| import OAI.Geometry.CylinderCovering.Footprints.ProjectionDefinitions |
| import OAI.Geometry.CylinderCovering.Radial.SetProjection |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace RadialFootprints |
| open Set MeasureTheory TriangularCovering |
| open scoped InnerProductSpace |
|
|
| theorem projectedCoordinateFootprint_area {e w : Space} |
| (p₀ a b : Space) (S : Set (ℝ × ℝ)) (he : ‖e‖ = 1) |
| (ha : inner ℝ e a = 0) (hb : inner ℝ e b = 0) (hw : inner ℝ e w = 0) : |
| area (projectedCoordinateFootprint e w p₀ a b S) = |
| (Real.sqrt (inner ℝ a a * inner ℝ b b - (inner ℝ a b)^2) / |
| Real.sqrt (1 + ‖w‖^2)) * volume.real S := by |
| change area (projection (unit (e + w)) '' |
| (RadialProjection.coordinateAffine p₀ a b '' S)) = _ |
| exact RadialProjection.radial_affine_projection_area p₀ e a b w he ha hb hw S |
|
|
| theorem projectedCoordinateFootprint_isCompact (e w p₀ a b : Space) |
| {S : Set (ℝ × ℝ)} (hS : IsCompact S) : |
| IsCompact (projectedCoordinateFootprint e w p₀ a b S) := |
| RadialProjection.affine_projection_isCompact p₀ (unit (e + w)) a b hS |
|
|
| theorem projectedCoordinateFootprint_finite {e w : Space} |
| (p₀ a b : Space) {S : Set (ℝ × ℝ)} (hS : IsCompact S) |
| (he : ‖e‖ = 1) (hw : inner ℝ e w = 0) : |
| Measure.euclideanHausdorffMeasure 2 |
| (projectedCoordinateFootprint e w p₀ a b S) ≠ ⊤ := |
| RadialProjection.affine_projection_finite p₀ (unit (e + w)) a b |
| (unit_norm (RadialProjection.sideways_ne_zero he hw)) hS |
|
|
| theorem projectedCoordinateFootprint_perpendicular {e w : Space} |
| (p₀ a b : Space) (S : Set (ℝ × ℝ)) (he : ‖e‖ = 1) |
| (hw : inner ℝ e w = 0) : |
| ∀ z ∈ projectedCoordinateFootprint e w p₀ a b S, |
| inner ℝ (unit (e + w)) z = 0 := by |
| rintro z ⟨x, hx, rfl⟩ |
| exact projection_perp (unit_norm (RadialProjection.sideways_ne_zero he hw)) x |
|
|
| end RadialFootprints |
|
|
| end |
|
|
| end OAI |
|
|
