openAI-math-dataset / lean /OAI /Geometry /CylinderCovering /Convexity /GeneralCapAreaDefinitions.lean
| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| namespace CylinderCovering.GeneralCapArea | |
| def profile (q : ℝ) : ℝ := (q ^ 2 + q ^ 4) / 16 | |
| def idealSpeed (q : ℝ) : ℝ := ((1 + q ^ 2) / 4) ^ 2 + 2 * (q / 2) ^ 2 | |
| def idealDensity (ε η q : ℝ) : ℝ := | |
| 1 / 4 + ε ^ 2 * (profile q + η - idealSpeed q / 8) | |
| def capDensity (ε α β : ℝ) (R : ℝ → ℝ) (q : ℝ) : ℝ := | |
| (R q) ^ 2 / Real.sqrt (1 + ε ^ 2 * (α ^ 2 + 2 * β ^ 2)) | |
| end CylinderCovering.GeneralCapArea | |
| end | |
| end OAI | |