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.

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

• Smith, Warren J. (2008). Modern Optical Engineering, 4th ed.
  ISBN 978-0-07-147687-4
  .

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