Medial Spheres for Shape Representation - CIM - McGill University

Definition 1.4. The vectors from a medial point p to its closest points on B are the spoke
vectors of p. 2
The set of all medial points defines the following object:
Definition 1.5. The medial surface MS of Ω is the closure of the set of medial points of
Ω.
We consider the closure in the above definition because the set of medial points may
not be closed; see [31] for a 3D example.
When Ω is a 2D solid, the 2D counterpart of the medial surface is called the me-
dial axis and is the closure of the set centres of maximal inscribed disks in Ω. Many
authors use the term ‘medial axis’ to refer to the medial surface of a 3D solid, as well
as higher-dimensional variants. Figure 1–4 presents examples of a medial axis and the
medial surface of non-trivial 2D and 3D solids, respectively.
The medial surface we have thus defined is also known as the interior medial surface
of Ω. The exterior medial surface of Ω is the closure of the set of centres of all maximal
balls exterior to Ω, i.e., it is the medial surface of the closure of the complement of Ω. This
object can be infinite (e.g., for non-convex Ω with no cavities) and empty (for convex Ω
with no cavities). The exterior and interior medial axes are defined analogously for 2D
solids. Figure 1–5(left) shows an example of both the interior and the exterior medial axis
of a 2D solid.
2 We follow the terminology of [106]. These vectors are also called pannormals in the
work of Harry Blum [18].
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