Surface Projection

The Projection of a Polyhedron Face onto the Surface of a Sphere

Prerequisite Knowledge

When going through this section one must consistently be imagining any of the polyhedra enclosed within its circumscribing sphere and then select one face and realise its relationship with that sphere. By a process of gnomonic (central) projection, the edges of the polyhedron generate a set of arcs of whose centre points are the centre of the polyhedron and thus the centre of the circumscribing sphere. These arcs belong to what are known as great circles. What now results is a spherical regular polygonal face on the surface of the sphere. When this process is completed for all the surfaces of the polyhedron, great circles project through all of the vertices of each face and intersect at the centre of each spherical face.

Cube face containing arcs of two of

the cubes great circles

Circumscribing and Inscribing Spheres of Polyhedra

Circumscribing Sphere of the Regular Tetrahedron

When one is refering to spherical geometry with regard to polyhedra, it is primarily concerned with the projection of the flat faces of the polyhedra onto the globular surface of the sphere. There are two spheres concerned with any polyhedron, its inscribed sphere and its circumscribed sphere. The above image shows half of the circumscribed sphere and the image below shows the inscribed sphere of a regular tetrahedron. It is onto the circumscribed sphere that the projection of the polyhedra faces will occur.

Inscribed Sphere of the Regular Tetrahedron

The above image shows the inscribed sphere of a regular tetrahedron, with one face of the tetrahedron missing to illustrate the effect. Note that the sphere is tangential to the faces of the tetrahedron. It touches each face at its centre.

Circumscribing Sphere of the Cube

The above image shows half of the circumscribed sphere of a cube in which the eight vertices of the cube touch the inside surface of the sphere.

Only triangular faces can be projected onto the surface of a sphere in the context of polyhedra. This is why when referring to the five regular polyhedra the faces of the cube and the dodecahedron have to be broken up into triangles of equal dimensions. The triangular faces of the tetrahedron, octahedron and the icosahedron are broken into smaller sub-triangles by projecting lines from each vertex to the centre of its opposite edge. The breaking up of each face concerned with regular polyhedra is shown in the images below.

                Breaking up of faces for projection onto sphere



                 Triangular Face                                         Square Face

Pentagonal Face

The Projection of Great Circles Through the Vertices of a Cube

Every section of a spherical surface made by a plane is a circle. Its like cutting an orange with a knife, no matter where one takes a straight cut on the surface of the orange the resulting surface after the cut will always be a circle, (try it if you don't believe). In particular this means that each face of a polyhedron inscribed in a sphere lies on a plane that cuts the sphere in a circle. This circle is the circumcircle of that face and it is called a small circle. Its centre is the centre of the polyhedral face related to it and its radius is less than that of the sphere itself.

If a plane passes through the centre of the sphere, the circle is called a great circle. Its centre and radius are the same as those of the sphere itself.

A unique great circle is determined by the centre of the sphere and any two points on its suface provided these points are not the extremeties of a diameter. In particular this means that each edge of a polyhedron determines an arc of a great circle, since the two end points of an edge lie on the surface of the sphere.