Medial Spheres for Shape Representation - CIM - McGill University

Figure 2–2: Medial axes, in black, for two solids with red boundaries. The boundary on
the right is obtained by smoothing the boundary on the left. Produced with [84].
original object. The nature of significance, or “salience”, is application dependent: for
motion planning, an exact medial surface is sometimes necessary, while for matching and
animation, a medial surface of minimal geometric complexity that describes the object
shape is sought.
Two common measures of significance of a medial point include
• the radius of the associated medial sphere, and
• the object angle of the medial point, defined as follows:
Definition 2.1. For a smooth medial point p, the object angle θ is the angle between the
vector from p to either of its two closest points on B and the tangent plane to MS at p.
Figure 2–3(Left) gives an example of how the object angle θ is evaluated. As shown
in [53], removal of medial points that have a small object angle has a small impact on the
volume of the union of medial balls (refer to Figure 2–3(Right)). When it is desirable to
preserve the volume of the solid being approximated, eliminating medial points with small
object angle is an appropriate simplification measure.
26