Conic constant

Conic constant

Ten different conic sections which open to the right from a common intersection point, at which point they have a common radius of curvature
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. 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.  
  1. ^ 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.