| import OAI.Geometry.CylinderCovering.Rational.FieldDefinitions |
| import OAI.Geometry.CylinderCovering.Rational.SimilarityDefinitions |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace RationalDictionary |
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|
| abbrev Plane := EuclideanSpace ℝ (Fin 2) |
| abbrev Space := EuclideanSpace ℝ (Fin 3) |
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|
| def vec2 (x y : ℝ) : Plane := !₂[x, y] |
| def vec3 (x y z : ℝ) : Space := !₂[x, y, z] |
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| def c0 : ℝ := 1 / Real.sqrt 2 |
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| def ell : ℝ := 1 / (2 * Real.sqrt 2) |
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| def g (d0 s : ℝ) : ℝ := (1 + d0) / (s + d0) |
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| def gPrime (d0 s : ℝ) : ℝ := -(1 + d0) / (s + d0) ^ 2 |
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| def v1 (d0 u s : ℝ) : ℝ := |
| c0 ^ 2 * s * g d0 s + (1 / 2) * (c0 ^ 2 * s ^ 2 - u ^ 2) * gPrime d0 s |
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|
| def v2 (d0 u s : ℝ) : ℝ := u * g d0 s |
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|
| def velocity (d0 : ℝ) (p : Plane) : Plane := vec2 (v1 d0 (p 0) (p 1)) (v2 d0 (p 0) (p 1)) |
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| def A0 (d0 : ℝ) : ℝ := c0 ^ 2 * (1 - 1 / (2 * (1 + d0))) |
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| def B0 (d0 : ℝ) : ℝ := 1 / (2 * (1 + d0)) |
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| def a0 (d0 u : ℝ) : ℝ := A0 d0 * u ^ 2 + B0 d0 * u ^ 4 |
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| def uncenteredDomain (d0 eta0 k : ℝ) : Set Plane := |
| {p | 0 ≤ p 1 ∧ p 1 ≤ 3 / 2 ∧ |p 0| ≤ c0 * p 1 ∧ |
| p 1 ≤ 1 + k ^ 2 * (a0 d0 (p 0) + eta0)} |
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|
| def labelMap (p : Plane) : Plane := vec2 (ell * p 0) ((1 - p 1) / 2) |
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| def centeredRegularization (d0 : ℝ) : ℝ := d0 / 2 |
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| def centeredTilt (d0 k : ℝ) : ℝ := 4 * (1 + d0) * k |
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| def centeredPadding (eta0 k : ℝ) : ℝ := eta0 * k ^ 2 / 2 |
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| def centeredMargin (d0 eta0 : ℝ) : ℝ := eta0 / (32 * (1 + d0) ^ 2) |
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| def lineMap (d0 k r : ℝ) (p : Plane) : Plane := p + (r * k) • velocity d0 p |
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| def inverseLabelMap (p : Plane) : Plane := vec2 (p 0 / ell) (1 - 2 * p 1) |
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| def reflectLabel (p : Plane) : Plane := vec2 (-p 0) (p 1) |
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| def uncenteredFirstSweep (d0 eta0 k : ℝ) : Set Space := |
| {z | ∃ p ∈ uncenteredDomain d0 eta0 k, ∃ r : ℝ, |r| ≤ 2 * c0 ∧ |
| z = vec3 r (lineMap d0 k r p 0) (lineMap d0 k r p 1)} |
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|
| def uncenteredSecondSweep (d0 eta0 k : ℝ) : Set Space := |
| {z | ∃ p ∈ uncenteredDomain d0 eta0 k, ∃ r : ℝ, |r| ≤ 2 * c0 ∧ |
| z = vec3 (lineMap d0 k (-r) p 0) r (2 - lineMap d0 k (-r) p 1)} |
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|
| def centeredFirstPoint (d eps x : ℝ) (p : Plane) : Space := |
| vec3 x (p 0 + eps * x * SlopeRational.a d (p 0) (p 1)) |
| (c0 * (p 1 + eps * x * SlopeRational.b d (p 0) (p 1))) |
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|
| def centeredSecondPoint (d eps y : ℝ) (p : Plane) : Space := |
| vec3 (p 0 - eps * y * SlopeRational.a d (p 0) (p 1)) y |
| (-c0 * (p 1 - eps * y * SlopeRational.b d (p 0) (p 1))) |
|
|
| def centeredFirstSweep (d eps q : ℝ) : Set Space := |
| {z | ∃ p ∈ SlopeRational.labelDomain d eps q, ∃ x : ℝ, |x| ≤ 1 / 2 ∧ |
| z = centeredFirstPoint d eps x p} |
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| def centeredSecondSweep (d eps q : ℝ) : Set Space := |
| {z | ∃ p ∈ SlopeRational.labelDomain d eps q, ∃ y : ℝ, |y| ≤ 1 / 2 ∧ |
| z = centeredSecondPoint d eps y p} |
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| def threshold (d0 k r u : ℝ) : ℝ := |
| 1 + k * r * u + k ^ 2 * (a0 d0 u - r ^ 2 * v1 d0 u 1) |
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| def uncenteredWeight (d0 k : ℝ) (p : Plane) : ℝ := |
| 1 / Real.sqrt (1 + k ^ 2 * (v1 d0 (p 0) (p 1) ^ 2 + v2 d0 (p 0) (p 1) ^ 2)) |
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|
| def centeredWeight (d eps : ℝ) (p : Plane) : ℝ := |
| c0 / Real.sqrt (1 + eps ^ 2 * |
| (SlopeRational.a d (p 0) (p 1) ^ 2 + c0 ^ 2 * SlopeRational.b d (p 0) (p 1) ^ 2)) |
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| def uncenteredCost (d0 eta0 k : ℝ) : ℝ := |
| ∫ p in uncenteredDomain d0 eta0 k, uncenteredWeight d0 k p |
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| def centeredCost (d eps q : ℝ) : ℝ := |
| ∫ p in SlopeRational.labelDomain d eps q, centeredWeight d eps p |
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| end RationalDictionary |
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| end |
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| end OAI |
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|
