midsphere

That is, the circle is the horizon of the midsphere, as viewed from the vertex.

An intersphere or midsphere, tangent to all edges.

The restriction of this projective transformation to the midsphere is a Möbius transformation.

In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric.

In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron.

The midsphere is so-called because it is between the inscribed sphere (which is tangent to every face of a polyhedron) and the circumscribed sphere (which touches every vertex).

Not every polyhedron has a midsphere, but for every polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere.

One stronger form of the circle packing theorem, on representing planar graphs by systems of tangent circles, states that every polyhedral graph can be represented by a polyhedron with a midsphere.

A packing of this type can be used to construct a convex polyhedron that represents the given graph and that has a midsphere, a sphere tangent to all of the edges of the polyhedron.

Any two polyhedra with the same face lattice and the same midsphere can be transformed into each other by a projective transformation of three-dimensional space that leaves the midsphere in the same position.

Any convex polyhedron can be distorted into a canonical form, in which a midsphere (or intersphere) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.

Alternatively, a transformed polyhedron that maximizes the minimum distance of a vertex from the midsphere can be found in linear time; the canonical polyhedron chosen in this way has maximal symmetry among all choices of the canonical polyhedron.

Conversely, if a polyhedron has a midsphere, then the circles formed by the intersections of the sphere with the polyhedron faces and the circles formed by the horizons on the sphere as viewed from each polyhedron vertex form a dual packing of this type.