Triangulation_3: Store subdeterminant for in_sphere predicate

[afabri]

Summary of Changes

Following [Marot &al] we store four sub determinants in each cell to accelerate the in sphere test needed for the Delaunay property.

This is only a draft pull request as currently I store the four sub determinants as info() and the code is in an #ifdef macro. We should test at compile time for the presence of a member function sub_determinants() to conditionally activate code.

Todo

• Nicer design than #ifdef
• We probably need another epsilon for the static filter.
• Benchmarking
• See if we can do something better for infinite cells
• Documentation

Release Management

• Affected package(s): Triangulation_3
• Feature/Small Feature (if any):
• Link to compiled documentation (obligatory for small feature) wrong link name to be changed
• License and copyright ownership: unchanged

[afabri]

Rather frustrating it seems to be 10% slower. Although with CGAL_PROFILE we see the same ratio of in sphere tests to updates of sub determinants. This holds for points on a surface as well as random points in a sphere.

[mglisse]

Rather frustrating it seems to be 10% slower.

The "true" computations are just pt* and det, all the rest is overhead in some sense.
If the special point was not t but any other, say p, then qt* and others would not depend on the extra point and their max could be stored in the cached array, that would reduce the work needed at each evaluation.
I think the cited paper uses a more static filter, the bound for det is precomputed once globally for the scene. That's obviously faster, at the cost of more filter failures.

[sloriot]

Successfully tested in CGAL-6.3-Ic-6

[afabri]

I think the cited paper uses a more static filter, the bound for det is precomputed once globally for the scene. That's obviously faster, at the cost of more filter failures.

I gave a quick try what @sloriot suggested me today: The max values computed in the static filter can be upper bounded by the span of the bounding box of the input points. That leads to 10% filter failures in the in-sphere predicate for a random point cloud in a unit cube and to 75% filter failures for points forming a surface Nefertiti (rescaled to be also in the unit cube).