Ophthalmic Optical Formulas & Thickness Mathematics

Mathematical derivations and standard equations for ophthalmic lens surfacing, sagitta curves, oblique powers, and prism decentration.

1. Exact Sagitta Equation (Lens Surface Depth)

The sagitta ($s$) of a spherical lens surface is the perpendicular distance from the vertex of the curve to the chord joining the edge endpoints across half-diameter ($y$):

          Exact: s = r - √(r² - y²)  |  Approximate: s ≈ y² / (2 · r)
        

Where r is the surface radius of curvature in millimeters (r = (n - 1) / F), and y is the radial distance from optical center to frame perimeter point. OptiLens 3D implements the exact square-root formulation across all 360 radial meridians.

2. Martin's Oblique Meridian Power Equation (Astigmatism)

In astigmatic prescriptions, the dioptric power F(θ) along any arbitrary meridian angle θ is calculated by Martin's law:

          F(θ) = Fsph + Fcyl · sin²(θ - α)
        

Where α is the prescribed cylinder axis angle. Maximum dioptric power occurs at θ = α + 90°, producing the greatest back surface sagitta and edge thickness.

3. Thick Lens Power & Center Thickness

When accounting for finite lens center thickness ($t$) and refractive index ($n$), total equivalent power ($F$) is governed by the full Gullstrand thick-lens equation:

          F = F1 + F2 - (t / n) · F1 · F2
        

4. Prentice's Rule for Decentration & Induced Prism

Decentering an optical center away from the visual axis induces prismatic deviation ($P$) proportional to decentration distance ($c$ in centimeters) and meridian power ($F$ in diopters):

          P (Δ) = c (cm) · F (D)  |  Δt (mm) = (P · d) / (100 · (n - 1))
        

5. Aspheric & Atoric Lens Geometry (Conic Section Surfacing)

Standard spherical lenses maintain a constant radius ($R$). In contrast, aspheric (AS) geometries flatten progressively toward the periphery according to the general conic sagitta equation:

          s(y) = (c · y²) / (1 + √(1 - (1 + k) · c² · y²))
        

Where c = 1 / R is apex curvature, y is radial distance, and k is the conic constant. Plus lenses experience 12% to 18% center thickness reduction; minus lenses gain 8% to 12% edge thinning with reduced minification.