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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 | import OAI.Geometry.ProjectionBody.GraphJacobian
import OAI.Geometry.ProjectionBody.HyperplaneJacobian
import OAI.Geometry.ProjectionBody.Definitions
import Mathlib.Tactic.FieldSimp
namespace OAI
noncomputable section
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
/-- Projection of a graph chart cancels the length of its unnormalized normal. -/
theorem graphProjection_normDet {n : ℕ} (i : Fin (n + 1)) (a : E n)
{u : E (n + 1)} (hu : ‖u‖ = 1) :
((project u).toLinearMap.comp (graphLinear i a)).normDet =
|⟪graphNormal i a, u⟫| := by
let w := graphNormal i a
have hw : ‖w‖ ≠ 0 := norm_ne_zero_iff.mpr (graphNormal_ne_zero i a)
let v := ‖w‖⁻¹ • w
have hv : ‖v‖ = 1 := by
simp [v, norm_smul, hw]
have hmem (x : E n) : ⟪v, graphLinear i a x⟫ = 0 := by
rw [show v = ‖w‖⁻¹ • graphNormal i a from rfl, real_inner_smul_left,
graphNormal_inner_graphLinear, mul_zero]
change ((perpendicularTo u).starProjection.toLinearMap.comp (graphLinear i a)).normDet = _
rw [normDet_project_comp (graphLinear i a) hv hu hmem (by simp),
graphLinear_normDet]
change |⟪‖w‖⁻¹ • w, u⟫| * ‖w‖ = |⟪w, u⟫|
rw [real_inner_smul_left, abs_mul, abs_inv, abs_of_nonneg (norm_nonneg w)]
field_simp [hw]
theorem coordinateProjection_normDet {n : ℕ} (i : Fin (n + 1))
{u : E (n + 1)} (hu : ‖u‖ = 1) :
((project u).toLinearMap.comp (insertZero i).toLinearMap).normDet = |u i| := by
have hL : graphLinear i 0 = (insertZero i).toLinearMap := by
ext x j
simp [graphLinear]
have h := graphProjection_normDet i 0 hu
simpa [hL, graphNormal, EuclideanSpace.inner_single_left] using h
end ProjectionCounterexample
end
end OAI
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