Use Eriksson's formula for acute triangle areas - #670
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Addresses #190.
Summary
atan2spherical excess formula when all three triangle edges are at most 90 degrees.(b - a) x (c - a)to retain relative accuracy for small triangles.Numerical validation
A 100,000-triangle comparison against a 100-decimal-digit oracle covered well-shaped, tiny, randomly rotated skinny, and near-90-degree triangles. The largest observed absolute candidate error was
8.88e-16, below the documented5e-15maximum error.An unguarded formula was also tested on near-antipodal inputs and produced errors as large as
0.65steradians. The pairwise non-negative dot-product guard avoids that unstable region and leaves those triangles on the existing implementation.Benchmark results
Release build, LLVM 23.1.0, x86-64 Windows; five repetitions:
The fallback overhead is the cost of the three pairwise dot-product checks.
Tests
s2measures_tests2loop_measures_tests2shape_measures_tests2shape_index_measures_testExactFloatTest.frexpcase expects glibc's NaN/Infinity exponent behavior and fails with the Windows MinGW C runtime.