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, after Karl Schwarzschild) is a quantity describing conic sections, and is represented by the letter K. The constant is given by K=−e2, 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 y2−2Rx+(K+1)x2=0

or alternately x=y2R+R2−(K+1)y2

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.