# Documentation

Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation

# Rotations by oriented angles. #

This file defines rotations by oriented angles in real inner product spaces.

## Main definitions #

• Orientation.rotation is the rotation by an oriented angle with respect to an orientation.
def Orientation.rotationAux {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) :

Auxiliary construction to build a rotation by the oriented angle θ.

Instances For
@[simp]
theorem Orientation.rotationAux_apply {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) (x : V) :
↑() x = +
def Orientation.rotation {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) :

A rotation by the oriented angle θ.

Instances For
theorem Orientation.rotation_apply {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) (x : V) :
↑() x = +
theorem Orientation.rotation_symm_apply {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) (x : V) :
↑() x = -
theorem Orientation.rotation_eq_matrix_toLin {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) {x : V} (hx : x 0) :
().toLinearEquiv = ↑() (Matrix.of ![![, ], ![, ]])
@[simp]
theorem Orientation.det_rotation {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) :
LinearMap.det ().toLinearEquiv = 1

The determinant of rotation (as a linear map) is equal to 1.

@[simp]
theorem Orientation.linearEquiv_det_rotation {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) :
LinearEquiv.det ().toLinearEquiv = 1

The determinant of rotation (as a linear equiv) is equal to 1.

@[simp]
theorem Orientation.rotation_symm {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) :

The inverse of rotation is rotation by the negation of the angle.

@[simp]
theorem Orientation.rotation_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) :

Rotation by 0 is the identity.

@[simp]
theorem Orientation.rotation_pi {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) :

Rotation by π is negation.

theorem Orientation.rotation_pi_apply {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) :
↑() x = -x

Rotation by π is negation.

theorem Orientation.rotation_pi_div_two {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) :

Rotation by π / 2 is the "right-angle-rotation" map J.

@[simp]
theorem Orientation.rotation_rotation {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ₁ : Real.Angle) (θ₂ : Real.Angle) (x : V) :
↑() (↑() x) = ↑(Orientation.rotation o (θ₁ + θ₂)) x

Rotating twice is equivalent to rotating by the sum of the angles.

@[simp]
theorem Orientation.rotation_trans {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ₁ : Real.Angle) (θ₂ : Real.Angle) :
= Orientation.rotation o (θ₂ + θ₁)

Rotating twice is equivalent to rotating by the sum of the angles.

@[simp]
theorem Orientation.kahler_rotation_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) (θ : Real.Angle) :
↑(↑() (↑() x)) y = ↑() * ↑(↑() x) y

Rotating the first of two vectors by θ scales their Kahler form by cos θ - sin θ * I.

theorem Orientation.neg_rotation {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) (x : V) :
-↑() x = ↑(Orientation.rotation o ( + θ)) x

Negating a rotation is equivalent to rotation by π plus the angle.

@[simp]
theorem Orientation.neg_rotation_neg_pi_div_two {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) :
-↑() x = ↑(Orientation.rotation o ↑()) x

Negating a rotation by -π / 2 is equivalent to rotation by π / 2.

theorem Orientation.neg_rotation_pi_div_two {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) :
-↑(Orientation.rotation o ↑()) x = ↑() x

Negating a rotation by π / 2 is equivalent to rotation by -π / 2.

theorem Orientation.kahler_rotation_left' {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) (θ : Real.Angle) :
↑(↑() (↑() x)) y = ↑() * ↑(↑() x) y

Rotating the first of two vectors by θ scales their Kahler form by cos (-θ) + sin (-θ) * I.

@[simp]
theorem Orientation.kahler_rotation_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) (θ : Real.Angle) :
↑(↑() x) (↑() y) = * ↑(↑() x) y

Rotating the second of two vectors by θ scales their Kahler form by cos θ + sin θ * I.

@[simp]
theorem Orientation.oangle_rotation_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} (hx : x 0) (hy : y 0) (θ : Real.Angle) :
Orientation.oangle o (↑() x) y = - θ

Rotating the first vector by θ subtracts θ from the angle between two vectors.

@[simp]
theorem Orientation.oangle_rotation_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} (hx : x 0) (hy : y 0) (θ : Real.Angle) :
Orientation.oangle o x (↑() y) = + θ

Rotating the second vector by θ adds θ to the angle between two vectors.

@[simp]
theorem Orientation.oangle_rotation_self_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} (hx : x 0) (θ : Real.Angle) :
Orientation.oangle o (↑() x) x = -θ

The rotation of a vector by θ has an angle of -θ from that vector.

@[simp]
theorem Orientation.oangle_rotation_self_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} (hx : x 0) (θ : Real.Angle) :
Orientation.oangle o x (↑() x) = θ

A vector has an angle of θ from the rotation of that vector by θ.

@[simp]
theorem Orientation.oangle_rotation_oangle_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) :
Orientation.oangle o (↑() x) y = 0

Rotating the first vector by the angle between the two vectors results in an angle of 0.

@[simp]
theorem Orientation.oangle_rotation_oangle_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) :
Orientation.oangle o y (↑() x) = 0

Rotating the first vector by the angle between the two vectors and swapping the vectors results in an angle of 0.

@[simp]
theorem Orientation.oangle_rotation {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) (θ : Real.Angle) :
Orientation.oangle o (↑() x) (↑() y) =

Rotating both vectors by the same angle does not change the angle between those vectors.

@[simp]
theorem Orientation.rotation_eq_self_iff_angle_eq_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} (hx : x 0) (θ : Real.Angle) :
↑() x = x θ = 0

A rotation of a nonzero vector equals that vector if and only if the angle is zero.

@[simp]
theorem Orientation.eq_rotation_self_iff_angle_eq_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} (hx : x 0) (θ : Real.Angle) :
x = ↑() x θ = 0

A nonzero vector equals a rotation of that vector if and only if the angle is zero.

theorem Orientation.rotation_eq_self_iff {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (θ : Real.Angle) :
↑() x = x x = 0 θ = 0

A rotation of a vector equals that vector if and only if the vector or the angle is zero.

theorem Orientation.eq_rotation_self_iff {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (θ : Real.Angle) :
x = ↑() x x = 0 θ = 0

A vector equals a rotation of that vector if and only if the vector or the angle is zero.

@[simp]
theorem Orientation.rotation_oangle_eq_iff_norm_eq {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (y : V) :
↑() x = y x = y

Rotating a vector by the angle to another vector gives the second vector if and only if the norms are equal.

theorem Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} (hx : x 0) (hy : y 0) (θ : Real.Angle) :
= θ y = (y / x) ↑() x

The angle between two nonzero vectors is θ if and only if the second vector is the first rotated by θ and scaled by the ratio of the norms.

theorem Orientation.oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} (hx : x 0) (hy : y 0) (θ : Real.Angle) :
= θ r, 0 < r y = r ↑() x

The angle between two nonzero vectors is θ if and only if the second vector is the first rotated by θ and scaled by a positive real.

theorem Orientation.oangle_eq_iff_eq_norm_div_norm_smul_rotation_or_eq_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} (θ : Real.Angle) :
= θ x 0 y 0 y = (y / x) ↑() x θ = 0 (x = 0 y = 0)

The angle between two vectors is θ if and only if they are nonzero and the second vector is the first rotated by θ and scaled by the ratio of the norms, or θ and at least one of the vectors are zero.

theorem Orientation.oangle_eq_iff_eq_pos_smul_rotation_or_eq_zero {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} (θ : Real.Angle) :
= θ (x 0 y 0 r, 0 < r y = r ↑() x) θ = 0 (x = 0 y = 0)

The angle between two vectors is θ if and only if they are nonzero and the second vector is the first rotated by θ and scaled by a positive real, or θ and at least one of the vectors are zero.

theorem Orientation.exists_linearIsometryEquiv_eq_of_det_pos {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {f : V ≃ₗᵢ[] V} (hd : 0 < LinearMap.det f.toLinearEquiv) :
θ,

Any linear isometric equivalence in V with positive determinant is rotation.

theorem Orientation.rotation_map {V : Type u_1} {V' : Type u_2} [] [] [] [Fact ()] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) (f : V ≃ₗᵢ[] V') (x : V') :
↑(Orientation.rotation (↑(Orientation.map (Fin 2) f.toLinearEquiv) o) θ) x = f (↑() (↑() x))
@[simp]
theorem Complex.rotation (θ : Real.Angle) (z : ) :
= * z
theorem Orientation.rotation_map_complex {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) (f : ) (hf : ↑(Orientation.map (Fin 2) f.toLinearEquiv) o = Complex.orientation) (x : V) :
f (↑() x) = * f x

Rotation in an oriented real inner product space of dimension 2 can be evaluated in terms of a complex-number representation of the space.

theorem Orientation.rotation_neg_orientation_eq_neg {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (θ : Real.Angle) :
=

Negating the orientation negates the angle in rotation.

@[simp]
theorem Orientation.inner_rotation_pi_div_two_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) :
inner (↑(Orientation.rotation o ↑()) x) x = 0

The inner product between a π / 2 rotation of a vector and that vector is zero.

@[simp]
theorem Orientation.inner_rotation_pi_div_two_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) :
inner x (↑(Orientation.rotation o ↑()) x) = 0

The inner product between a vector and a π / 2 rotation of that vector is zero.

@[simp]
theorem Orientation.inner_smul_rotation_pi_div_two_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (r : ) :
inner (r ↑(Orientation.rotation o ↑()) x) x = 0

The inner product between a multiple of a π / 2 rotation of a vector and that vector is zero.

@[simp]
theorem Orientation.inner_smul_rotation_pi_div_two_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (r : ) :
inner x (r ↑(Orientation.rotation o ↑()) x) = 0

The inner product between a vector and a multiple of a π / 2 rotation of that vector is zero.

@[simp]
theorem Orientation.inner_rotation_pi_div_two_left_smul {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (r : ) :
inner (↑(Orientation.rotation o ↑()) x) (r x) = 0

The inner product between a π / 2 rotation of a vector and a multiple of that vector is zero.

@[simp]
theorem Orientation.inner_rotation_pi_div_two_right_smul {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (r : ) :
inner (r x) (↑(Orientation.rotation o ↑()) x) = 0

The inner product between a multiple of a vector and a π / 2 rotation of that vector is zero.

@[simp]
theorem Orientation.inner_smul_rotation_pi_div_two_smul_left {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (r₁ : ) (r₂ : ) :
inner (r₁ ↑(Orientation.rotation o ↑()) x) (r₂ x) = 0

The inner product between a multiple of a π / 2 rotation of a vector and a multiple of that vector is zero.

@[simp]
theorem Orientation.inner_smul_rotation_pi_div_two_smul_right {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) (x : V) (r₁ : ) (r₂ : ) :
inner (r₂ x) (r₁ ↑(Orientation.rotation o ↑()) x) = 0

The inner product between a multiple of a vector and a multiple of a π / 2 rotation of that vector is zero.

theorem Orientation.inner_eq_zero_iff_eq_zero_or_eq_smul_rotation_pi_div_two {V : Type u_1} [] [Fact ()] (o : Orientation V (Fin 2)) {x : V} {y : V} :
inner x y = 0 x = 0 r, r ↑(Orientation.rotation o ↑()) x = y

The inner product between two vectors is zero if and only if the first vector is zero or the second is a multiple of a π / 2 rotation of that vector.