Conic constant

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The conic constant is a quantity describing conic sections, and is represented by the letter K. It is given by

K = - e 2

where e is the eccentricity of the conic section.

The equation for a conic section defined by the conic constant and radius (R) of curvature at x = 0 is:

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

This formulation is used in geometric optics to specify elliptical (-1 < K < 0), 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.