Unit sphere

In mathematics, a unit sphere is a sphere of unit radius: the set of points at Euclidean distance 1 from some center point in three-dimensional space.

An (open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center.

The unit sphere is often used as a model for spherical geometry because it has constant sectional curvature of 1, which simplifies calculations.

In trigonometry, circular arc length on the unit circle is called radians and used for measuring angular distance; in spherical trigonometry surface area on the unit sphere is called steradians and used for measuring solid angle.

In more general contexts, a unit sphere is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance", and an (open) unit ball is the region inside.

-dimensional unit sphere is the set of all points

which satisfy the equation The open unit

-ball is the set of all points satisfying the inequality and closed unit

-ball is the set of all points satisfying the inequality The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the

- axes: The volume of the unit ball in Euclidean

-space, and the surface area of the unit sphere, appear in many important formulas of analysis.

can be expressed by making use of the gamma function.

-dimensional unit sphere (i.e., the "area" of the boundary of the

is the "area" of the boundary of the unit ball

is the area of the boundary of the unit ball

, which is the surface area of the unit sphere

The surface areas and the volumes for some values of

for the two-dimensional surface of the three-dimensional ball of radius

The open unit ball of a normed vector space

is given by It is the topological interior of the closed unit ball of

The latter is the disjoint union of the former and their common border, the unit sphere of

The "shape" of the unit ball is entirely dependent on the chosen norm; it may well have "corners", and for example may look like

One obtains a naturally round ball as the unit ball pertaining to the usual Hilbert space norm, based in the finite-dimensional case on the Euclidean distance; its boundary is what is usually meant by the unit sphere.

is the usual Hilbert space norm.

norm, as the unit ball in any normed space must be convex as a consequence of the triangle inequality.

of the two-dimensional unit balls, we have: All three of the above definitions can be straightforwardly generalized to a metric space, with respect to a chosen origin.

However, topological considerations (interior, closure, border) need not apply in the same way (e.g., in ultrametric spaces, all of the three are simultaneously open and closed sets), and the unit sphere may even be empty in some metric spaces.

is a linear space with a real quadratic form

, when set equal to one, produces the unit hyperbola, which plays the role of the "unit circle" in the plane of split-complex numbers.

yields a pair of lines for the unit sphere in the dual number plane.