Fáry's theorem, that every graph that can be drawn without crossings in the plane using curved edges can also be drawn without crossings using straight line segment edges, follows as a simple corollary of the circle packing theorem: by placing vertices at the centers of the circles and drawing straight edges between them, a straight-line planar embedding is obtained.
A variation of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same point of the plane. 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. 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.
Read more about this topic: Circle Packing Theorem
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