Conic constant

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 $${\displaystyle K=-e^{2},}$$ where e is the eccentricity of the conic section.

An illustration of various conic constants

The equation for a conic section with apex at the origin and tangent to the y axis is $${\displaystyle y^{2}-2Rx+(K+1)x^{2}=0}$$

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

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.