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Torus
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Torus

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Torus[{x,y,z},{rinner,router}]

represents a torus centered at {x,y,z} with inner radius rinner and outer radius router.

Details and Options

  • Torus is also known as torus of revolution.
  • Torus can be used as a geometric region and a 3D graphics primitive.
  • Torus[] is equivalent to Torus[{0,0,0},{1/2,1}].
  • Torus represents the shell .
  • Torus can be used in Graphics3D.
  • Graphics rendering is affected by directives such as FaceForm, Specularity, Opacity and color.

Examples

open allclose all

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

A standard torus at the origin:

In[2]:=2
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https://wolfram.com/xid/0j41r02s-c7o
Out[2]=2

Area and centroid:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-7wedhk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0j41r02s-kxw6nj
Out[2]=2

Scope  (18)Survey of the scope of standard use cases

Graphics  (9)

Specification  (4)

The standard torus:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-btz1q2
Out[1]=1

Filled tori with different outer radii:

In[1]:=1
✖
https://wolfram.com/xid/0j41r02s-l0hn2
Out[1]=1

Filled tori with different inner radii:

In[1]:=1
✖
https://wolfram.com/xid/0j41r02s-hky91r
Out[1]=1

Short form for a torus with radii at the origin:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-37iz6
Out[1]=1

Styling  (4)

Colored tori:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-cdxdbh
Out[1]=1

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

In[1]:=1
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https://wolfram.com/xid/0j41r02s-vb6wx
Out[1]=1

Filled tori with different specular exponents:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-d1tyd0
Out[1]=1

White torus that glows red:

In[2]:=2
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https://wolfram.com/xid/0j41r02s-ca9zt
Out[2]=2

Opacity specifies the face opacity:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-v6995
Out[1]=1

Coordinates  (1)

Points can be Dynamic:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-6gl1zz
Out[1]=1

Regions  (9)

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

In[1]:=1
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https://wolfram.com/xid/0j41r02s-y220
Out[1]=1

Geometric dimension is the dimension of the shape itself:

In[2]:=2
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https://wolfram.com/xid/0j41r02s-bx9tom
Out[2]=2

Membership testing:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-c7lq97
In[2]:=2
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https://wolfram.com/xid/0j41r02s-f70gib
Out[2]=2

Get conditions for point membership:

In[3]:=3
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https://wolfram.com/xid/0j41r02s-inebf9
Out[3]=3

Area:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-se0twe
In[2]:=2
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https://wolfram.com/xid/0j41r02s-e06l44
Out[2]=2

Centroid:

In[3]:=3
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https://wolfram.com/xid/0j41r02s-gwq4b4
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0j41r02s-oknxhk
Out[4]=4

Distance from a point:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-jcsb4b
In[2]:=2
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https://wolfram.com/xid/0j41r02s-8aexdg
Out[2]=2

The equidistance contours for a torus:

In[3]:=3
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https://wolfram.com/xid/0j41r02s-ew8anh
Out[3]=3

Signed distance from a point:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-kjgbyj
In[2]:=2
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https://wolfram.com/xid/0j41r02s-zognbt
Out[2]=2

Nearest point in the region:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-d7g53y
In[2]:=2
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https://wolfram.com/xid/0j41r02s-mtue
Out[2]=2

Nearest points to an enclosing sphere:

In[3]:=3
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https://wolfram.com/xid/0j41r02s-e29k5d
In[4]:=4
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https://wolfram.com/xid/0j41r02s-5ksoo8
In[5]:=5
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https://wolfram.com/xid/0j41r02s-uv1cfm
Out[5]=5

A torus is bounded:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-ypd96t
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0j41r02s-po0eks
Out[2]=2
In[3]:=3
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https://wolfram.com/xid/0j41r02s-dym4fu
In[4]:=4
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https://wolfram.com/xid/0j41r02s-i3tfrr
Out[4]=4

Find its range:

In[5]:=5
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https://wolfram.com/xid/0j41r02s-l3exhn
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0j41r02s-23060u
Out[6]=6

Optimize over a torus region:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-nf9ton
In[2]:=2
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https://wolfram.com/xid/0j41r02s-hyz4dq
Out[2]=2

Solve equations in a torus region:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-bnrw6
In[2]:=2
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https://wolfram.com/xid/0j41r02s-lhnpf3
Out[2]=2

Applications  (2)Sample problems that can be solved with this function

Bubbles:

In[15]:=15
✖
https://wolfram.com/xid/0j41r02s-fp53yw
Out[15]=15

Use Torus to render nodes in a GraphPlot3D:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-u52ea
Out[1]=1

Properties & Relations  (3)Properties of the function, and connections to other functions

An implicit specification of a torus generated by ContourPlot3D:

In[2]:=2
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https://wolfram.com/xid/0j41r02s-be10oz
Out[2]=2

A parametric specification of a torus generated by ParametricPlot3D:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-i7028d
Out[1]=1

Torus is the RegionBoundary of FilledTorus:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-h6x47m
Out[1]=1

Neat Examples  (3)Surprising or curious use cases

Random torus collections:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-coltip
Out[1]=1

Double helix:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-clnpvv
Out[1]=1

Nested tori:

In[1]:=1
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https://wolfram.com/xid/0j41r02s-w17054
Out[1]=1
Wolfram Research (2021), Torus, Wolfram Language function, https://reference.wolfram.com/language/ref/Torus.html.
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Wolfram Research (2021), Torus, Wolfram Language function, https://reference.wolfram.com/language/ref/Torus.html.

Text

Wolfram Research (2021), Torus, Wolfram Language function, https://reference.wolfram.com/language/ref/Torus.html.

✖
Wolfram Research (2021), Torus, Wolfram Language function, https://reference.wolfram.com/language/ref/Torus.html.

CMS

Wolfram Language. 2021. "Torus." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Torus.html.

✖
Wolfram Language. 2021. "Torus." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Torus.html.

APA

Wolfram Language. (2021). Torus. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Torus.html

✖
Wolfram Language. (2021). Torus. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Torus.html

BibTeX

@misc{reference.wolfram_2025_torus, author="Wolfram Research", title="{Torus}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Torus.html}", note=[Accessed: 09-July-2025 ]}

✖
@misc{reference.wolfram_2025_torus, author="Wolfram Research", title="{Torus}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Torus.html}", note=[Accessed: 09-July-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_torus, organization={Wolfram Research}, title={Torus}, year={2021}, url={https://reference.wolfram.com/language/ref/Torus.html}, note=[Accessed: 09-July-2025 ]}

✖
@online{reference.wolfram_2025_torus, organization={Wolfram Research}, title={Torus}, year={2021}, url={https://reference.wolfram.com/language/ref/Torus.html}, note=[Accessed: 09-July-2025 ]}