VORONOI DIAGRAM BASED DIMENSIONAL ANCHOR ASSESSMENT FOR RADIAL VISUALIZATIONS - Information Visualisation: Techniques, Usability and Evaluation - page 263

B
Fig. 13.12. A Uniform Co-circular Barycentric Voronoi Diagram (UCBVD) for 20
co-circular sites. The barycentre is marked with a 'B'.
We identify the following UCBVD properties, where B denotes the
barycentre, BR is the barycentre region, C is the unit circle, and d is finite
except where noted:
Property 1: In the case of the circle being the unit circle then the
barycentre is the origin.
Property 2: The UCBVD barycentre region (BR) is a polygon with d
edges.
Property 3: In a UCBVD, line segments drawn from the barycentre to
any Dimensional Anchor are perpendicular to the intervening
Voronoi edge of the BR.
Property 4: In a UCBVD, the BR is a regular polygon.
Property 5: A UCBVD has d +1 regions.
Property 6: As d ĺ the UCBVD BR tends towards a circle of radius
½ that of the enclosing circle C.
We establish these properties rigorously:
Proof of Property 1 : Construct vectors, based at the origin, and in the
direction of the DAs, of length 1/ d . Pick any one of these vectors as a
starting point. Proceed counter-clockwise and add the next vector. This
gives us two edges of equal length, and the angle between them is 180 ޤ
(360/ d ) because we turn by 360/ d as we traverse the edges. Continuing this
process yields d edges, all of equal length. The total of the internal angles is
equal to d (180 - (360/ d )) = 180 d - 360 = ( d -2)180. Thus, traversing in this
manner produces a regular polygon that brings us back to the origin. Thus,
the sum of the vectors is the origin which is equivalent to the barycentre.