English
For a simplex s with barycentric weights w summing to 1, the signedInfDist of the affine combination of the vertices is equal to w_i times the norm of the displacement from vertex i to the opposite face.
Русский
Для простейшего конуса s с весами барицентрика w, сумма которых равна 1, s.signedInfDist i на аффинной комбинации вершин равна w_i умножить на норму вектора смещения от вершины i к противоположной грани.
LaTeX
$$$s.signedInfDist i (\text{affineCombination}(s.points, w)) = w_i \cdot \|s.points i -\! (s.faceOpposite i).orthogonalProjectionSpan(s.points i)\|$$$
Lean4
/-- The signed distance between the face of `s` excluding point `i` of that simplex and a point,
in the direction of the reference point `i`. This is expected to be used when the point at which
the distance is evaluated lies in the affine span of the simplex (any component of the distance
orthogonal to that span is disregarded). In the case of a triangle, these distances are
trilinear coordinates; in a tetrahedron, they are quadriplanar coordinates. -/
noncomputable def signedInfDist : P →ᴬ[ℝ] ℝ :=
AffineSubspace.signedInfDist (affineSpan ℝ (s.points '' { i }ᶜ)) (s.points i)
