openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Singular /EdgeOneGeometryDefinitions.lean
| import OAI.Geometry.CylinderCovering.Singular.EdgeOneDefinitions | |
| import OAI.Geometry.RuledCovering.Model | |
| namespace OAI | |
| noncomputable section | |
| namespace SingularEdgeOne | |
| def edgeDomain0 : Set RuledApproximation.Plane := | |
| {z | 0 ≤ z 1 ∧ z 1 ≤ (1 / 2 : ℝ) ∧ |z 0| ≤ z 1 / 2} | |
| def edgeField (p : ℝ) (z : RuledApproximation.Plane) : RuledApproximation.Plane := | |
| RuledApproximation.vec2 (edgeU p (z 0) (z 1)) (edgeV p (z 0) (z 1)) | |
| def physicalSimilarity (z : RuledApproximation.Space) : RuledApproximation.Space := | |
| RuledApproximation.vec3 (z 1 / 2) (z 0 / 2) (z 2 / 2) | |
| def labelMap (z : RuledApproximation.Plane) : RuledApproximation.Plane := | |
| RuledApproximation.vec2 (z 0 / 2) (z 1) | |
| def edgeTetrahedron : Set RuledApproximation.Space := | |
| {z | ∃ s : ℝ, 0 ≤ s ∧ s ≤ 1 ∧ |z 0| ≤ s / 2 ∧ |z 1| ≤ (1 - s) / 2 ∧ | |
| z 2 = (1 / Real.sqrt 2) * s} | |
| def edgeLower (p k : ℝ) (D : Set RuledApproximation.Plane) : Set RuledApproximation.Space := | |
| {z | ∃ b ∈ D, ∃ y : ℝ, |y| ≤ (1 / 2 : ℝ) ∧ | |
| z = RuledApproximation.vec3 (b 0 + k * y * edgeU p (b 0) (b 1)) y | |
| ((1 / Real.sqrt 2) * (b 1 + k * y * edgeV p (b 0) (b 1)))} | |
| def edgeUpper (p k : ℝ) (D : Set RuledApproximation.Plane) : Set RuledApproximation.Space := | |
| {z | ∃ b ∈ D, ∃ x : ℝ, |x| ≤ (1 / 2 : ℝ) ∧ | |
| z = RuledApproximation.vec3 x (b 0 - k * x * edgeU p (b 0) (b 1)) | |
| ((1 / Real.sqrt 2) * (1 - (b 1 - k * x * edgeV p (b 0) (b 1))))} | |
| def upperSegments (p c : ℝ) (D : Set RuledApproximation.Plane) : Set RuledApproximation.Space := | |
| {z | ∃ b ∈ D, ∃ y : ℝ, |y| ≤ 1 ∧ | |
| z = RuledApproximation.vec3 | |
| (b 0 - c * y * RuledApproximation.singularField p b 0) y | |
| (Real.sqrt 2 * (1 - (b 1 - c * y * RuledApproximation.singularField p b 1)))} | |
| end SingularEdgeOne | |
| end | |
| end OAI | |