# 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

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## Basic Examples(2)

An insphere in 2D:

And in 3D:

The insphere of the regular octahedron:

Its surface area:

## Scope(17)

### Graphics(6)

#### Specification(2)

Inspheres in different dimensions:

Insphere evaluates to a Sphere:

#### Styling(4)

Colored circumspheres:

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

Inspheres with different specular exponents:

Black circumsphere that glows red:

Opacity specifies the face opacity:

### Regions(11)

Insphere 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 an Insphere:

Optimize over it:

Solve equations over an Insphere:

## Applications(3)

Recursively construct inscribed triangles and disks:

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

Use Insphere to compute a circle for each triangle:

Compute the packing density:

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

Use Insphere to compute spheres for each tetrahedron:

Compute the packing density:

## Properties & Relations(2)

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

In 3D:

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

In 3D:

## Neat Examples(1)

Random insphere collections: