File size: 2,624 Bytes
2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 | import OAI.Geometry.ProjectionBody.FacetCoordinates
import Mathlib.Tactic.Abel
namespace OAI
noncomputable section
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
/-- A graph chart with one coordinate determined by a linear equation. -/
def graphLinear {n : ℕ} (i : Fin (n + 1)) (a : E n) : E n →ₗ[ℝ] E (n + 1) where
toFun x := insertZero i x - ⟪a, x⟫ • EuclideanSpace.single i 1
map_add' x y := by
simp only [map_add, inner_add_right, add_smul]
abel
map_smul' c x := by
simp only [map_smul, real_inner_smul_right, smul_sub, smul_smul, RingHom.id_apply]
/-- The corresponding unnormalized normal has coordinate `i` equal to one. -/
def graphNormal {n : ℕ} (i : Fin (n + 1)) (a : E n) : E (n + 1) :=
insertZero i a + EuclideanSpace.single i 1
@[simp] theorem graphNormal_same {n : ℕ} (i : Fin (n + 1)) (a : E n) :
graphNormal i a i = 1 := by simp [graphNormal]
theorem graphNormal_ne_zero {n : ℕ} (i : Fin (n + 1)) (a : E n) :
graphNormal i a ≠ 0 := by
intro h
have := congrArg (fun x : E (n + 1) => x i) h
simp at this
@[simp] theorem graphLinear_same {n : ℕ} (i : Fin (n + 1)) (a x : E n) :
graphLinear i a x i = -⟪a, x⟫ := by simp [graphLinear]
@[simp] theorem graphLinear_succAbove {n : ℕ} (i : Fin (n + 1))
(a x : E n) (j : Fin n) :
graphLinear i a x (i.succAbove j) = x j := by simp [graphLinear]
theorem graphLinear_inner {n : ℕ} (i : Fin (n + 1)) (a x y : E n) :
⟪graphLinear i a x, graphLinear i a y⟫ = ⟪x, y⟫ + ⟪a, x⟫ * ⟪a, y⟫ := by
simp [graphLinear, inner_sub_left, inner_sub_right, real_inner_smul_left,
real_inner_smul_right, EuclideanSpace.inner_single_left,
EuclideanSpace.inner_single_right, (insertZero i).inner_map_map, mul_comm]
theorem graphNormal_inner_graphLinear {n : ℕ} (i : Fin (n + 1)) (a x : E n) :
⟪graphNormal i a, graphLinear i a x⟫ = 0 := by
simp [graphNormal, graphLinear, inner_add_left, inner_sub_right,
real_inner_smul_right, EuclideanSpace.inner_single_left,
EuclideanSpace.inner_single_right, (insertZero i).inner_map_map]
theorem graphNormal_norm_sq {n : ℕ} (i : Fin (n + 1)) (a : E n) :
‖graphNormal i a‖ ^ 2 = 1 + ‖a‖ ^ 2 := by
calc
‖graphNormal i a‖ ^ 2 = ⟪graphNormal i a, graphNormal i a⟫ :=
(real_inner_self_eq_norm_sq _).symm
_ = 1 + ‖a‖ ^ 2 := by
unfold graphNormal
rw [inner_add_left, inner_add_right, inner_add_right]
simp [EuclideanSpace.inner_single_left, EuclideanSpace.inner_single_right,
add_comm]
end ProjectionCounterexample
end
end OAI
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