# mathlib3documentation

geometry.euclidean.angle.oriented.rotation

# Rotations by oriented angles. #

THIS FILE IS SYNCHRONIZED WITH MATHLIB4. Any changes to this file require a corresponding PR to mathlib4.

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.
noncomputable def orientation.rotation_aux {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) :

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

Equations
@[simp]
theorem orientation.rotation_aux_apply {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) (x : V) :
@[irreducible]
noncomputable def orientation.rotation {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) :

A rotation by the oriented angle θ.

Equations
theorem orientation.rotation_apply {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) (x : V) :
theorem orientation.rotation_symm_apply {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) (x : V) :
theorem orientation.rotation_eq_matrix_to_lin {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) {x : V} (hx : x 0) :
= (matrix.to_lin hx) hx)) (matrix.of ![![θ.cos, -θ.sin], ![θ.sin, θ.cos]])
@[simp]
theorem orientation.det_rotation {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) :

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

@[simp]
theorem orientation.linear_equiv_det_rotation {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) :

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

@[simp]
theorem orientation.rotation_symm {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) :
(o.rotation θ).symm = o.rotation (-θ)

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

@[simp]
theorem orientation.rotation_zero {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) :

Rotation by 0 is the identity.

@[simp]
theorem orientation.rotation_pi {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) :

Rotation by π is negation.

theorem orientation.rotation_pi_apply {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (x : V) :

Rotation by π is negation.

theorem orientation.rotation_pi_div_two {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) :

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

@[simp]
theorem orientation.rotation_rotation {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ₁ θ₂ : real.angle) (x : V) :
(o.rotation θ₁) ((o.rotation θ₂) x) = (o.rotation (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 (fdim V = 2)] (o : (fin 2)) (θ₁ θ₂ : real.angle) :
(o.rotation θ₁).trans (o.rotation θ₂) = o.rotation (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 (fdim V = 2)] (o : (fin 2)) (x y : V) (θ : real.angle) :
((o.kahler) ((o.rotation θ) x)) y = * ((o.kahler) 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 (fdim V = 2)] (o : (fin 2)) (θ : real.angle) (x : V) :
-(o.rotation θ) x = (o.rotation (real.pi + 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 (fdim V = 2)] (o : (fin 2)) (x : V) :

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 (fdim V = 2)] (o : (fin 2)) (x : V) :

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

theorem orientation.kahler_rotation_left' {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (x y : V) (θ : real.angle) :
((o.kahler) ((o.rotation θ) x)) y = ((-θ).exp_map_circle) * ((o.kahler) 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 (fdim V = 2)] (o : (fin 2)) (x y : V) (θ : real.angle) :
((o.kahler) x) ((o.rotation θ) y) = (θ.exp_map_circle) * ((o.kahler) 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 (fdim V = 2)] (o : (fin 2)) {x y : V} (hx : x 0) (hy : y 0) (θ : real.angle) :
o.oangle ((o.rotation θ) x) y = o.oangle 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 (fdim V = 2)] (o : (fin 2)) {x y : V} (hx : x 0) (hy : y 0) (θ : real.angle) :
o.oangle x ((o.rotation θ) y) = o.oangle 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 (fdim V = 2)] (o : (fin 2)) {x : V} (hx : x 0) (θ : real.angle) :
o.oangle ((o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) {x : V} (hx : x 0) (θ : real.angle) :
o.oangle x ((o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) (x y : V) :
o.oangle ((o.rotation (o.oangle x y)) x) y = 0

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

@[simp]
theorem orientation.oangle_rotation_oangle_right {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (x y : V) :
o.oangle y ((o.rotation (o.oangle x y)) x) = 0

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

@[simp]
theorem orientation.oangle_rotation {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (x y : V) (θ : real.angle) :
o.oangle ((o.rotation θ) x) ((o.rotation θ) y) = o.oangle 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 (fdim V = 2)] (o : (fin 2)) {x : V} (hx : x 0) (θ : real.angle) :
(o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) {x : V} (hx : x 0) (θ : real.angle) :
x = (o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) (x : V) (θ : real.angle) :
(o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) (x : V) (θ : real.angle) :
x = (o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) (x y : V) :
(o.rotation (o.oangle x y)) 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 (fdim V = 2)] (o : (fin 2)) {x y : V} (hx : x 0) (hy : y 0) (θ : real.angle) :
o.oangle x y = θ y = (y / x) (o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) {x y : V} (hx : x 0) (hy : y 0) (θ : real.angle) :
o.oangle x y = θ (r : ), 0 < r y = r (o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) {x y : V} (θ : real.angle) :
o.oangle x y = θ x 0 y 0 y = (y / x) (o.rotation θ) 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 (fdim V = 2)] (o : (fin 2)) {x y : V} (θ : real.angle) :
o.oangle x y = θ (x 0 y 0 (r : ), 0 < r y = r (o.rotation θ) 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_linear_isometry_equiv_eq_of_det_pos {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) {f : V ≃ₗᵢ[] V} (hd : 0 < ) :
(θ : real.angle), f = o.rotation θ

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

theorem orientation.rotation_map {V : Type u_1} {V' : Type u_2} [ V'] [fact (fdim V = 2)] [fact (fdim V' = 2)] (o : (fin 2)) (θ : real.angle) (f : V ≃ₗᵢ[] V') (x : V') :
@[protected, simp]
theorem complex.rotation (θ : real.angle) (z : ) :
theorem orientation.rotation_map_complex {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (θ : real.angle) (f : V ≃ₗᵢ[] ) (hf : (orientation.map (fin 2) f.to_linear_equiv) o = complex.orientation) (x : V) :
f ((o.rotation θ) x) = (θ.exp_map_circle) * 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 (fdim V = 2)] (o : (fin 2)) (θ : real.angle) :
(-o).rotation θ = o.rotation (-θ)

Negating the orientation negates the angle in rotation.

@[simp]
theorem orientation.inner_rotation_pi_div_two_left {V : Type u_1} [fact (fdim V = 2)] (o : (fin 2)) (x : V) :

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 (fdim V = 2)] (o : (fin 2)) (x : V) :
((o.rotation (real.pi / 2)) 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 (fdim V = 2)] (o : (fin 2)) (x : V) (r : ) :

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 (fdim V = 2)] (o : (fin 2)) (x : V) (r : ) :
(r (o.rotation (real.pi / 2)) 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 (fdim V = 2)] (o : (fin 2)) (x : V) (r : ) :

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 (fdim V = 2)] (o : (fin 2)) (x : V) (r : ) :

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 (fdim V = 2)] (o : (fin 2)) (x : V) (r₁ r₂ : ) :
has_inner.inner (r₁ (o.rotation (real.pi / 2)) 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 (fdim V = 2)] (o : (fin 2)) (x : V) (r₁ r₂ : ) :
has_inner.inner (r₂ x) (r₁ (o.rotation (real.pi / 2)) 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 (fdim V = 2)] (o : (fin 2)) {x y : V} :
= 0 x = 0 (r : ), r (o.rotation (real.pi / 2)) 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.