| import OAI.Geometry.CylinderCovering.Barycentric.Cost |
| import OAI.Geometry.CylinderCovering.Barycentric.Correspondence |
|
|
| namespace OAI |
|
|
| open scoped InnerProductSpace |
|
|
| noncomputable section |
| namespace RadialCoordinates |
|
|
| theorem intercept_gram : |
| inner ℝ interceptTangentA interceptTangentA = (3 / 2 : ℝ) ∧ |
| inner ℝ interceptTangentB interceptTangentB = (3 / 2 : ℝ) ∧ |
| inner ℝ interceptTangentA interceptTangentB = (1 / 2 : ℝ) := by |
| norm_num [interceptTangentA, interceptTangentB, EuclideanSpace.inner_eq_star_dotProduct, |
| EuclideanSpace.real_norm_sq_eq, dotProduct, Fin.sum_univ_succ] |
|
|
| theorem intercept_gram_determinant : |
| inner ℝ interceptTangentA interceptTangentA * |
| inner ℝ interceptTangentB interceptTangentB - |
| (inner ℝ interceptTangentA interceptTangentB) ^ 2 = (2 : ℝ) := by |
| rw [intercept_gram.1, intercept_gram.2.1, intercept_gram.2.2] |
| norm_num |
|
|
| theorem firstDirection_sum (k t : ℝ) : ∑ i, firstDirection k t i = 0 := by |
| simp [firstDirection, angular, Fin.sum_univ_succ] |
| ring |
|
|
| theorem firstSweep_line_direction (k S t p p' : ℝ) : |
| firstSweepPoint k t S p - firstSweepPoint k t S p' = |
| (p - p') • firstDirection k t := by |
| ext i |
| fin_cases i <;> simp [firstSweepPoint, barycentricPoint, firstDirection, angular] <;> ring |
|
|
| theorem firstDirection_norm_squared (k t : ℝ) : |
| ‖firstDirection k t‖ ^ 2 = (1 + k ^ 2 * speedSquared t) / 2 := by |
| rw [EuclideanSpace.real_norm_sq_eq] |
| simp [firstDirection, speedSquared, angular, Fin.sum_univ_succ] |
| ring |
|
|
| theorem firstDirection_ne_zero (k t : ℝ) : firstDirection k t ≠ 0 := by |
| intro h |
| have hn := firstDirection_norm_squared k t |
| rw [h, norm_zero] at hn |
| have hE : 0 ≤ speedSquared t := by unfold speedSquared; positivity |
| nlinarith [mul_nonneg (sq_nonneg k) hE] |
|
|
| theorem interceptNormal_norm : ‖interceptNormal‖ = 1 := by |
| apply (sq_eq_sq₀ (norm_nonneg _) (by norm_num)).mp |
| rw [EuclideanSpace.real_norm_sq_eq] |
| have hsqrt : (Real.sqrt 2) ^ 2 = (2 : ℝ) := Real.sq_sqrt (by norm_num) |
| norm_num [interceptNormal, Fin.sum_univ_succ, div_pow, hsqrt] |
|
|
| theorem interceptNormal_tangent_orthogonal : |
| inner ℝ interceptNormal interceptTangentA = 0 ∧ |
| inner ℝ interceptNormal interceptTangentB = 0 := by |
| simp [interceptNormal, interceptTangentA, interceptTangentB, |
| EuclideanSpace.inner_eq_star_dotProduct, dotProduct, Fin.sum_univ_succ] |
| ring |
|
|
| theorem firstDirection_normal_inner (k t : ℝ) : |
| inner ℝ interceptNormal (firstDirection k t) = 1 / Real.sqrt 2 := by |
| simp [interceptNormal, firstDirection, |
| EuclideanSpace.inner_eq_star_dotProduct, dotProduct, Fin.sum_univ_succ] |
| ring |
|
|
| theorem firstDirection_projection_cosine (k t : ℝ) : |
| inner ℝ interceptNormal (firstDirection k t) / ‖firstDirection k t‖ = |
| 1 / Real.sqrt (1 + k ^ 2 * speedSquared t) := by |
| rw [firstDirection_normal_inner] |
| have hE : 0 ≤ speedSquared t := by unfold speedSquared; positivity |
| have hx : 0 < 1 + k ^ 2 * speedSquared t := by positivity |
| have hn : 0 < ‖firstDirection k t‖ := norm_pos_iff.mpr (firstDirection_ne_zero k t) |
| have hs : 0 < Real.sqrt 2 := Real.sqrt_pos.2 (by norm_num) |
| have hxroot : 0 < Real.sqrt (1 + k ^ 2 * speedSquared t) := Real.sqrt_pos.2 hx |
| apply (sq_eq_sq₀ (by positivity) (by positivity)).mp |
| simp only [div_pow, one_pow] |
| rw [firstDirection_norm_squared, Real.sq_sqrt (by norm_num : (0 : ℝ) ≤ 2), |
| Real.sq_sqrt hx.le] |
| field_simp |
|
|
| end RadialCoordinates |
|
|
| end |
|
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| end OAI |
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|
