Formulas

Paraboloid of revolution with apex at origin and axis along z: z = r² / (4f). Focal point at (0, 0, f).

• Sag ↔ aperture ↔ focal length: a² = 4 f h, so h = a²/(4f), f = a²/(4h), a = 2√(f h).
• Rim half-angle (from focal point): tan(φ/2) = a/(2f) = D/(4f) where D = 2a. Equivalently φ = 2 arctan(a/(2f)).
• Focal ratio (f-number): f/D = f/(2a) = 1 / (4 tan(φ/2)).
• Curved surface area: A = (π/(3f)) · [(4f²+a²)3/2 − 8 f3], equivalently A = (8π/3) √f · [(f+h)3/2 − f3/2]. Shallow limit (a << f): A → π a² (flat-disk area).
• Closed-form pairs: every combination except (h, A) and (a, A) is direct. Those two are solved numerically for f by bisection on a strictly monotonic function.

Validity: only A > π a² admits a real cap (curved surface cannot be smaller than its flat aperture). The (h, A) pair always has a solution for A > 0.