{{Short description|Parameter describing conic sections}} thumb|350px|alt=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''',<ref>{{Cite journal| last=Rakich|first=Andrew | editor1-first=Jose M | editor1-last=Sasian | editor2-first=R. John|editor2-last=Koshel | editor3-first=Richard C|editor3-last=Juergens | date=2005-08-18 | title=The 100th birthday of the conic constant and Schwarzschild's revolutionary papers in optics | url=https://www.spiedigitallibrary.org/conference-proceedings-of-spie/5875/587501/The-100th-birthday-of-the-conic-constant-and-Schwarzschilds-revolutionary/10.1117/12.635041.short | journal=Novel Optical Systems Design and Optimization VIII| publisher=International Society for Optics and Photonics| volume=5875|pages=587501 | doi=10.1117/12.635041|bibcode=2005SPIE.5875....1R |s2cid=119718303}}</ref> after Karl Schwarzschild) is a quantity describing conic sections, and is represented by the letter&nbsp;''K''. The constant is given by <math display="block">K = -e^2, </math> where {{math|''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 <math display="block">y^2-2Rx+(K+1)x^2 = 0</math>

or alternately <math display="block"> x = \dfrac{y^2}{R+\sqrt{R^2-(K+1)y^2}}</math>

where ''R'' is the radius of curvature at&nbsp;{{math|1=''x'' = 0}}.

This formulation is used in geometric optics to specify oblate elliptical ({{math|''K'' > 0}}), spherical ({{math|1=''K'' = 0}}), prolate elliptical ({{math|0 > ''K'' > −1}}), parabolic ({{math|1=''K'' = −1}}), and hyperbolic ({{math|1=''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== {{reflist|1}} *{{cite book|last=Smith |first=Warren J. |title=Modern Optical Engineering, 4th ed |publisher=McGraw-Hill Professional |year=2008 |isbn=978-0-07-147687-4 |pages=512–515}}

Category:Mathematical constants Category:Conic sections Category:Geometrical optics

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