Theorems · Theorem · geometry

Affine.Simplex.faceOppositeCentroid_eq_smul_vsub_vadd_point

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
  [inst_3 : AddTorsor V P] {n : ℕ} [inst_4 : NeZero n] [CharZero k] (s : Affine.Simplex k P n) (i : Fin (n + 1)),
  s.faceOppositeCentroid i = (↑n)⁻¹ • (s.centroid -ᵥ s.points i) +ᵥ s.centroid

The point faceOppositeCentroid of an n-simplex can be obtained from the centroid by adding n⁻¹ times the vector from the vertex to the centroid.

Defined in

Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Cited by

2 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

DivisionRingAddCommGroupModuleAddTorsorNeZeroCharZero