English
For a simplex s, the excenter weights unnormalized are defined by signs and heights; they form the unnormalized barycentric weights.
Русский
Для симплекса s неопределённые нормированные веса эксенцеров задаются знаками и высотами вершин; это ненормированные барицентрические веса.
LaTeX
$$$\text{excenterWeightsUnnorm}(signs,i) = (\text{if } i \in signs \text{ then } -1 \text{ else } 1) \cdot (s.height(i))^{-1}$$$
Lean4
/-- The unnormalized weights of the vertices in an affine combination that gives an excenter with
signs determined by the given set of indices (for the empty set, this is the incenter; for a
singleton set, this is the excenter opposite a vertex). An excenter with those signs exists if
and only if the sum of these weights is nonzero (so the normalized weights sum to 1). -/
def excenterWeightsUnnorm (signs : Finset (Fin (n + 1))) (i : Fin (n + 1)) : ℝ :=
(if i ∈ signs then -1 else 1) * (s.height i)⁻¹
