In geometry, the conic constant (or Schwarzschild constant, 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}
or 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.
References
Smith, Warren J. (2008). Modern Optical Engineering, 4th ed. McGraw-Hill Professional. pp. 512–515. ISBN 978-0-07-147687-4.