Theorems · Theorem · geometry

Affine.Simplex.inner_mongePoint_vsub_face_centroid_vsub

∀ {V : Type u_1} {P : Type u_2} [inst : NormedAddCommGroup V] [inst_1 : InnerProductSpace ℝ V] [inst_2 : MetricSpace P]
  [inst_3 : NormedAddTorsor V P] {n : ℕ} (s : Affine.Simplex ℝ P (n + 2)) {i₁ i₂ : Fin (n + 3)},
  inner ℝ (s.mongePoint -ᵥ Finset.centroid ℝ {i₁, i₂}ᶜ s.points) (s.points i₁ -ᵥ s.points i₂) = 0

The Monge point of an (n+2)-simplex, minus the centroid of an n-dimensional face, is orthogonal to the difference of the two vertices not in that face.

Defined in

Mathlib.Geometry.Euclidean.MongePoint

Cited by

1 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor