Unit sphere

Summary

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. More generally, the unit -sphere is an -sphere of unit radius in -dimensional Euclidean space; the unit circle is a special case, the unit -sphere in the plane. An (open) unit ball is the region inside of a unit sphere, the set of points of distance less than 1 from the center.

Some 1-spheres: x2 is the norm for Euclidean space.

A sphere or ball with unit radius and center at the origin of the space is called the unit sphere or the unit ball. Any arbitrary sphere can be transformed to the unit sphere by a combination of translation and scaling, so the study of spheres in general can often be reduced to the study of the unit sphere.

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.

Unit spheres and balls in Euclidean space edit

In Euclidean space of   dimensions, the  -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

 

Volume and area edit

 
Graphs of volumes (V) and surface areas (S) of unit n-balls

The classical equation of a unit sphere is that of the ellipsoid with a radius of 1 and no alterations to the  -,  -, or  - 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. The volume of the unit  -ball, which we denote   can be expressed by making use of the gamma function. It is

 

where   is the double factorial.

The hypervolume of the  -dimensional unit sphere (i.e., the "area" of the boundary of the  -dimensional unit ball), which we denote   can be expressed as

 

For example,   is the "area" of the boundary of the unit ball  , which simply counts the two points. Then   is the "area" of the boundary of the unit disc, which is the circumference of the unit circle.   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   are as follows:

    (surface area)   (volume)
0   1
1   2   2
2   6.283   3.141
3   12.57   4.189
4   19.74   4.935
5   26.32   5.264
6   31.01   5.168
7   33.07   4.725
8   32.47   4.059
9   29.69   3.299
10   25.50   2.550

where the decimal expanded values for   are rounded to the displayed precision.

Recursion edit

The   values satisfy the recursion:

 
 
  for  .

The   values satisfy the recursion:

 
 
  for  .

Non-negative real-valued dimensions edit

The value   at non-negative real values of   is sometimes used for normalization of Hausdorff measure.[1][2]

Other radii edit

The surface area of an  -sphere with radius   is   and the volume of an  - ball with radius   is   For instance, the area is   for the two-dimensional surface of the three-dimensional ball of radius   The volume is   for the three-dimensional ball of radius  .

Unit balls in normed vector spaces edit

The open unit ball of a normed vector space   with the norm   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   in the case of the max-norm in  . 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.

Let   Define the usual  -norm for   as:

 

Then   is the usual Hilbert space norm.   is called the Hamming norm, or  -norm. The condition   is necessary in the definition of the   norm, as the unit ball in any normed space must be convex as a consequence of the triangle inequality. Let   denote the max-norm or  -norm of  .

Note that for the one-dimensional circumferences   of the two-dimensional unit balls, we have:

  is the minimum value.
 
  is the maximum value.

Generalizations edit

Metric spaces edit

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.

Quadratic forms edit

If   is a linear space with a real quadratic form   then   may be called the unit sphere[3][4] or unit quasi-sphere of   For example, the 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. Similarly, the quadratic form   yields a pair of lines for the unit sphere in the dual number plane.

See also edit

Notes and references edit

  1. ^ The Chinese University of Hong Kong, Math 5011, Chapter 3, Lebesgue and Hausdorff Measures
  2. ^ Manin, Yuri I. (2006). "The notion of dimension in geometry and algebra" (PDF). Bulletin of the American Mathematical Society. 43 (2): 139–161. doi:10.1090/S0273-0979-06-01081-0. Retrieved 17 December 2021.
  3. ^ Takashi Ono (1994) Variations on a Theme of Euler: quadratic forms, elliptic curves, and Hopf maps, chapter 5: Quadratic spherical maps, page 165, Plenum Press, ISBN 0-306-44789-4
  4. ^ F. Reese Harvey (1990) Spinors and calibrations, "Generalized Spheres", page 42, Academic Press, ISBN 0-12-329650-1
  • Mahlon M. Day (1958) Normed Linear Spaces, page 24, Springer-Verlag.
  • Deza, E.; Deza, M. (2006), Dictionary of Distances, Elsevier, ISBN 0-444-52087-2. Reviewed in Newsletter of the European Mathematical Society 64 (June 2007), p. 57. This book is organized as a list of distances of many types, each with a brief description.

External links edit