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. The constant is given by

K = − e 2 , {\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

y 2 − 2 R x + ( K + 1 ) x 2 = 0 {\displaystyle y^{2}-2Rx+(K+1)x^{2}=0}
alternately
x = y 2 R + R 2 − ( K + 1 ) y 2 {\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). Sasian, Jose M; Koshel, R. John; Juergens, Richard C (eds.). "The 100th birthday of the conic constant and Schwarzschild's revolutionary papers in optics". Novel Optical Systems Design and Optimization VIII. 5875. International Society for Optics and Photonics: 587501. Bibcode:2005SPIE.5875....1R. doi:10.1117/12.635041. S2CID 119718303.
  • Smith, Warren J. (2008). Modern Optical Engineering, 4th ed. McGraw-Hill Professional. pp. 512–515. ISBN 978-0-07-147687-4.


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