Filtration values of the alpha complex (squared radius)

[CharlesArnal]

I assume there are good reasons why the alpha complex uses the squared radius (rather than the radius) of the smallest circumspheres as filtration values, but this is very counter-intuitive, and does not agree with the definition most often found in the literature.
I know of several researchers who were unpleasantly surprised upon discovering that. Do the benefits of this convention really outweigh its disadvantages ?

In any case, the convention is explained in the reference manual, but not in the user manual https://gudhi.inria.fr/python/latest/alpha_complex_user.html# - it might be worth it to add a line or two about it.
Moreover, it says that the alpha complex is equivalent to the Cech complex - this will only be true if the Cech complex also uses the squared radius convention. Is it the case ?

Best,

Charles

[mglisse]

See also #80. If we use squared radii, people are surprised. If we do something different for unweighted and for weight 0, people are surprised. If we apply a square root to negative numbers for the weighted case, people are surprised. Something has to give... Squared radius is also what CGAL produces. But yes, improving the doc is always a good idea.
We can compute an alpha-complex using rational numbers, which would not support sqrt. However, when we convert to double, there is less reason not to do the sqrt.
The Cech uses sqrt, that's inconsistent indeed.
The equivalence could be in a loose sense, if you could deduce one from the other with a factor of 2, a sqrt, or some other similar transformation, I would still call them equivalent.

[CharlesArnal]

But I think that if someone reads that two filtrations are equivalent, they will expect e.g. the persistence diagrams to be same.

I understand your point about other use cases and parameters configurations, but I would have expected the standard case to be the one that is used most often - I don't have any solid ground for thinking that, though.

[VincentRouvreau]

Fixed by #1172