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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. For negative K it 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 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.
References

Smith, Warren J. (2008). Modern Optical Engineering, 4th ed. McGraw-Hill Professional. pp. 513–515. ISBN 978-0-07-147687-4.

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. edit

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