Angles and orthogonal projection.

This file proves lemmas relating to angles involving orthogonal projections.

@[simp]

theorem EuclideanGeometry.angle_self_orthogonalProjection {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p : P) {p' : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (h : p' ∈ s) : angle p (↑((orthogonalProjection s) p)) p' = Real.pi / 2

@[simp]

theorem EuclideanGeometry.angle_orthogonalProjection_self {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p : P) {p' : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (h : p' ∈ s) : angle p' (↑((orthogonalProjection s) p)) p = Real.pi / 2

theorem EuclideanGeometry.dist_orthogonalProjection_eq_sin_mul_dist {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p : P) {q : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (h : q ∈ s) : dist p ↑((orthogonalProjection s) p) = Real.sin (angle p q ↑((orthogonalProjection s) p)) * dist p q

theorem EuclideanGeometry.dist_orthogonalProjection_eq_sin_mul_dist_of_collinear {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p : P) {q r : P} {s : AffineSubspace ℝ P} [s.direction.HasOrthogonalProjection] (hq : q ∈ s) (hr : r ∈ s) (hcollinear : Collinear ℝ {↑((orthogonalProjection s) p), q, r}) (hqr : q ≠ r) : dist p ↑((orthogonalProjection s) p) = Real.sin (angle p q r) * dist p q