]=2

Find its range:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-vr4sfb
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-z4q5c
Out[4]=4

Integrate over an Insphere:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-izew4u
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-xs2ioc
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-sl101w
Out[3]=3

Optimize over it:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-5qhh0w
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-31c9br
Out[2]=2

Solve equations over an Insphere:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-i0dix4
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-70rdaw
Out[2]=2

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

Recursively construct inscribed triangles and disks:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-umomr3
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-5b5xxm
In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-b9gako
In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-cfvtlb
Out[4]=4

Use Insphere to generate a circle packing for a triangulated region. First triangulate the region:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-84njj
Out[1]=1

Use Insphere to compute a circle for each triangle:

In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-mtktm
In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-buhkmy
Out[3]=3

Compute the packing density:

In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-dz7d5y
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0d6gh876r6-eq582
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0d6gh876r6-cpl3qf
Out[6]=6

Use Insphere to generate a sphere packing for a triangulated region. First discretize and triangulate the region:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-ul7xd
Out[1]=1

Use Insphere to compute spheres for each tetrahedron:

In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-gvmsw
In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-dv8hkw
Out[3]=3

Compute the packing density:

In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-bk32o2
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0d6gh876r6-jozy
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0d6gh876r6-jnid5
Out[6]=6

Properties & Relations  (2)Properties of the function, and connections to other functions

Insphere is the largest Sphere that can be inscribed in a Simplex:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-k265g7
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-tg9ijn
Out[2]=2

In 3D:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-3ftiwq
In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-pwi1ou
Out[4]=4

Use Circumsphere to get a Sphere (blue) that circumscribes a Simplex:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-gp2exh
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-3w20tp
Out[2]=2

In 3D:

In[3]:=3
✖
https://wolfram.com/xid/0d6gh876r6-ud8x8p
In[4]:=4
✖
https://wolfram.com/xid/0d6gh876r6-zdbjwv
Out[4]=4

Neat Examples  (1)Surprising or curious use cases

Random insphere collections:

In[1]:=1
✖
https://wolfram.com/xid/0d6gh876r6-k9pv2f
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0d6gh876r6-cbidmu
Out[2]=2
Wolfram Research (2015), Insphere, Wolfram Language function, https://reference.wolfram.com/language/ref/Insphere.html (updated 2019).
✖
Wolfram Research (2015), Insphere, Wolfram Language function, https://reference.wolfram.com/language/ref/Insphere.html (updated 2019).

Text

Wolfram Research (2015), Insphere, Wolfram Language function, https://reference.wolfram.com/language/ref/Insphere.html (updated 2019).

✖
Wolfram Research (2015), Insphere, Wolfram Language function, https://reference.wolfram.com/language/ref/Insphere.html (updated 2019).

CMS

Wolfram Language. 2015. "Insphere." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Insphere.html.

✖
Wolfram Language. 2015. "Insphere." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Insphere.html.

APA

Wolfram Language. (2015). Insphere. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Insphere.html

✖
Wolfram Language. (2015). Insphere. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Insphere.html

BibTeX

@misc{reference.wolfram_2025_insphere, author="Wolfram Research", title="{Insphere}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Insphere.html}", note=[Accessed: 21-May-2025 ]}

✖
@misc{reference.wolfram_2025_insphere, author="Wolfram Research", title="{Insphere}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Insphere.html}", note=[Accessed: 21-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_insphere, organization={Wolfram Research}, title={Insphere}, year={2019}, url={https://reference.wolfram.com/language/ref/Insphere.html}, note=[Accessed: 21-May-2025 ]}

✖
@online{reference.wolfram_2025_insphere, organization={Wolfram Research}, title={Insphere}, year={2019}, url={https://reference.wolfram.com/language/ref/Insphere.html}, note=[Accessed: 21-May-2025 ]}