The Sagitta (Sag) Depth Formulas: Exact Pythagorean Theorem vs Approximate $s = y^2 / 2r$ in Lens Thickness

Consider a circular lens surface with radius of curvature $r$ and semi-diameter (half-chord) $y$. Applying the Pythagorean theorem to the right-angled triangle formed by the center of curvature:

$$(r - s)^2 + y^2 = r^2 \implies r - s = \sqrt{r^2 - y^2} \implies \mathbf{s_{\text{exact}} = r - \sqrt{r^2 - y^2}}$$

Alternatively expressed using binomial expansion:

$$s = r \left[ 1 - \left(1 - \frac{y^2}{r^2}\right)^{1/2} \right] = \mathbf{\frac{y^2}{2r} + \frac{y^4}{8r^3} + \frac{y^6}{16r^5} + \dots}$$

By truncating the binomial expansion to its first term, optical dispensers use the Approximate Sag Formula:

$$s_{\text{approx}} \approx \frac{y^2}{2r} \approx \frac{y^2 \cdot F}{2000 \cdot (n - 1)} \quad (\text{where } y \text{ in mm}, \ F \text{ in Diopters})$$

For shallow, low-power curves ($<2.00\ \text{D}$ on small frames), the error is negligible ($<0.02\ \text{mm}$). However, for a $+8.00\ \text{D}$ lens ($r = 66.25\ \text{mm}$) across a wide $65\ \text{mm}$ blank ($y = 32.5\ \text{mm}$):

• $s_{\text{approx}} = \frac{(32.5)^2}{2 \times 66.25} = \mathbf{7.97\ \text{mm}}$.
• $s_{\text{exact}} = 66.25 - \sqrt{(66.25)^2 - (32.5)^2} = \mathbf{8.51\ \text{mm}}$ (A dangerous $0.54\ \text{mm}$ error that causes high-plus lenses to crack or poke out of frames!).

The master thickness equations linking front sag ($s_1$) and back sag ($s_2$):

1. Minus Lens (Thinnest at Center): $t_{\text{edge}} = t_{\text{center}} + (s_2 - s_1)$. Minimum center substance: $1.0\text{ to } 1.5\ \text{mm}$.
2. Plus Lens (Thinnest at Edge): $t_{\text{center}} = t_{\text{edge}} + (s_1 - s_2)$. Minimum edge substance: $1.0\text{ to } 1.2\ \text{mm}$.