| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace RadialCoordinates |
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| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
| abbrev Space := EuclideanSpace ℝ (Fin 3) |
| abbrev BarycentricSpace := EuclideanSpace ℝ (Fin 4) |
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| def barycentricSimplex : Set BarycentricSpace := |
| {z | (∀ i, 0 ≤ z i) ∧ ∑ i, z i = 1} |
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| def barycentricPoint (s q p : ℝ) : BarycentricSpace := |
| !₂[(s + q) / 2, (s - q) / 2, (1 - s + p) / 2, (1 - s - p) / 2] |
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| def canonicalPoint (s q p : ℝ) : Space := |
| !₂[2 * q, 2 * p, Real.sqrt 2 * (2 * s - 1)] |
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| def canonicalTetrahedron : Set Space := |
| {z | ∃ X Y Z : ℝ, -1 ≤ Z ∧ Z ≤ 1 ∧ |X| ≤ 1 + Z ∧ |
| |Y| ≤ 1 - Z ∧ z = !₂[X, Y, Real.sqrt 2 * Z]} |
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| def angular (t : ℝ) : ℝ := 2 * t - 1 |
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| def velocity (t : ℝ) : Plane := !₂[t ^ 2, -(1 - t) ^ 2] |
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| def ray (t : ℝ) : Plane := !₂[t, 1 - t] |
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| def capPolynomial (v : ℝ) : ℝ := v ^ 2 + 4 * v ^ 4 + 1 / 1000 |
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| def cap (k t : ℝ) : ℝ := 1 / 2 + k ^ 2 * capPolynomial (angular t / 2) |
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| def speedSquared (t : ℝ) : ℝ := |
| 2 * angular t ^ 2 + (t ^ 2 + (1 - t) ^ 2) ^ 2 |
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| def firstSweepImagePoint (k S t p : ℝ) : BarycentricSpace := |
| barycentricPoint (S + k * p * angular t) |
| (S * angular t + k * p * (t ^ 2 + (1 - t) ^ 2)) p |
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| def secondSweepImagePoint (k T r q : ℝ) : BarycentricSpace := |
| barycentricPoint (1 - T + k * q * angular r) q |
| (T * angular r - k * q * (r ^ 2 + (1 - r) ^ 2)) |
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| def firstSweepImage (k : ℝ) : Set BarycentricSpace := |
| {v | ∃ t ∈ Set.Icc (0 : ℝ) 1, ∃ S ∈ Set.Icc (0 : ℝ) (cap k t), |
| ∃ p ∈ Set.Icc (-1 : ℝ) 1, v = firstSweepImagePoint k S t p} |
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| def secondSweepImage (k : ℝ) : Set BarycentricSpace := |
| {v | ∃ r ∈ Set.Icc (0 : ℝ) 1, ∃ T ∈ Set.Icc (0 : ℝ) (cap k r), |
| ∃ q ∈ Set.Icc (-1 : ℝ) 1, v = secondSweepImagePoint k T r q} |
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| def J (k : ℝ) : ℝ := Real.sqrt 2 * |
| ∫ t in (0 : ℝ)..1, (cap k t) ^ 2 / (2 * Real.sqrt (1 + k ^ 2 * speedSquared t)) |
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| def interceptTangentA : BarycentricSpace := !₂[1, 0, -1 / 2, -1 / 2] |
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| def interceptTangentB : BarycentricSpace := !₂[0, 1, -1 / 2, -1 / 2] |
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| def interceptNormal : BarycentricSpace := !₂[0, 0, 1 / Real.sqrt 2, -1 / Real.sqrt 2] |
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| def firstDirection (k t : ℝ) : BarycentricSpace := |
| !₂[k * t ^ 2, -k * (1 - t) ^ 2, |
| (1 - k * angular t) / 2, (-1 - k * angular t) / 2] |
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| end RadialCoordinates |
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| end |
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| end OAI |
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