Calculating Lens Edge Thickness for High Minus Prescriptions: The Sagittal Depth Formula

The sagittal depth ($s$) represents the perpendicular distance from the vertex of a spherical curve to the chord joining its edges. For an optical lens with semi-diameter $y$ (half of the lens diameter, $y = d/2$) and radius of curvature $r$:

$$\text{Exact Sagitta Formula: } s = r - \sqrt{r^2 - y^2}$$

$$\text{Approximate Sagitta Formula: } s \approx \frac{y^2}{2r} = \frac{y^2 \times F}{2000(n - 1)}$$

Where $F$ is surface dioptric power in diopters, $y$ is semi-diameter in millimeters, and $n$ is refractive index of the monomer material. For a minus lens, total edge thickness ($t_e$) is equal to center thickness ($t_c$) plus the sagittal depth of the back surface minus the front surface:

$$t_e = t_c + (s_2 - s_1)$$

Notice the quadratic variable $y^2$ in the numerator! Edge thickness increases with the square of the lens aperture radius.

• In a small 46 mm eye size round frame ($y = 23\ \text{mm}$), $y^2 = 529$.
• In a wide 56 mm eye size rectangular frame ($y = 28\ \text{mm}$), $y^2 = 784$.

Simply increasing frame width from 46 mm to 56 mm increases edge thickness by nearly 50%, completely obliterating any cosmetic gains achieved by paying for an expensive high-index material!

Consider a $-6.00\ \text{D}$ prescription with a $1.0\ \text{mm}$ center thickness across common refractive indices:

1. Standard CR-39 ($n = 1.50$): $t_e \approx 8.5\ \text{mm}$ (Extremely thick, heavy rim overhang).
2. Mid-Index 1.56: $t_e \approx 7.7\ \text{mm}$.
3. High-Index MR-8 ($n = 1.60$): $t_e \approx 6.0\ \text{mm}$ (30% thinner, outstanding optical clarity).
4. Ultra-High Index MR-7 ($n = 1.67$): $t_e \approx 5.3\ \text{mm}$ (38% thinner).
5. Extreme Index MR-174 ($n = 1.74$): $t_e \approx 4.7\ \text{mm}$ (45% thinner).