In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the conic constant (or Schwarzschild constant, after Karl Schwarzschild

Karl Schwarzschild

Karl Schwarzschild was a German physicist. He is also the father of astrophysicist Martin Schwarzschild.He is best known for providing the first exact solution to the Einstein field equations of general relativity, for the limited case of a single spherical non-rotating mass, which he accomplished...

) is a quantity describing conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

s, and is represented by the letter K. It is given by


where e is the eccentricity

Eccentricity (mathematics)

In mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...

 of the conic section.

The equation for a conic section with apex at the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

 and tangent to the y axis is


where K is the conic constant and R is the radius of curvature

Radius of curvature (optics)

Radius of curvature has specific meaning and sign convention in optical design. A spherical lens or mirror surface has a center of curvature located in either along or decentered from the system local optical axis. The vertex of the lens surface is located on the local optical axis...

 at x = 0.

This formulation is used in geometric optics to specify oblate elliptical (K > 0), spherical

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

 (K = 0), prolate elliptical

Prolate spheroid

A prolate spheroid is a spheroid in which the polar axis is greater than the equatorial diameter. Prolate spheroids stand in contrast to oblate spheroids...

 (0 > K > −1), parabolic

Paraboloid

In mathematics, a paraboloid is a quadric surface of special kind. There are two kinds of paraboloids: elliptic and hyperbolic. The elliptic paraboloid is shaped like an oval cup and can have a maximum or minimum point....

 (K = −1), and hyperbolic (K < −1) lens and mirror surfaces. When the paraxial approximation

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 ....

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, p = K + 1.