Midsphere

In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point. Not every polyhedron has a midsphere, but for every convex polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere. The radius of the midsphere is called the midradius.

A polyhedron and its midsphere. The red circles are the boundaries of spherical caps within which the surface of the sphere can be seen from each vertex.
Cube and dual octahedron with common midsphere

Examples

The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric,[1] and the midsphere touches each edge at its midpoint.[2]

Not every irregular tetrahedron has a midsphere. The tetrahedra that have a midsphere have been called "Crelle's tetrahedra"; they form a four-dimensional subfamily of the six-dimensional space of all tetrahedra (as parameterized by their six edge lengths).[3]

Tangent circles

If O is the midsphere of a convex polyhedron P, then the intersection of O with any face of P is a circle that lies within the face, and is tangent to its edges at the same points where the midsphere is tangent. The circles formed in this way on all of the faces of P form a system of circles on O that are tangent to each other exactly when the faces they lie in share an edge.

Dually, if v is a vertex of P, then there is a cone that has its apex at v and that is tangent to O in a circle; this circle forms the boundary of a spherical cap within which the sphere's surface is visible from the vertex. That is, the circle is the horizon of the midsphere, as viewed from the vertex. The circles formed in this way are tangent to each other exactly when the vertices they correspond to are connected by an edge.

Duality

If a polyhedron P has a midsphere O, then the polar polyhedron with respect to O also has O as its midsphere. The face planes of the polar polyhedron pass through the circles on O that are tangent to cones having the vertices of P as their apexes.[4] The edges of the polar polyhedron have the same points of tangency with the midsphere, at which they are perpendicular to the edges of P.[5]

Edge lengths

For a polyhedron with a midsphere, it is possible to assign a real number to each vertex (the power of the vertex with respect to the midsphere) that equals the distance from that vertex to the point of tangency of each edge that touches it. For each edge, the sum of the two numbers assigned to its endpoints is just the edge's length. For instance, Crelle's tetrahedra can be parameterized by the four numbers assigned in this way to their four vertices, showing that they form a four-dimensional family.[6]

When a polyhedron with a midsphere has a Hamiltonian cycle, the sum of the lengths of the edges in the cycle can be subdivided in the same way into twice the sum of the powers of the vertices. Because this sum of powers of vertices does not depend on the choice of edges in the cycle, all Hamiltonian cycles have equal lengths.[7]

Canonical polyhedron

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. The horizon circles of a canonical polyhedron can be transformed, by stereographic projection, into a collection of circles in the Euclidean plane that do not cross each other and are tangent to each other exactly when the vertices they correspond to are adjacent.[8] In contrast, there exist polyhedra that do not have an equivalent form with an inscribed sphere or circumscribed sphere.[9]

Any two convex 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. The restriction of this projective transformation to the midsphere is a Möbius transformation.[10] There is a unique way of performing this transformation so that the midsphere is the unit sphere and so that the centroid of the points of tangency is at the center of the sphere; this gives a representation of the given polyhedron that is unique up to congruence, the canonical polyhedron.[11] 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.[12] For polyhedra with a non-cyclic group of orientation-preserving symmetries, the two choices of transformation coincide.[13]

See also

Notes
  1. Coxeter (1973) states this for regular polyhedra; Cundy & Rollett 1961 for Archimedean polyhedra.
  2. Pugh (1976).
  3. László (2017). The irregular tetrahedra with a midsphere provide a counterexample to an incorrect claim of Pugh (1976): it is not true that only the regular polyhedra have all three of a midsphere, insphere, and circumsphere.
  4. Coxeter (1973).
  5. Cundy & Rollett (1961).
  6. László (2017).
  7. Fetter (2012).
  8. Schramm (1992); Sachs (1994). Schramm states that the existence of an equivalent polyhedron with a midsphere was claimed by Koebe (1936), but that Koebe only proved this result for polyhedra with triangular faces. Schramm credits the full result to William Thurston, but the relevant portion of Thurston's lecture notes again only states the result explicitly for triangulated polyhedra.
  9. Schramm (1992); Steinitz (1928).
  10. Sachs (1994).
  11. Ziegler (1995).
  12. Bern & Eppstein (2001).
  13. Springborn (2005).

References
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