# Affine isometries #

In this file we define AffineIsometry 𝕜 P P₂ to be an affine isometric embedding of normed add-torsors P into P₂ over normed 𝕜-spaces and AffineIsometryEquiv to be an affine isometric equivalence between P and P₂.

We also prove basic lemmas and provide convenience constructors. The choice of these lemmas and constructors is closely modelled on those for the LinearIsometry and AffineMap theories.

Since many elementary properties don't require ‖x‖ = 0 → x = 0 we initially set up the theory for SeminormedAddCommGroup and specialize to NormedAddCommGroup only when needed.

## Notation #

We introduce the notation P →ᵃⁱ[𝕜] P₂ for AffineIsometry 𝕜 P P₂, and P ≃ᵃⁱ[𝕜] P₂ for AffineIsometryEquiv 𝕜 P P₂. In contrast with the notation →ₗᵢ for linear isometries, ≃ᵢ for isometric equivalences, etc., the "i" here is a superscript. This is for aesthetic reasons to match the superscript "a" (note that in mathlib →ᵃ is an affine map, since →ₐ has been taken by algebra-homomorphisms.)

structure AffineIsometry (𝕜 : Type u_1) {V : Type u_2} {V₂ : Type u_5} (P : Type u_10) (P₂ : Type u_11) [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] extends :
Type (max (max (max u_10 u_11) u_2) u_5)

A 𝕜-affine isometric embedding of one normed add-torsor over a normed 𝕜-space into another.

• toFun : PP₂
• linear : V →ₗ[𝕜] V₂
• map_vadd' : ∀ (p : P) (v : V), self.toFun (v +ᵥ p) = self.linear v +ᵥ self.toFun p
• norm_map : ∀ (x : V), self.linear x = x
Instances For
theorem AffineIsometry.norm_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (self : P →ᵃⁱ[𝕜] P₂) (x : V) :
self.linear x = x

A 𝕜-affine isometric embedding of one normed add-torsor over a normed 𝕜-space into another.

Equations
• One or more equations did not get rendered due to their size.
Instances For
def AffineIsometry.linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
V →ₗᵢ[𝕜] V₂

The underlying linear map of an affine isometry is in fact a linear isometry.

Equations
• f.linearIsometry = let __src := f.linear; { toLinearMap := __src, norm_map' := }
Instances For
@[simp]
theorem AffineIsometry.linear_eq_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
f.linear = f.linearIsometry.toLinearMap
instance AffineIsometry.instFunLike {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] :
FunLike (P →ᵃⁱ[𝕜] P₂) P P₂
Equations
• AffineIsometry.instFunLike = { coe := fun (f : P →ᵃⁱ[𝕜] P₂) => f.toFun, coe_injective' := }
@[simp]
theorem AffineIsometry.coe_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
f.toAffineMap = f
theorem AffineIsometry.toAffineMap_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] :
Function.Injective AffineIsometry.toAffineMap
theorem AffineIsometry.coeFn_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] :
Function.Injective DFunLike.coe
theorem AffineIsometry.ext {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] {f : P →ᵃⁱ[𝕜] P₂} {g : P →ᵃⁱ[𝕜] P₂} (h : ∀ (x : P), f x = g x) :
f = g
def LinearIsometry.toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) :
V →ᵃⁱ[𝕜] V₂

Reinterpret a linear isometry as an affine isometry.

Equations
• f.toAffineIsometry = let __src := f.toAffineMap; { toAffineMap := __src, norm_map := }
Instances For
@[simp]
theorem LinearIsometry.coe_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) :
f.toAffineIsometry = f
@[simp]
theorem LinearIsometry.toAffineIsometry_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) :
f.toAffineIsometry.linearIsometry = f
@[simp]
theorem LinearIsometry.toAffineIsometry_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂) :
f.toAffineIsometry.toAffineMap = f.toAffineMap
@[simp]
theorem AffineIsometry.map_vadd {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p : P) (v : V) :
f (v +ᵥ p) = f.linearIsometry v +ᵥ f p
@[simp]
theorem AffineIsometry.map_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p1 : P) (p2 : P) :
f.linearIsometry (p1 -ᵥ p2) = f p1 -ᵥ f p2
@[simp]
theorem AffineIsometry.dist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x : P) (y : P) :
dist (f x) (f y) = dist x y
@[simp]
theorem AffineIsometry.nndist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x : P) (y : P) :
nndist (f x) (f y) = nndist x y
@[simp]
theorem AffineIsometry.edist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x : P) (y : P) :
edist (f x) (f y) = edist x y
theorem AffineIsometry.isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
theorem AffineIsometry.injective {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂) :
@[simp]
theorem AffineIsometry.map_eq_iff {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂) {x : P₁'} {y : P₁'} :
f₁ x = f₁ y x = y
theorem AffineIsometry.map_ne {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂) {x : P₁'} {y : P₁'} (h : x y) :
f₁ x f₁ y
theorem AffineIsometry.lipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
theorem AffineIsometry.antilipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
theorem AffineIsometry.continuous {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
theorem AffineIsometry.ediam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (s : Set P) :
EMetric.diam (f '' s) =
theorem AffineIsometry.ediam_range {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
= EMetric.diam Set.univ
theorem AffineIsometry.diam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (s : Set P) :
Metric.diam (f '' s) =
theorem AffineIsometry.diam_range {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
@[simp]
theorem AffineIsometry.comp_continuous_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) {α : Type u_14} [] {g : αP} :
Continuous (f g)
def AffineIsometry.id {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
P →ᵃⁱ[𝕜] P

The identity affine isometry.

Equations
• AffineIsometry.id = { toAffineMap := , norm_map := }
Instances For
@[simp]
theorem AffineIsometry.coe_id {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
AffineIsometry.id = id
@[simp]
theorem AffineIsometry.id_apply {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) :
AffineIsometry.id x = x
@[simp]
theorem AffineIsometry.id_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
AffineIsometry.id.toAffineMap =
instance AffineIsometry.instInhabited {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
Equations
• AffineIsometry.instInhabited = { default := AffineIsometry.id }
def AffineIsometry.comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) :
P →ᵃⁱ[𝕜] P₃

Composition of affine isometries.

Equations
• g.comp f = { toAffineMap := g.comp f.toAffineMap, norm_map := }
Instances For
@[simp]
theorem AffineIsometry.coe_comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) :
(g.comp f) = g f
@[simp]
theorem AffineIsometry.id_comp {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
AffineIsometry.id.comp f = f
@[simp]
theorem AffineIsometry.comp_id {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) :
f.comp AffineIsometry.id = f
theorem AffineIsometry.comp_assoc {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {V₄ : Type u_7} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} {P₄ : Type u_13} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] [NormedSpace 𝕜 V₄] [] [NormedAddTorsor V₄ P₄] (f : P₃ →ᵃⁱ[𝕜] P₄) (g : P₂ →ᵃⁱ[𝕜] P₃) (h : P →ᵃⁱ[𝕜] P₂) :
(f.comp g).comp h = f.comp (g.comp h)
instance AffineIsometry.instMonoid {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
Equations
• AffineIsometry.instMonoid = Monoid.mk npowRec
@[simp]
theorem AffineIsometry.coe_one {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
1 = id
@[simp]
theorem AffineIsometry.coe_mul {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (f : P →ᵃⁱ[𝕜] P) (g : P →ᵃⁱ[𝕜] P) :
(f * g) = f g
def AffineSubspace.subtypeₐᵢ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (s : ) [Nonempty s] :
s →ᵃⁱ[𝕜] P

AffineSubspace.subtype as an AffineIsometry.

Equations
• s.subtypeₐᵢ = let __src := s.subtype; { toAffineMap := __src, norm_map := }
Instances For
theorem AffineSubspace.subtypeₐᵢ_linear {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (s : ) [Nonempty s] :
s.subtypeₐᵢ.linear = s.direction.subtype
@[simp]
theorem AffineSubspace.subtypeₐᵢ_linearIsometry {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (s : ) [Nonempty s] :
s.subtypeₐᵢ.linearIsometry = s.direction.subtypeₗᵢ
@[simp]
theorem AffineSubspace.coe_subtypeₐᵢ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (s : ) [Nonempty s] :
s.subtypeₐᵢ = s.subtype
@[simp]
theorem AffineSubspace.subtypeₐᵢ_toAffineMap {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (s : ) [Nonempty s] :
s.subtypeₐᵢ.toAffineMap = s.subtype
structure AffineIsometryEquiv (𝕜 : Type u_1) {V : Type u_2} {V₂ : Type u_5} (P : Type u_10) (P₂ : Type u_11) [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] extends :
Type (max (max (max u_10 u_11) u_2) u_5)

An affine isometric equivalence between two normed vector spaces.

Instances For
theorem AffineIsometryEquiv.norm_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (self : P ≃ᵃⁱ[𝕜] P₂) (x : V) :
self.linear x = x
Equations
• One or more equations did not get rendered due to their size.
Instances For
def AffineIsometryEquiv.linearIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
V ≃ₗᵢ[𝕜] V₂

The underlying linear equiv of an affine isometry equiv is in fact a linear isometry equiv.

Equations
• e.linearIsometryEquiv = let __src := e.linear; { toLinearEquiv := __src, norm_map' := }
Instances For
@[simp]
theorem AffineIsometryEquiv.linear_eq_linear_isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.linear = e.linearIsometryEquiv.toLinearEquiv
instance AffineIsometryEquiv.instEquivLike {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] :
EquivLike (P ≃ᵃⁱ[𝕜] P₂) P P₂
Equations
• AffineIsometryEquiv.instEquivLike = { coe := fun (f : P ≃ᵃⁱ[𝕜] P₂) => f.toFun, inv := fun (f : P ≃ᵃⁱ[𝕜] P₂) => f.invFun, left_inv := , right_inv := , coe_injective' := }
@[simp]
theorem AffineIsometryEquiv.coe_mk {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃ[𝕜] P₂) (he : ∀ (x : V), e.linear x = x) :
{ toAffineEquiv := e, norm_map := he } = e
@[simp]
theorem AffineIsometryEquiv.coe_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.toAffineEquiv = e
theorem AffineIsometryEquiv.toAffineEquiv_injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] :
Function.Injective AffineIsometryEquiv.toAffineEquiv
theorem AffineIsometryEquiv.ext {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] {e : P ≃ᵃⁱ[𝕜] P₂} {e' : P ≃ᵃⁱ[𝕜] P₂} (h : ∀ (x : P), e x = e' x) :
e = e'
def AffineIsometryEquiv.toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
P →ᵃⁱ[𝕜] P₂

Reinterpret an AffineIsometryEquiv as an AffineIsometry.

Equations
• e.toAffineIsometry = { toAffineMap := e.toAffineEquiv, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.toAffineIsometry = e
def AffineIsometryEquiv.mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [] [NormedSpace 𝕜 V₁] [] [NormedAddTorsor V₁ P₁] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P₁P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) :
P₁ ≃ᵃⁱ[𝕜] P₂

Construct an affine isometry equivalence by verifying the relation between the map and its linear part at one base point. Namely, this function takes a map e : P₁ → P₂, a linear isometry equivalence e' : V₁ ≃ᵢₗ[k] V₂, and a point p such that for any other point p' we have e p' = e' (p' -ᵥ p) +ᵥ e p.

Equations
• = let __src := AffineEquiv.mk' e e'.toLinearEquiv p h; { toAffineEquiv := __src, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [] [NormedSpace 𝕜 V₁] [] [NormedAddTorsor V₁ P₁] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P₁P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) :
() = e
@[simp]
theorem AffineIsometryEquiv.linearIsometryEquiv_mk' {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [] [NormedSpace 𝕜 V₁] [] [NormedAddTorsor V₁ P₁] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P₁P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p : P₁) (h : ∀ (p' : P₁), e p' = e' (p' -ᵥ p) +ᵥ e p) :
().linearIsometryEquiv = e'
def LinearIsometryEquiv.toAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) :
V ≃ᵃⁱ[𝕜] V₂

Reinterpret a linear isometry equiv as an affine isometry equiv.

Equations
• e.toAffineIsometryEquiv = let __src := e.toAffineEquiv; { toAffineEquiv := __src, norm_map := }
Instances For
@[simp]
theorem LinearIsometryEquiv.coe_toAffineIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) :
e.toAffineIsometryEquiv = e
@[simp]
theorem LinearIsometryEquiv.toAffineIsometryEquiv_linearIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) :
e.toAffineIsometryEquiv.linearIsometryEquiv = e
@[simp]
theorem LinearIsometryEquiv.toAffineIsometryEquiv_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) :
e.toAffineIsometryEquiv.toAffineEquiv = e.toAffineEquiv
@[simp]
theorem LinearIsometryEquiv.toAffineIsometryEquiv_toAffineIsometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [] [] [NormedSpace 𝕜 V₂] (e : V ≃ₗᵢ[𝕜] V₂) :
e.toAffineIsometryEquiv.toAffineIsometry = e.toLinearIsometry.toAffineIsometry
theorem AffineIsometryEquiv.isometry {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
def AffineIsometryEquiv.toIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
P ≃ᵢ P₂

Reinterpret an AffineIsometryEquiv as an IsometryEquiv.

Equations
• e.toIsometryEquiv = { toEquiv := e.toEquiv, isometry_toFun := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_toIsometryEquiv {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.toIsometryEquiv = e
theorem AffineIsometryEquiv.range_eq_univ {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
= Set.univ
def AffineIsometryEquiv.toHomeomorph {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
P ≃ₜ P₂

Reinterpret an AffineIsometryEquiv as a Homeomorph.

Equations
• e.toHomeomorph = e.toIsometryEquiv.toHomeomorph
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_toHomeomorph {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.toHomeomorph = e
theorem AffineIsometryEquiv.continuous {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
theorem AffineIsometryEquiv.continuousAt {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {x : P} :
ContinuousAt (e) x
theorem AffineIsometryEquiv.continuousOn {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {s : Set P} :
ContinuousOn (e) s
theorem AffineIsometryEquiv.continuousWithinAt {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {s : Set P} {x : P} :
def AffineIsometryEquiv.refl (𝕜 : Type u_1) {V : Type u_2} (P : Type u_10) [] [] [] :
P ≃ᵃⁱ[𝕜] P

Identity map as an AffineIsometryEquiv.

Equations
• = { toAffineEquiv := , norm_map := }
Instances For
instance AffineIsometryEquiv.instInhabited {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
Equations
• AffineIsometryEquiv.instInhabited = { default := }
@[simp]
theorem AffineIsometryEquiv.coe_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
() = id
@[simp]
theorem AffineIsometryEquiv.toAffineEquiv_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
().toAffineEquiv =
@[simp]
theorem AffineIsometryEquiv.toIsometryEquiv_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
().toIsometryEquiv =
@[simp]
theorem AffineIsometryEquiv.toHomeomorph_refl {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
().toHomeomorph =
def AffineIsometryEquiv.symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
P₂ ≃ᵃⁱ[𝕜] P

The inverse AffineIsometryEquiv.

Equations
• e.symm = let __src := e.symm; { toAffineEquiv := __src, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.apply_symm_apply {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P₂) :
e (e.symm x) = x
@[simp]
theorem AffineIsometryEquiv.symm_apply_apply {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) :
e.symm (e x) = x
@[simp]
theorem AffineIsometryEquiv.symm_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.symm.symm = e
@[simp]
theorem AffineIsometryEquiv.toAffineEquiv_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.symm = e.symm.toAffineEquiv
@[simp]
theorem AffineIsometryEquiv.toIsometryEquiv_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.toIsometryEquiv.symm = e.symm.toIsometryEquiv
@[simp]
theorem AffineIsometryEquiv.toHomeomorph_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.toHomeomorph.symm = e.symm.toHomeomorph
def AffineIsometryEquiv.trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] (e : P ≃ᵃⁱ[𝕜] P₂) (e' : P₂ ≃ᵃⁱ[𝕜] P₃) :
P ≃ᵃⁱ[𝕜] P₃

Composition of AffineIsometryEquivs as an AffineIsometryEquiv.

Equations
• e.trans e' = { toAffineEquiv := e.trans e'.toAffineEquiv, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) :
(e₁.trans e₂) = e₂ e₁
@[simp]
theorem AffineIsometryEquiv.trans_refl {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.trans () = e
@[simp]
theorem AffineIsometryEquiv.refl_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
().trans e = e
@[simp]
theorem AffineIsometryEquiv.self_trans_symm {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.trans e.symm =
@[simp]
theorem AffineIsometryEquiv.symm_trans_self {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
e.symm.trans e =
@[simp]
theorem AffineIsometryEquiv.coe_symm_trans {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] (e₁ : P ≃ᵃⁱ[𝕜] P₂) (e₂ : P₂ ≃ᵃⁱ[𝕜] P₃) :
(e₁.trans e₂).symm = e₁.symm e₂.symm
theorem AffineIsometryEquiv.trans_assoc {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {V₄ : Type u_7} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} {P₄ : Type u_13} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] [NormedSpace 𝕜 V₃] [] [NormedAddTorsor V₃ P₃] [NormedSpace 𝕜 V₄] [] [NormedAddTorsor V₄ P₄] (ePP₂ : P ≃ᵃⁱ[𝕜] P₂) (eP₂G : P₂ ≃ᵃⁱ[𝕜] P₃) (eGG' : P₃ ≃ᵃⁱ[𝕜] P₄) :
ePP₂.trans (eP₂G.trans eGG') = (ePP₂.trans eP₂G).trans eGG'
instance AffineIsometryEquiv.instGroup {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
Group (P ≃ᵃⁱ[𝕜] P)

The group of affine isometries of a NormedAddTorsor, P.

Equations
• AffineIsometryEquiv.instGroup =
@[simp]
theorem AffineIsometryEquiv.coe_one {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
1 = id
@[simp]
theorem AffineIsometryEquiv.coe_mul {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (e : P ≃ᵃⁱ[𝕜] P) (e' : P ≃ᵃⁱ[𝕜] P) :
(e * e') = e e'
@[simp]
theorem AffineIsometryEquiv.coe_inv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (e : P ≃ᵃⁱ[𝕜] P) :
e⁻¹ = e.symm
@[simp]
theorem AffineIsometryEquiv.map_vadd {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (p : P) (v : V) :
e (v +ᵥ p) = e.linearIsometryEquiv v +ᵥ e p
@[simp]
theorem AffineIsometryEquiv.map_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (p1 : P) (p2 : P) :
e.linearIsometryEquiv (p1 -ᵥ p2) = e p1 -ᵥ e p2
@[simp]
theorem AffineIsometryEquiv.dist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) (y : P) :
dist (e x) (e y) = dist x y
@[simp]
theorem AffineIsometryEquiv.edist_map {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (x : P) (y : P) :
edist (e x) (e y) = edist x y
theorem AffineIsometryEquiv.bijective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
theorem AffineIsometryEquiv.injective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
theorem AffineIsometryEquiv.surjective {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
theorem AffineIsometryEquiv.map_eq_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {x : P} {y : P} :
e x = e y x = y
theorem AffineIsometryEquiv.map_ne {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {x : P} {y : P} (h : x y) :
e x e y
theorem AffineIsometryEquiv.lipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
theorem AffineIsometryEquiv.antilipschitz {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) :
@[simp]
theorem AffineIsometryEquiv.ediam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (s : Set P) :
EMetric.diam (e '' s) =
@[simp]
theorem AffineIsometryEquiv.diam_image {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) (s : Set P) :
Metric.diam (e '' s) =
@[simp]
theorem AffineIsometryEquiv.comp_continuousOn_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {α : Type u_14} [] {f : αP} {s : Set α} :
ContinuousOn (e f) s
@[simp]
theorem AffineIsometryEquiv.comp_continuous_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (e : P ≃ᵃⁱ[𝕜] P₂) {α : Type u_14} [] {f : αP} :
Continuous (e f)
def AffineIsometryEquiv.vaddConst (𝕜 : Type u_1) {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
V ≃ᵃⁱ[𝕜] P

The map v ↦ v +ᵥ p as an affine isometric equivalence between V and P.

Equations
• = let __src := ; { toAffineEquiv := __src, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_vaddConst {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
= fun (v : V) => v +ᵥ p
@[simp]
theorem AffineIsometryEquiv.coe_vaddConst' {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
() = fun (v : V) => v +ᵥ p
@[simp]
theorem AffineIsometryEquiv.coe_vaddConst_symm {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
.symm = fun (p' : P) => p' -ᵥ p
@[simp]
theorem AffineIsometryEquiv.vaddConst_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
.toAffineEquiv =
def AffineIsometryEquiv.constVSub (𝕜 : Type u_1) {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
P ≃ᵃⁱ[𝕜] V

p' ↦ p -ᵥ p' as an affine isometric equivalence.

Equations
• = let __src := ; { toAffineEquiv := __src, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_constVSub {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
= fun (x : P) => p -ᵥ x
@[simp]
theorem AffineIsometryEquiv.symm_constVSub {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (p : P) :
.symm = .toAffineIsometryEquiv.trans
def AffineIsometryEquiv.constVAdd (𝕜 : Type u_1) {V : Type u_2} (P : Type u_10) [] [] [] (v : V) :
P ≃ᵃⁱ[𝕜] P

Translation by v (that is, the map p ↦ v +ᵥ p) as an affine isometric automorphism of P.

Equations
• = let __src := ; { toAffineEquiv := __src, norm_map := }
Instances For
@[simp]
theorem AffineIsometryEquiv.coe_constVAdd {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (v : V) :
() = fun (x : P) => v +ᵥ x
@[simp]
theorem AffineIsometryEquiv.constVAdd_zero {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] :
theorem AffineIsometryEquiv.vadd_vsub {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] {f : PP₂} (hf : ) {p : P} {g : VV₂} (hg : ∀ (v : V), g v = f (v +ᵥ p) -ᵥ f p) :

The map g from V to V₂ corresponding to a map f from P to P₂, at a base point p, is an isometry if f is one.

def AffineIsometryEquiv.pointReflection (𝕜 : Type u_1) {V : Type u_2} {P : Type u_10} [] [] [] (x : P) :
P ≃ᵃⁱ[𝕜] P

Point reflection in x as an affine isometric automorphism.

Equations
Instances For
theorem AffineIsometryEquiv.pointReflection_apply {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) (y : P) :
= x -ᵥ y +ᵥ x
@[simp]
theorem AffineIsometryEquiv.pointReflection_toAffineEquiv {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) :
.toAffineEquiv =
@[simp]
theorem AffineIsometryEquiv.pointReflection_self {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) :
= x
theorem AffineIsometryEquiv.pointReflection_involutive {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) :
@[simp]
theorem AffineIsometryEquiv.pointReflection_symm {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) :
@[simp]
theorem AffineIsometryEquiv.dist_pointReflection_fixed {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) (y : P) :
dist () x = dist y x
theorem AffineIsometryEquiv.dist_pointReflection_self' {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) (y : P) :
dist () y = bit0 (x -ᵥ y)
theorem AffineIsometryEquiv.dist_pointReflection_self {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] (x : P) (y : P) :
dist () y = 2 * dist x y
theorem AffineIsometryEquiv.pointReflection_fixed_iff {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [] [] [] [] {x : P} {y : P} :
= y y = x
theorem AffineIsometryEquiv.dist_pointReflection_self_real {V : Type u_2} {P : Type u_10} [] [] (x : P) (y : P) :
dist () y = 2 * dist x y
@[simp]
theorem AffineIsometryEquiv.pointReflection_midpoint_left {V : Type u_2} {P : Type u_10} [] [] (x : P) (y : P) :
x = y
@[simp]
theorem AffineIsometryEquiv.pointReflection_midpoint_right {V : Type u_2} {P : Type u_10} [] [] (x : P) (y : P) :
y = x
theorem AffineMap.continuous_linear_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] {f : P →ᵃ[𝕜] P₂} :
Continuous f.linear

If f is an affine map, then its linear part is continuous iff f is continuous.

theorem AffineMap.isOpenMap_linear_iff {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [] [] [] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] {f : P →ᵃ[𝕜] P₂} :
IsOpenMap f.linear

If f is an affine map, then its linear part is an open map iff f is an open map.

@[simp]
theorem AffineSubspace.linear_equivMapOfInjective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [] [NormedSpace 𝕜 V₁] [] [NormedAddTorsor V₁ P₁] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (E : ) [Nonempty E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : ) :
(E.equivMapOfInjective φ ).linear = (Submodule.equivMapOfInjective φ.linear E.direction).trans (LinearEquiv.ofEq (Submodule.map φ.linear E.direction) ().direction )
@[simp]
theorem AffineSubspace.equivMapOfInjective_toFun {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [] [NormedSpace 𝕜 V₁] [] [NormedAddTorsor V₁ P₁] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (E : ) [Nonempty E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : ) (p : E) :
(E.equivMapOfInjective φ ) p = φ p,
noncomputable def AffineSubspace.equivMapOfInjective {𝕜 : Type u_1} {V₁ : Type u_3} {V₂ : Type u_5} {P₁ : Type u_8} {P₂ : Type u_11} [] [NormedSpace 𝕜 V₁] [] [NormedAddTorsor V₁ P₁] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (E : ) [Nonempty E] (φ : P₁ →ᵃ[𝕜] P₂) (hφ : ) :
E ≃ᵃ[𝕜] ()

An affine subspace is isomorphic to its image under an injective affine map. This is the affine version of Submodule.equivMapOfInjective.

Equations
• One or more equations did not get rendered due to their size.
Instances For
noncomputable def AffineSubspace.isometryEquivMap {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty E] :
E ≃ᵃⁱ[𝕜] (AffineSubspace.map φ.toAffineMap E)

Restricts an affine isometry to an affine isometry equivalence between a nonempty affine subspace E and its image.

This is an isometry version of AffineSubspace.equivMap, having a stronger premise and a stronger conclusion.

Equations
• = { toAffineEquiv := E.equivMapOfInjective φ.toAffineMap , norm_map := }
Instances For
@[simp]
theorem AffineSubspace.isometryEquivMap.apply_symm_apply {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] {E : AffineSubspace 𝕜 P₁'} [Nonempty E] {φ : P₁' →ᵃⁱ[𝕜] P₂} (x : (AffineSubspace.map φ.toAffineMap E)) :
φ (.symm x) = x
@[simp]
theorem AffineSubspace.isometryEquivMap.coe_apply {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty E] (g : E) :
() = φ g
@[simp]
theorem AffineSubspace.isometryEquivMap.toAffineMap_eq {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [] [] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [NormedSpace 𝕜 V₂] [] [NormedAddTorsor V₂ P₂] (φ : P₁' →ᵃⁱ[𝕜] P₂) (E : AffineSubspace 𝕜 P₁') [Nonempty E] :
.toAffineEquiv = (E.equivMapOfInjective φ.toAffineMap )