Theorems · Theorem · geometry

EuclideanGeometry.dist_affineCombination

∀ {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] {ι : Type u_3} {s : Finset ι} {w₁ w₂ : ι → ℝ} (p : ι → P),
  ∑ i ∈ s, w₁ i = 1 →
    ∑ i ∈ s, w₂ i = 1 →
      have a₁ := (Finset.affineCombination ℝ s p) w₁;
      have a₂ := (Finset.affineCombination ℝ s p) w₂;
      dist a₁ a₂ * dist a₁ a₂ =
        (-∑ i₁ ∈ s, ∑ i₂ ∈ s, (w₁ - w₂) i₁ * (w₁ - w₂) i₂ * (dist (p i₁) (p i₂) * dist (p i₁) (p i₂))) / 2

The distance between two points given with affineCombination, in terms of the pairwise distances between the points in that combination.

Defined in

Mathlib.Geometry.Euclidean.Basic

Cited by

1 results in Mathlib

Foundations

Depth 168 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor