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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 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | import OAI.Geometry.RuledCovering.Model
namespace OAI
noncomputable section
open Set MeasureTheory
open scoped InnerProductSpace
namespace CylinderCovering.BeamStrip
abbrev Plane := EuclideanSpace ℝ (Fin 2)
abbrev Space := EuclideanSpace ℝ (Fin 3)
def vec (x y z : ℝ) : Space := !₂[x, y, z]
def vertical : Space := vec 0 0 1
def denominator (q : ℝ) : ℝ := Real.sqrt (2 * (1 + q ^ 2))
def normal (q : ℝ) : Space :=
vec ((1 + q) / denominator q) ((1 - q) / denominator q) 0
def axis (q : ℝ) : Space :=
vec ((1 - q) / denominator q) (-(1 + q) / denominator q) 0
def rectangle (q A l r : ℝ) : Set Space :=
{z | ∃ s ∈ Icc (-A) A, ∃ t ∈ Icc l r, z = s • normal q + t • vertical}
def cylinder (q A l r : ℝ) : Set Space :=
{z | ∃ b ∈ rectangle q A l r, ∃ s : ℝ, z = b + s • axis q}
def area (S : Set Space) : ℝ := (Measure.euclideanHausdorffMeasure 2 S).toReal
def strip (β M a b : ℝ) (W q : ℝ → ℝ) : Set Space :=
{z | ∃ x y t : ℝ, z = vec (β*x) (β*y) t ∧ t ∈ Icc a b ∧
|x| ≤ M ∧ |y| ≤ M ∧ |(1+q t)*x+(1-q t)*y| ≤ W t}
def IsHorizontalRectangle (C : RuledApproximation.Cylinder) : Prop :=
∃ q A l r : ℝ, 0 < A ∧ l < r ∧ C.axis = axis q ∧
((fun z : ((Submodule.span ℝ ({C.axis} : Set Space))ᗮ) => (z : Space)) '' C.base) = rectangle q A l r
def IsRectangle (B : Set Space) : Prop :=
∃ c u v : Space, inner ℝ u v = 0 ∧ u ≠ 0 ∧ v ≠ 0 ∧
B = {z | ∃ s ∈ Icc (0 : ℝ) 1, ∃ t ∈ Icc (0 : ℝ) 1,
z = c + s • u + t • v}
def HasRectangularBase (C : RuledApproximation.Cylinder) : Prop :=
IsRectangle ((fun z : ((Submodule.span ℝ ({C.axis} : Set Space))ᗮ) =>
(z : Space)) '' C.base)
end CylinderCovering.BeamStrip
end
end OAI
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