# What kind of transformations can I do on a 2D Voronoi diagram and have it remain valid? [closed]

I'm a programmer trying to implement a graphical effect using 2D Voronoi diagrams, and I'm wondering what kind of basic geometric transformations I can apply to it while having it still remain a valid Voronoi diagram.

Intuitively, I think I should be able to translate, rotate, reflect and scale it, as long as the scale is uniform along both axes. I can't scale it non-uniformely or shear it. Is that correct?

## closed as off-topic by Dima Pasechnik, Stefan Kohl, YCor, Alex Degtyarev, Ben BarberNov 7 '17 at 21:09

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• that's right; basically you need to preserve the distances between points on the plane (or scale them uniformly). – Dima Pasechnik Nov 7 '17 at 12:42
• Right, of course, that criterion makes perfect sense. Thanks! – Oskar Nov 7 '17 at 12:48
• I don't know what Voronoi diagrams are, but Dima's comment makes me think that your transformations should be from the group ${\mathbb R}^2\rtimes ({\rm O}(2)\times {\mathbb R}^*)$, i.e. all transformations should be a product $T.R.S$ where $T$ is a translation, $R$ is a rotation/reflection, and $S$ is a scalar matrix. – Nick Gill Nov 7 '17 at 17:27
• This question does not appear off-topic. Perhaps the preamble "I'm a programmer..." led some to this conclusion. The question can be understood in one of two senses. (a) Under what transformations of a set of points and their Voronoi diagram does the transformed VD remain the VD for the transformed set of points? (b) Under what transformations does a VD remain a VD for some, possibly different, set of points. It's not obvious to me that the answers to these question must be the same. – Menachem Nov 8 '17 at 15:13
• There are certainly (somewhat generic) cases in which non-uniform scaling transforms a point set and its VD in a consistent manner. For example, Z^n and its VD can be scaled non-uniformly in orthogonal directions and remain consistent. Another interesting question (though probably not @Oskar's question) might consider which transformations preserve the VD's combinatorial structure. This does not seem obvious to me, even for the simple case of a lattice. If answers to these questions are already known, perhaps the question can be considered merely asking for a reference. – Menachem Nov 8 '17 at 15:14