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# Conic constant

An illustration of various conic constants

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

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

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

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

alternately

${\displaystyle x={\dfrac {y^{2}}{R+{\sqrt {R^{2}-(K+1)y^{2}}}}}}$

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

1. ^ Rakich, Andrew (2005-08-18). "The 100th birthday of the conic constant and Schwarzschild's revolutionary papers in optics". Novel Optical Systems Design and Optimization VIII. International Society for Optics and Photonics. 5875: 587501. doi:10.1117/12.635041.