| import OAI.Geometry.ProjectionBody.GraphChart |
| import Mathlib.Analysis.InnerProductSpace.NormDet |
| import Mathlib.LinearAlgebra.Matrix.SchurComplement |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped RealInnerProductSpace |
|
|
| namespace ProjectionCounterexample |
|
|
| /-- The Jacobian of a graph chart is the length of its unnormalized normal. -/ |
| theorem graphLinear_normDet {n : ℕ} (i : Fin (n + 1)) (a : E n) : |
| (graphLinear i a).normDet = ‖graphNormal i a‖ := by |
| classical |
| let b := EuclideanSpace.basisFun (Fin n) ℝ |
| have hGram : Matrix.gram ℝ (fun j => graphLinear i a (b j)) = |
| 1 + Matrix.replicateCol Unit (fun j => a j) * |
| Matrix.replicateRow Unit (fun j => a j) := by |
| ext j k |
| rw [Matrix.gram_apply, graphLinear_inner] |
| simp [b, EuclideanSpace.basisFun_apply, eq_comm, |
| EuclideanSpace.inner_single_right, Matrix.add_apply, Matrix.one_apply, |
| Matrix.mul_apply, Matrix.replicateCol_apply, Matrix.replicateRow_apply] |
| have hsquare : (graphLinear i a).normDet ^ 2 = ‖graphNormal i a‖ ^ 2 := by |
| calc |
| (graphLinear i a).normDet ^ 2 = |
| (Matrix.gram ℝ (fun j => graphLinear i a (b j))).det := |
| (graphLinear i a).normDet_sq_eq_det_gram b |
| _ = ‖graphNormal i a‖ ^ 2 := by |
| rw [hGram, Matrix.det_one_add_replicateCol_mul_replicateRow, |
| graphNormal_norm_sq, EuclideanSpace.real_norm_sq_eq] |
| simp [dotProduct, pow_two] |
| exact (sq_eq_sq₀ (graphLinear i a).normDet_nonneg (norm_nonneg _)).mp hsquare |
|
|
| end ProjectionCounterexample |
|
|
| end |
|
|
| end OAI |
|
|
