[DO NOT MERGE] Calculate bezier curve length

[danieldouglas92]

Add the functionality within bezier_curve.cc to determine the total arc-length of the bezier curve across all user-specified coordinates, and the length of the bezier curve between each incremental pair of user-specified coordinates.

[github-actions]

Benchmark	Main	Feature	Difference (99.9% CI)
Slab interpolation simple none	1.114 ± 0.009 (s=386)	1.117 ± 0.009 (s=424)	+0.1% .. +0.4%
Slab interpolation curved simple none	1.116 ± 0.011 (s=393)	1.116 ± 0.008 (s=416)	-0.1% .. +0.3%
Spherical slab interpolation simple none	1.094 ± 0.008 (s=407)	1.095 ± 0.010 (s=418)	-0.1% .. +0.3%
Slab interpolation simple curved CMS	1.165 ± 0.013 (s=404)	1.168 ± 0.015 (s=370)	-0.0% .. +0.5%
Spherical slab interpolation simple CMS	1.442 ± 0.012 (s=331)	1.421 ± 0.013 (s=300)	-1.6% .. -1.2%
Spherical fault interpolation simple none	1.100 ± 0.009 (s=385)	1.103 ± 0.009 (s=435)	+0.1% .. +0.4%
Cartesian min max surface	2.535 ± 0.017 (s=181)	2.545 ± 0.020 (s=176)	+0.1% .. +0.6%
Spherical min max surface	7.155 ± 0.066 (s=69)	7.179 ± 0.080 (s=59)	-0.3% .. +1.0%

[danieldouglas92]

At this point I think the code that I have is correct, there was an issue with the existing implementation of the Bezier curve when dealing with points that were co-linear.

If you have points P1 and P2, to compute the Bezier curve two additional control points CP1 and CP2 must be determined which control the shape of the curve. Sometimes, the control points would be outside of the bounds of Points P1 and P2, as illustrated below:

x----o--------x----o
P1 CP1 P2 CP2

However, the control points should lie within on the vector P1P2, not beyond it, like this:
x----o--------o----x
P1 CP1 CP2 P2

This is important because when the control points lie beyond the vector P1P2, the arclength of the Bezier curve is artificially made longer.

Currently, some of the tests are still failing but they are mostly related to tests which utilize coordinates that are colinear (with the exception of the test slab_interpolation_simple_curved_CMS.dat
@MFraters this is ready for a look when you have the time!

[MFraters]

This looks good to me. thanks for finding and fixing the bug! I have a few small comments, but otherwise I think it is good to go.

[MFraters]

What are you doing here?

[danieldouglas92]

I'm not sure what is happening here? This is the same as the current file and I made no changes to it.

[danieldouglas92]

Thanks for the review! I've addressed your comments, it turns out it wasn't slab_interpolation_simple_curved_CMS that was failing, but actually slab_interpolation_simple_CMS that was failing. This means that all the tests that are actually failing currently involve coordinates that are colinear. If you agree that it is ok to update the test results then I will do that in a follow up commit and hopefully that will wrap this up.

[danieldouglas92]

After talking to @MFraters I've tentatively updated the test results. The test results should be more correct now, because the tests that were failing involved colinear points. However, we want to do a little more checking to make sure that the new code still produces correct results, so do not merge this quite yet.