In geometry, the **conic constant** (or **Schwarzschild constant**,^{[1]} after Karl Schwarzschild) is a quantity describing conic sections, and is represented by the letter *K*. The constant is given by

- $K=-e^{2},$

where *e* is the eccentricity of the conic section.

The equation for a conic section with apex at the origin and tangent to the y axis is

- $y^{2}-2Rx+(K+1)x^{2}=0$

where *R* is the radius of curvature at *x* = 0.

This formulation is used in geometric optics to specify oblate elliptical (*K* > 0), spherical (*K* = 0), prolate elliptical (0 > *K* > −1), parabolic (*K* = −1), and hyperbolic (*K* < −1) lens and mirror surfaces. When the paraxial approximation is valid, the optical surface can be treated as a spherical surface with the same radius.

Some^{[which?]} non-optical design references use the letter *p* as the conic constant. In these cases, *p* = *K* + 1.

## References

**^** Chan, L.; Tse, M.; Chim, M.; Wong, W.; Choi, C.; Yu, J.; Zhang, M.; Sung, J. (May 2005). Sasian, Jose M; Koshel, R. John; Juergens, Richard C (eds.). "The 100th birthday of the conic constant and Schwarzschild's revolutionary papers in optics". *Proceedings of SPIE*. Novel Optical Systems Design and Optimization VIII. **5875**: 587501. doi:10.1117/12.635041. ISSN 0277-786X.