Each Voronoi cell is a convex
polyhedron; their vertices are called Voronoi vertices.
In 3D a Voronoi vertex v is shared by the cells of at
least four samples, which are all closest to v.
The Voronoi ball at v is the ball centered at v passing
through its closest samples.
The power diagram is a kind of weighted Voronoi diagram with the
convenient property that the cells continue to be convex polyhedra.
Here is a picture of a 2D power diagram.
Each weighted point is represented by a ball,
where the point is the center of the ball and the weight
is represented by the radius.
The medial axis of an object is the closure of the
set of points with more than
one closest point on the surface of the object.
Notice that a point of the medial axis is the center of a ball touching
the surface in at least two points, but completely contained in the object.
The union of all of these balls completely fill up the object.
The medial axis transform is the representation of the object
by this set of balls.
Here is an example of a two-dimensional medial axis.
Notice that the medial axis is the continuous cousin of the Voronoi
diagram - the set of points with more than one closest point on the
input point set S gives the Voronoi diagram.
Power Diagram
Medial Axis Transform
