Conic constant

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