- 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
where K is the conic constant and 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 non-optical design references use the letter p as the conic constant. In these cases, p = K + 1.
- Smith, Warren J. (2008). Modern Optical Engineering, 4th ed.
- 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.