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785c727 | 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 | import OAI.Geometry.CylinderCovering.Radial.CoordinatesDefinitions
import OAI.Geometry.CylinderCovering.Angular.CapsDefinitions
namespace OAI
noncomputable section
namespace RadialCoordinates
def firstSweep (k : ℝ) : Set BarycentricSpace :=
{v | (∑ i, v i = 1) ∧ ∃ t S : ℝ,
t ∈ Set.Icc 0 1 ∧ 0 ≤ S ∧ S ≤ cap k t ∧ |v 2 - v 3| ≤ 1 ∧
v 0 = S * t + k * (v 2 - v 3) * t ^ 2 ∧
v 1 = S * (1 - t) - k * (v 2 - v 3) * (1 - t) ^ 2}
def secondSweep (k : ℝ) : Set BarycentricSpace :=
{v | (∑ i, v i = 1) ∧ ∃ r T : ℝ,
r ∈ Set.Icc 0 1 ∧ 0 ≤ T ∧ T ≤ cap k r ∧ |v 0 - v 1| ≤ 1 ∧
v 2 = T * r - k * (v 0 - v 1) * r ^ 2 ∧
v 3 = T * (1 - r) + k * (v 0 - v 1) * (1 - r) ^ 2}
def firstSweepPoint (k t S p : ℝ) : BarycentricSpace :=
barycentricPoint (S + k * p * angular t)
(S * angular t + k * p * (1 + angular t ^ 2) / 2) p
def secondSweepPoint (k r T q : ℝ) : BarycentricSpace :=
barycentricPoint (1 - T + k * q * angular r) q
(T * angular r - k * q * (1 + angular r ^ 2) / 2)
def firstSweepWithCap (k : ℝ) (R : ℝ → ℝ) : Set BarycentricSpace :=
(fun z : (ℝ × ℝ) × ℝ => firstSweepPoint k z.1.1 z.1.2 z.2) ''
(AngularCaps.radialParameterSet 0 1 R ×ˢ Set.Icc (-1) 1)
def secondSweepWithCap (k : ℝ) (R : ℝ → ℝ) : Set BarycentricSpace :=
(fun z : (ℝ × ℝ) × ℝ => secondSweepPoint k z.1.1 z.1.2 z.2) ''
(AngularCaps.radialParameterSet 0 1 R ×ˢ Set.Icc (-1) 1)
end RadialCoordinates
end
end OAI
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