I want to understand the structure of a multiplicatively weighted voronoi diagram
. I found that the bisector between 2 sites is circle shaped, but couldn't formally see it. can someone explain?
Thanks, Ohad.
why does multiplicatively weighted voronoi diagram (mwvd) with 2 sites create a circle?
1 Answers
About "multiplicatively weighted Voronoi diagrams" see here:
http://www.nirarebakun.com/voro/emwvoro.html
For the benefit of the readers I give a short description: You have a set of cities $p_i$ in the euclidean plane $E:={\mathbb R}^2$, and each of these cities has a given weight $w_i>0$. The "felt distance" $d(z,p_i)$ from an arbitrary point $z\in E$ to the city $p_i$ is defined to be $d(z,p_i)\ :=\ {|z-p_i| \over w_i}\ ,$ where |z-z'| denotes the euclidean distance. This means that the "felt distance" to a city with large weight is comparatively small, and conversely, that the $d$-unit-disk of a city with large weight has a large euclidean radius.
Cosider now two cities $p$ and p' with respective weights $w$ and w'. The $d$-Voronoi-boundary $\partial$ between these two cities consists of the points $z\in E$ which have the same $d$-distance to $p$ and to p', i.e., it is defined by the equation {|z-p|\over |z-p'|} \ =\ {w\over w'}\ .\qquad (*) This says that $\partial$ is the locus of all points $z\in E$ for which the ratio |z-p|/ |z-p'| has the a priori given value \lambda:=w/w'. It is a theorem of elementary geometry that such a locus is a circle, called the ${\it Apollonian\ circle}$ for the given data $p$, p' and $\lambda$. The simplest proof is by choosing $p=(0,0)$, p'=(c,0). Then the condition $(*)$ becomes $x^2+y^2=\lambda^2\bigl((x-c)^2 + y^2 \bigr)$, which can be simplified to the equation of a circle (or a line).
-
0Belated "thanks" for that answer from another interested reader (me). I was googling for an answer to exactly that same question, and your explanation is the clearest I found (indeed, **so** clear that I'm now embarrassed to have needed to ask the question in the first place:). – 2016-03-05