# Torus

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

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## Basic Examples(2)

A standard torus at the origin:

Area and centroid:

## Scope(18)

### Graphics(9)

#### Specification(4)

The standard torus:

Filled tori with different outer radii:

Filled tori with different inner radii:

Short form for a torus with radii at the origin:

#### Styling(4)

Colored tori:

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

Filled tori with different specular exponents:

White torus that glows red:

Opacity specifies the face opacity:

#### Coordinates(1)

Points can be Dynamic:

### Regions(9)

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

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for point membership:

Area:

Centroid:

Distance from a point:

The equidistance contours for a torus:

Signed distance from a point:

Nearest point in the region:

Nearest points to an enclosing sphere:

A torus is bounded:

Find its range:

Optimize over a torus region:

Solve equations in a torus region:

## Applications(2)

Bubbles:

Use Torus to render nodes in a GraphPlot3D:

## Properties & Relations(3)

An implicit specification of a torus generated by ContourPlot3D:

A parametric specification of a torus generated by ParametricPlot3D:

Torus is the RegionBoundary of FilledTorus:

## Neat Examples(3)

Random torus collections:

Double helix:

Nested tori:

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.

#### CMS

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

#### BibTeX

@misc{reference.wolfram_2024_torus, author="Wolfram Research", title="{Torus}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/Torus.html}", note=[Accessed: 30-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_torus, organization={Wolfram Research}, title={Torus}, year={2021}, url={https://reference.wolfram.com/language/ref/Torus.html}, note=[Accessed: 30-May-2024 ]}