Oriented angles and orthogonal projection. #
This file proves lemmas relating to oriented angles involving orthogonal projections.
theorem
EuclideanGeometry.oangle_self_orthogonalProjection
{V : Type u_1}
{P : Type u_2}
[NormedAddCommGroup V]
[InnerProductSpace ℝ V]
[MetricSpace P]
[NormedAddTorsor V P]
[hd2 : Fact (Module.finrank ℝ V = 2)]
[Module.Oriented ℝ V (Fin 2)]
(p : P)
{p' : P}
{s : AffineSubspace ℝ P}
[s.direction.HasOrthogonalProjection]
(hp : p ∉ s)
(h : p' ∈ s)
(hp' : p' ≠ ↑((orthogonalProjection s) p))
:
oangle p (↑((orthogonalProjection s) p)) p' = ↑(Real.pi / 2) ∨ oangle p (↑((orthogonalProjection s) p)) p' = ↑(-Real.pi / 2)
theorem
EuclideanGeometry.oangle_orthogonalProjection_self
{V : Type u_1}
{P : Type u_2}
[NormedAddCommGroup V]
[InnerProductSpace ℝ V]
[MetricSpace P]
[NormedAddTorsor V P]
[hd2 : Fact (Module.finrank ℝ V = 2)]
[Module.Oriented ℝ V (Fin 2)]
(p : P)
{p' : P}
{s : AffineSubspace ℝ P}
[s.direction.HasOrthogonalProjection]
(hp : p ∉ s)
(h : p' ∈ s)
(hp' : p' ≠ ↑((orthogonalProjection s) p))
:
oangle p' (↑((orthogonalProjection s) p)) p = ↑(Real.pi / 2) ∨ oangle p' (↑((orthogonalProjection s) p)) p = ↑(-Real.pi / 2)
theorem
EuclideanGeometry.two_zsmul_oangle_self_orthogonalProjection
{V : Type u_1}
{P : Type u_2}
[NormedAddCommGroup V]
[InnerProductSpace ℝ V]
[MetricSpace P]
[NormedAddTorsor V P]
[hd2 : Fact (Module.finrank ℝ V = 2)]
[Module.Oriented ℝ V (Fin 2)]
(p : P)
{p' : P}
{s : AffineSubspace ℝ P}
[s.direction.HasOrthogonalProjection]
(hp : p ∉ s)
(h : p' ∈ s)
(hp' : p' ≠ ↑((orthogonalProjection s) p))
:
2 • oangle p (↑((orthogonalProjection s) p)) p' = ↑Real.pi
theorem
EuclideanGeometry.two_zsmul_oangle_orthogonalProjection_self
{V : Type u_1}
{P : Type u_2}
[NormedAddCommGroup V]
[InnerProductSpace ℝ V]
[MetricSpace P]
[NormedAddTorsor V P]
[hd2 : Fact (Module.finrank ℝ V = 2)]
[Module.Oriented ℝ V (Fin 2)]
(p : P)
{p' : P}
{s : AffineSubspace ℝ P}
[s.direction.HasOrthogonalProjection]
(hp : p ∉ s)
(h : p' ∈ s)
(hp' : p' ≠ ↑((orthogonalProjection s) p))
:
2 • oangle p' (↑((orthogonalProjection s) p)) p = ↑Real.pi