Theorems · Theorem · geometry

EuclideanGeometry.orthogonalProjection_vadd_smul_vsub_orthogonalProjection

∀ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup V]
  [inst_2 : InnerProductSpace 𝕜 V] [inst_3 : MetricSpace P] [inst_4 : NormedAddTorsor V P] {s : AffineSubspace 𝕜 P}
  [inst_5 : Nonempty ↥s] [inst_6 : s.direction.HasOrthogonalProjection] {p₁ : P} (p₂ : P) (r : 𝕜) (hp : p₁ ∈ s),
  (EuclideanGeometry.orthogonalProjection s) (r • (p₂ -ᵥ ↑((EuclideanGeometry.orthogonalProjection s) p₂)) +ᵥ p₁) =
    ⟨p₁, hp⟩

Adding a vector to a point in the given subspace, then taking the orthogonal projection, produces the original point if the vector is a multiple of the result of subtracting a point's orthogonal projection from that point.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

2 results in Mathlib

Foundations

Depth 183 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection