Theorems · Theorem · geometry

AffineSubspace.affineCombination_mem_shift

∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : AddTorsor V P]
  [inst_3 : Module k V] {ι : Type u_4} [inst_4 : Fintype ι] [Nontrivial ι] (p : ι → P) (i : ι) {w : ι → k},
  ∑ i, w i = 1 → (Finset.affineCombination k Finset.univ p) w ∈ (affineSpan k (p '' {i}ᶜ)).shift (p i) (1 - w i)

Consider a point A with barycentric coordinates associated to a collection of points P. If the coordinate associated to one of the points Pᵢ is r, then the point A is on the span of P \ {Pᵢ} shifted towards Pᵢ with parameter 1 - r.

Defined in

Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift

Cited by

1 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddTorsorModuleFintypeNontrivial