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

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

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

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