# Circumsphere

Circumsphere[{p1,,pn+1}]

gives the sphere that circumscribes the points pi in .

Circumsphere[poly]

gives the circumsphere of a polyhedron or polygon poly.

# Examples

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## Basic Examples(2)

A circumsphere in 2D:

And in 3D:

The circumsphere of the regular octahedron:

Its surface area:

## Scope(17)

### Graphics(6)

#### Specification(2)

Circumspheres in different dimensions:

Circumsphere evaluates to a Sphere:

#### Styling(4)

Colored circumspheres:

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

Circumspheres with different specular exponents:

Black circumsphere that glows red:

Opacity specifies the face opacity:

### Regions(11)

Circumsphere works in any number of dimensions:

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

Geometric dimension is the dimension of the shape itself:

Membership testing:

Get conditions for membership:

Area:

Centroid:

Distance from a point:

Plot it:

Signed distance from a point:

Plot it:

Nearest point in the region:

Nearest points to an enclosing sphere:

A sphere is bounded:

Find its range:

Integrate over a Circumsphere:

Optimize over it:

Solve equations over a Circumsphere:

## Applications(7)

Find the intersections of a Line and a Circumsphere:

Find the intersection of two circumspheres:

Find a perpendicular bisector of a triangle:

Visualize circumcenter and bisectors in red:

The defining property of a DelaunayMesh is that no input point is contained in the circumcircle of any Triangle in the mesh:

Use Circumsphere to approximate the radius of curvature of a function:

Compare the exact radius of curvature with the radius from the circumcircle approximation:

Plot it:

Use a circumsphere with symbolic input to derive a formula for the radius of curvature:

The result is identical to the radius formula:

Use Circumsphere to find a disk covering for any region with a triangulation. First triangulate the region:

Use Circumsphere to compute a circle for each triangle:

Compute its efficiency:

Use Circumsphere to generate a ball covering for a triangulated region. First discretize and triangulate the region:

Use Circumsphere to compute spheres for each tetrahedron:

Compute the packing density:

## Properties & Relations(1)

Circumsphere can represent any Sphere by picking three points on a sphere in 2D:

Picking four points on a sphere in 3D etc.:

Wolfram Research (2014), Circumsphere, Wolfram Language function, https://reference.wolfram.com/language/ref/Circumsphere.html (updated 2019).

#### Text

Wolfram Research (2014), Circumsphere, Wolfram Language function, https://reference.wolfram.com/language/ref/Circumsphere.html (updated 2019).

#### CMS

Wolfram Language. 2014. "Circumsphere." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2019. https://reference.wolfram.com/language/ref/Circumsphere.html.

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_circumsphere, author="Wolfram Research", title="{Circumsphere}", year="2019", howpublished="\url{https://reference.wolfram.com/language/ref/Circumsphere.html}", note=[Accessed: 22-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_circumsphere, organization={Wolfram Research}, title={Circumsphere}, year={2019}, url={https://reference.wolfram.com/language/ref/Circumsphere.html}, note=[Accessed: 22-July-2024 ]}