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If we consider a portion of the number line, say on the interval $[0,100]$, and split that into regions e.g. split at $80$ to create $2$ regions.

Now I want to subdivide the two regions. The region $[0,80]$ has $m$ partitions and the region $[80,100]$ has $n$ partitions. If $m = 2$ and $n = 10$ (for example), and the subpartitioning was done by splitting the region up equally, when we go from the first region to the second, the partition size is very different. What I want to do is blend the divisions between regions.

Suggestions on how I can do this (bearing in mind I'm not that good at maths) welcome!

EDIT: Sorry for being unclear. Picture should help. In this pic there's 2 regions divided into 2 and 4 (I couldn't easily draw 10 divisions!). The upper part shows equidistant splitting which is ignorant of what's happening in a neighbouring region. The lower part of the picture is roughly what I want. A key requirement is the nodes in the interval $[80,100]$ go from having larger spacing to smaller spacing the further they get away from the interval $[0,80]$. If there was 10 nodes within $[80,100]$, they'd be mainly bunched up towards 100 and the gaps between the nodes would get progressively smaller. The larger spacing is because of the large division size in the neighbouring interval.

The number line could be split up into any number of regions, and the number of divisions on each region is not chosen by me, but imposed on me. One region would affect adjacent regions only.

Blend pic

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    What do you mean by "blend the divisions"? Perhaps you could draw a picture to explain?2012-03-06
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    @Antonio: have added a picture and a bit more of an explanation.2012-03-08
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    Could you give some context on what problem you are trying to solve through this? I ask because it is a little unusual to do this sort of blending between divisions, and if we know more about what you are really trying to achieve, there may be a more natural solution for it.2012-03-09
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    At first glance, this looks like it could be solved by a variant of [monotone cubic interpolation](http://en.wikipedia.org/wiki/Monotone_cubic_interpolation), but I'd still like to hear your answer to my previous comment first.2012-03-09
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    @Rahul: I'm in CAD and have a model from a scanner. It is discrete i.e. point cloud/triangulated mesh. The task is to uv parameterise the surface. The regions allow the user to specify the regions where they want more patches. In one region they might have 20 horizontal patches, in another they might have 5, so I'd like the change in patch size when transitioning from one region to another to not be "drastic". Hope this gives more context, if not, I can supply pictures of the actual application.2012-03-09
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    No pictures necessary, I get what you mean. It sounds like you might also want control over how heavily you smooth the transition as well, yes? It's an interesting problem; I'll think about it and post an answer if I come up with something good.2012-03-09
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    @Rahul: Something to control the transition would be the icing on the cake. It's almost like for a given region, part of the left hand side region's spacing (width / divisions) is apportioned to the division in this region in diminishing quantities. Of course, there is only enough space in this region, so it's "normal" division size is smaller further away from the left hand side region. That alone is more complex than I can solve. Then, there's the right hand side at play too...2012-03-09

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