Theorems · Theorem · geometry

EuclideanGeometry.dist_smul_vadd_sq

∀ {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] (r : ℝ) (v : V) (p₁ p₂ : P),
  dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ =
    inner ℝ v v * r * r + 2 * inner ℝ v (p₁ -ᵥ p₂) * r + inner ℝ (p₁ -ᵥ p₂) (p₁ -ᵥ p₂)

The squared distance between points on a line (expressed as a multiple of a fixed vector added to a point) and another point, expressed as a quadratic.

Defined in

Mathlib.Geometry.Euclidean.Basic

Cited by

1 results in Mathlib

Foundations

Depth 170 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsor