geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

, the conic constant (or Schwarzschild constant, after

Karl Schwarzschild Karl Schwarzschild (; 9 October 1873 – 11 May 1916) was a German physicist and astronomer. Schwarzschild provided the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non ...

) is a quantity describing

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...

s, and is represented by the letter ''K''. The constant is given by

K = -e^2,

where is the

eccentricity Eccentricity or eccentric may refer to: * Eccentricity (behavior), odd behavior on the part of a person, as opposed to being "normal" Mathematics, science and technology Mathematics * Off-center, in geometry * Eccentricity (graph theory) of a ...

of the conic section. The equation for a conic section with apex at the

origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...

and tangent to the y axis is

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

alternately

x = \dfrac

where ''R'' is the

radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...

at . This formulation is used in

geometric optics Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

to specify oblate elliptical (),

spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

(), prolate elliptical (), parabolic (), and

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

() lens and mirror surfaces. When the

paraxial approximation In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens). A paraxial ray is a ray which makes a small angle (''θ'') to the optica ...

is valid, the optical surface can be treated as a spherical surface with the same radius. Some non-optical design references use the letter ''p'' as the conic constant. In these cases, .

References

* Mathematical constants Conic sections Geometrical optics {{geometry-stub