# 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.

# Details • Circumsphere is also known as circumcircle, circumscribed circle, or circumscribed disk.
• Circumsphere gives the Sphere of smallest measure (arc length, area, etc.) that circumscribes the points pi.
• Circumsphere evaluates to a Sphere[c,r] where the center c is known as the circumcenter, and radius r is known as the circumradius for the Simplex[{p1,,pn+1}].
• • Circumsphere is defined for and is affinely independent.
• For polyhedra, Circumsphere[poly] is equivalent to .
• For polygons, Circumsphere[poly] is equivalent to Circumsphere[PolygonCoordinates[poly]].
• Circumsphere can be used with symbolic points in GeometricScene.

# 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:

Get the center and radius:

#### 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:

Get the circumcenter and circumradius:

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.: