Conic constant

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 Template:Math 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 Template:Math.

This formulation is used in geometric optics to specify oblate elliptical (Template:Math), spherical (Template:Math), prolate elliptical (Template:Math), parabolic (Template:Math), and hyperbolic (Template:Math) 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

• Template:Cite book

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