Theorems · Definition · geometry

EuclideanGeometry.reflection

{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) → [Nonempty ↥s] → [s.direction.HasOrthogonalProjection] → P ≃ᵃⁱ[𝕜] P

Reflection in an affine subspace, which is expected to be nonempty and complete. The word "reflection" is sometimes understood to mean specifically reflection in a codimension-one subspace, and sometimes more generally to cover operations such as reflection in a point. The definition here, of reflection in an affine subspace, is a more general sense of the word that includes both those common cases.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

24 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection