Affine bases and barycentric coordinates

Suppose P is an affine space modelled on the module V over the ring k, and p : ι → P is an affine-independent family of points spanning P. Given this data, each point q : P may be written uniquely as an affine combination: q = w₀ p₀ + w₁ p₁ + ⋯ for some (finitely-supported) weights wᵢ. For each i : ι, we thus have an affine map P →ᵃ[k] k, namely q ↦ wᵢ. This family of maps is known as the family of barycentric coordinates. It is defined in this file.

The construction

Fixing i : ι, and allowing j : ι to range over the values j ≠ i, we obtain a basis bᵢ of V defined by bᵢ j = p j -ᵥ p i. Let fᵢ j : V →ₗ[k] k be the corresponding dual basis and let fᵢ = ∑ j, fᵢ j : V →ₗ[k] k be the corresponding "sum of all coordinates" form. Then the ith barycentric coordinate of q : P is 1 - fᵢ (q -ᵥ p i).

Main definitions

• AffineBasis: a structure representing an affine basis of an affine space.
• AffineBasis.coord: the map P →ᵃ[k] k corresponding to i : ι.
• AffineBasis.coord_apply_eq: the behaviour of AffineBasis.coord i on p i.
• AffineBasis.coord_apply_ne: the behaviour of AffineBasis.coord i on p j when j ≠ i.
• AffineBasis.coord_apply: the behaviour of AffineBasis.coord i on p j for general j.
• AffineBasis.coord_apply_combination: the characterisation of AffineBasis.coord i in terms of affine combinations, i.e., AffineBasis.coord i (w₀ p₀ + w₁ p₁ + ⋯) = wᵢ.

TODO

• Construct the affine equivalence between P and { f : ι →₀ k | f.sum = 1 }.

structure AffineBasis (ι : Type u₁) (k : Type u₂) {V : Type u₃} (P : Type u₄) [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] : Type (max u₁ u₄)

• toFun : ι → P
• ind' : AffineIndependent k s.toFun
• tot' : affineSpan k (Set.range s.toFun) = ⊤

An affine basis is a family of affine-independent points whose span is the top subspace.

instance AffineBasis.instInhabitedAffineBasisPUnitPUnitAddCommGroupAddGroupIsAddTorsorToAddGroupToAddGroupWithOneToRingCommRingModuleToSemiring {k : Type u_3} [Ring k] : Inhabited (AffineBasis PUnit.{u_6 + 1} k PUnit.{u_6 + 1} )

The unique point in a single-point space is the simplest example of an affine basis.

Equations

• One or more equations did not get rendered due to their size.

instance AffineBasis.funLike {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] : FunLike (AffineBasis ι k P) ι fun x => P

Equations

• AffineBasis.funLike = { coe := AffineBasis.toFun, coe_injective' := (_ : ∀ (f g : AffineBasis ι k P), f.toFun = g.toFun → f = g) }

theorem AffineBasis.ext {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] {b₁ : AffineBasis ι k P} {b₂ : AffineBasis ι k P} (h : ↑b₁ = ↑b₂) : b₁ = b₂

theorem AffineBasis.ind {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) : AffineIndependent k ↑b

theorem AffineBasis.tot {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) : affineSpan k (Set.range ↑b) = ⊤

theorem AffineBasis.nonempty {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) : Nonempty ι

def AffineBasis.reindex {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') : AffineBasis ι' k P

Composition of an affine basis and an equivalence of index types.

Equations

• AffineBasis.reindex b e = { toFun := ↑b ∘ ↑e.symm, ind' := (_ : AffineIndependent k (↑b ∘ ↑(Equiv.toEmbedding e.symm))), tot' := (_ : affineSpan k (Set.range (↑b ∘ ↑e.symm)) = ⊤) }

@[simp]

theorem AffineBasis.coe_reindex {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') : ↑(AffineBasis.reindex b e) = ↑b ∘ ↑e.symm

@[simp]

theorem AffineBasis.reindex_apply {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') (i' : ι') : ↑(AffineBasis.reindex b e) i' = ↑b (↑e.symm i')

@[simp]

theorem AffineBasis.reindex_refl {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) : AffineBasis.reindex b (Equiv.refl ι) = b

noncomputable def AffineBasis.basisOf {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι) : Basis { j // j ≠ i } k V

Given an affine basis for an affine space P, if we single out one member of the family, we obtain a linear basis for the model space V.

The linear basis corresponding to the singled-out member i : ι is indexed by {j : ι // j ≠ i} and its jth element is b j -ᵥ b i. (See basisOf_apply.)

Equations

• AffineBasis.basisOf b i = Basis.mk (_ : LinearIndependent k fun i => ↑b ↑i -ᵥ ↑b i) (_ : ⊤ ≤ Submodule.span k (Set.range fun i => ↑b ↑i -ᵥ ↑b i))

@[simp]

theorem AffineBasis.basisOf_apply {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι) (j : { j // j ≠ i }) : ↑(AffineBasis.basisOf b i) j = ↑b ↑j -ᵥ ↑b i

@[simp]

theorem AffineBasis.basisOf_reindex {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') (i : ι') : AffineBasis.basisOf (AffineBasis.reindex b e) i = Basis.reindex (AffineBasis.basisOf b (↑e.symm i)) (Equiv.subtypeEquiv e (_ : ∀ (x : ι), ¬x = ↑e.symm i ↔ ¬↑e x = i))

noncomputable def AffineBasis.coord {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι) : P →ᵃ[k] k

The ith barycentric coordinate of a point.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineBasis.linear_eq_sumCoords {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι) : (AffineBasis.coord b i).linear = -Basis.sumCoords (AffineBasis.basisOf b i)

@[simp]

theorem AffineBasis.coord_reindex {ι : Type u_1} {ι' : Type u_2} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (e : ι ≃ ι') (i : ι') : AffineBasis.coord (AffineBasis.reindex b e) i = AffineBasis.coord b (↑e.symm i)

@[simp]

theorem AffineBasis.coord_apply_eq {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (i : ι) : ↑(AffineBasis.coord b i) (↑b i) = 1

@[simp]

theorem AffineBasis.coord_apply_ne {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) {i : ι} {j : ι} (h : i ≠ j) : ↑(AffineBasis.coord b i) (↑b j) = 0

theorem AffineBasis.coord_apply {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [DecidableEq ι] (i : ι) (j : ι) : ↑(AffineBasis.coord b i) (↑b j) = if i = j then 1 else 0

@[simp]

theorem AffineBasis.coord_apply_combination_of_mem {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) {s : Finset ι} {i : ι} (hi : i ∈ s) {w : ι → k} (hw : Finset.sum s w = 1) : ↑(AffineBasis.coord b i) (↑(Finset.affineCombination k s ↑b) w) = w i

@[simp]

theorem AffineBasis.coord_apply_combination_of_not_mem {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) {s : Finset ι} {i : ι} (hi : ¬i ∈ s) {w : ι → k} (hw : Finset.sum s w = 1) : ↑(AffineBasis.coord b i) (↑(Finset.affineCombination k s ↑b) w) = 0

@[simp]

theorem AffineBasis.sum_coord_apply_eq_one {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Fintype ι] (q : P) : (Finset.sum Finset.univ fun i => ↑(AffineBasis.coord b i) q) = 1

@[simp]

theorem AffineBasis.affineCombination_coord_eq_self {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Fintype ι] (q : P) : (↑(Finset.affineCombination k Finset.univ ↑b) fun i => ↑(AffineBasis.coord b i) q) = q

@[simp]

theorem AffineBasis.linear_combination_coord_eq_self {ι : Type u_1} {k : Type u_3} {V : Type u_4} [AddCommGroup V] [Ring k] [Module k V] [Fintype ι] (b : AffineBasis ι k V) (v : V) : (Finset.sum Finset.univ fun i => ↑(AffineBasis.coord b i) v • ↑b i) = v

A variant of AffineBasis.affineCombination_coord_eq_self for the special case when the affine space is a module so we can talk about linear combinations.

theorem AffineBasis.ext_elem {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Finite ι] {q₁ : P} {q₂ : P} (h : ∀ (i : ι), ↑(AffineBasis.coord b i) q₁ = ↑(AffineBasis.coord b i) q₂) : q₁ = q₂

@[simp]

theorem AffineBasis.coe_coord_of_subsingleton_eq_one {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Subsingleton ι] (i : ι) : ↑(AffineBasis.coord b i) = 1

theorem AffineBasis.surjective_coord {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Nontrivial ι] (i : ι) : Function.Surjective ↑(AffineBasis.coord b i)

noncomputable def AffineBasis.coords {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) : P →ᵃ[k] ι → k

Barycentric coordinates as an affine map.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem AffineBasis.coords_apply {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (q : P) (i : ι) : ↑(AffineBasis.coords b) q i = ↑(AffineBasis.coord b i) q

@[simp]

theorem AffineBasis.coord_apply_centroid {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [DivisionRing k] [Module k V] [CharZero k] (b : AffineBasis ι k P) {s : Finset ι} {i : ι} (hi : i ∈ s) : ↑(AffineBasis.coord b i) (Finset.centroid k s ↑b) = (↑(Finset.card s))⁻¹

theorem AffineBasis.exists_affine_subbasis {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AddTorsor V P] [DivisionRing k] [Module k V] {t : Set P} (ht : affineSpan k t = ⊤) : ∃ s x b, ↑b = Subtype.val

theorem AffineBasis.exists_affineBasis (k : Type u_3) (V : Type u_4) (P : Type u_5) [AddCommGroup V] [AddTorsor V P] [DivisionRing k] [Module k V] : ∃ s b, ↑b = Subtype.val