Sagitta (optics)

In optics and especially telescope making, sagitta or sag is a measure of the glass removed to yield an optical curve. It is approximated by the formula


 * $$S(r) \approx \frac{r^2}{2 \times R}$$,

where $R$ is the radius of curvature of the optical surface. The sag $S(r)$ is the displacement along the optic axis of the surface from the vertex, at distance $$r$$ from the axis.

A good explanation both of this approximate formula and the exact formula can be found here.

Aspheric surfaces
Optical surfaces with non-spherical profiles, such as the surfaces of aspheric lenses, are typically designed such that their sag is described by the equation
 * $$S(r)=\frac{r^2}{R\left (1+\sqrt{1-(1+K)\frac{r^2}{R^2}}\right )}+\alpha_1 r^2+\alpha_2 r^4+\alpha_3 r^6+\cdots .$$

Here, $$K$$ is the conic constant as measured at the vertex (where $$r=0$$). The coefficients $$\alpha_i$$ describe the deviation of the surface from the axially symmetric quadric surface specified by $$R$$ and $$K$$.