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 source ↗

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

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

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.

References

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.

[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.

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