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Conic constant

Parameter describing conic sections

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Source updated September 16, 2025

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

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.

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This article is adapted from the Wikipedia article “Conic constant”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Conic constant.svg by 0x30114 at English Wikipedia, CC BY-SA 3.0

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