Approximate concordia with equal-length segments

[johnzeringue]

Currently, the concordia line is drawn by breaking it into equal lengths of t and approximating with bezier curves. As has been previously suggested by @noahmclean, it would be better to generate equal-length segments instead.

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@noahmclean, you mentioned that you had code for creating equal-length concordia segments in #27. Do you still have that code? I'm interested in implementing this in Topsoil before writing an overview of how we plot everything.

[noahmclean]

Here's a sketch for what I had in mind, though I can't find where I implemented this code originally. The document has an outline of the algorithm I was using, and then a worked problem with a particularly nasty Tera-Wasserburg view. It hasn't been proofread closely -- let me know if there are any typos/mistakes or if you think of a better way of doing this. And a last note: I wouldn't spend too much time looking for exponential-optimized algorithms, as we'll be using this frequently for functions that don't have simple exponential forms as well.

Apparently Github doesn't like pdf and tex attachments, and I don't know how to Git, so here are the docs in a Google Drive folder:
https://drive.google.com/folderview?id=0B8o2Mni1YcPFMkEzQnJSZllBQTQ&usp=sharing

[johnzeringue]

After playing with this on my machine in Python, it seems perfect. I used the same quality measure as you (max(dists) / min(dists)). After a few cases (most quite tough), I found the algorithm needed no more than 55 iterations to bring the quality below 1 + 10e-13. 😎

I'll add this in this week and there should be a big performance boost, since we'll need a lot less Bezier segments with this feature to attain the same (or higher!) visual quality.

[bowring]

yeah - progress through collaboration!

On Tue, Oct 27, 2015 at 12:07 AM, John Zeringue notifications@github.com
wrote:

After playing with this on my machine in Python, it seems perfect. I used
the same quality measure as you (max(dists) / min(dists)). After a few
cases (most quite tough), I found the algorithm needed no more than 55
iterations to bring the quality below 1 + 10e-13. [image: 😎]

I'll add this in this week and there should be a big performance boost,
since we'll need a lot less Bezier segments with this feature to attain the
same (or higher!) visual quality.

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