The cotangent weights divided by 2·√(t12·t23·t31·l123). For a needle/cap
(sliver) triangle one of the t-values is the difference of near-equal edge
lengths → catastrophic cancellation, and the area under the sqrt loses
precision, feeding large relative error into every cotangent weight, the
Hessian, and the linear solve (numerical-stability audit N5).
Replace the area computation with Kahan's stable side-length formula
(sort a≥b≥c, evaluate ¼·√[(a+(b+c))(c−(a−b))(c+(a−b))(a+(b−c))]). The
denominator is still exactly 8·Area for well-shaped triangles but accurate
for slivers. The triangle-inequality guard and the cotangent numerators are
unchanged.
Test: CotWeights_SliverMatchesHighPrecisionReference cross-checks a thin
triangle (apex ≈ 0.01 rad) against a long-double law-of-cosines reference.
292/292 CGAL tests pass.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
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