Unit sphere

Unit sphere (red) and sphere (blue) for the Euclidean norm in two dimensions

In mathematics, the unit sphere is the sphere with radius one around the zero point of a vector space . A generalized concept of distance is used as a basis, so that, depending on the context, the unit sphere no longer has to be similar to a conventional sphere. This unit sphere is the edge of the unit sphere, in the two-dimensional real vector space with the Euclidean norm this is the unit circle .

general definition

Let it be a normalized vector space . Then the set of points whose distance from the zero point is less than one is called the open unit sphere in : ${\displaystyle (X,\|{\cdot }\|)}$ ${\displaystyle X}$

${\displaystyle B_{X}:=\{x\in X:\|x\|<1\}.}$

Labeled accordingly

${\displaystyle {\overline {B_{X}}}:=\{x\in X:\|x\|\leq 1\}}$

the closed unit sphere in as well ${\displaystyle X}$

${\displaystyle \partial B_{X}:=\{x\in X:\|x\|=1\}}$

the unitary sphere in . ${\displaystyle X}$

By means of translation and scaling , any number of spheres in a room can be converted into the unit sphere. Therefore it is often sufficient to prove certain statements only for the unit sphere in order to infer the validity for any spheres.

Unit sphere in finite-dimensional spaces

Unit spheres in ${\displaystyle \mathbb {R} ^{2}}$

In the case of Euclidean space , the closed unit sphere is defined with respect to the Euclidean norm using ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \|x\|_{2}={\sqrt {x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}}}$ ${\displaystyle {\overline {B_{\mathbb {R} ^{n}}}}:=\{x\in \mathbb {R} ^{n}:\|x\|_{2}\leq 1\}.}$

Unit spheres can alternatively be defined in relation to other standards , for example the sum standard (1 standard) or the maximum standard . The geometric shape of the unit sphere depends on the selected standard and is only actually spherical with the Euclidean standard. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \|x\|_{1}=|x_{1}|+|x_{2}|+\cdots +|x_{n}|}$ ${\displaystyle \|x\|_{\infty }=\max\{|x_{1}|,|x_{2}|,\dots ,|x_{n}|\}}$

Volume and surface

The volume of a -dimensional, Euclidean unit sphere is ${\displaystyle n}$

${\displaystyle V_{n}={\frac {2\pi ^{n/2}}{n\Gamma ({\frac {n}{2}})}}={\frac {\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}}$

Here is the gamma function , an analytical continuation of the (shifted) factorial on the real numbers. For straight lines the formula is simplified to . The surface is ${\displaystyle \Gamma }$ ${\displaystyle n}$ ${\displaystyle V_{n}={\tfrac {\pi ^{n/2}}{(n/2)!}}}$

${\displaystyle A_{n}=nV_{n}={\frac {n\pi ^{n/2}}{\Gamma (1+n/2)}}={\frac {2\pi ^{n/2}}{\Gamma (n/2)}}}$

The following recursions apply :

${\displaystyle V_{n}={\frac {2\pi }{n}}V_{n-2}}$for . ${\displaystyle n>1}$

${\displaystyle A_{n}={\frac {2\pi }{n-2}}A_{n-2}}$for . ${\displaystyle n>2}$

It is noteworthy in this context that the volume of the unit sphere, depending on the space dimension to first increases and then decreases again - and even go against the 0th The surface increases from the spatial dimension to initially, and goes for towards 0. ${\displaystyle n}$ ${\displaystyle n=5}$ ${\displaystyle n\to \infty }$ ${\displaystyle n}$ ${\displaystyle n=7}$ ${\displaystyle n\to \infty }$

Volume and surface of the unit sphere
dimension	volume		surface
0	1	1	0	0
1	2	2	2	2
2	${\displaystyle \pi }$	3.141	${\displaystyle 2\pi }$	6.283
3	${\displaystyle {\frac {4}{3}}\pi }$	4.189	${\displaystyle 4\pi }$	12.57
4th	${\displaystyle {\frac {1}{2}}\pi ^{2}}$	4.935	${\displaystyle 2\pi ^{2}}$	19.74
5	${\displaystyle {\frac {8}{15}}\pi ^{2}}$	5.264	${\displaystyle {\frac {8}{3}}\pi ^{2}}$	26.32
6th	${\displaystyle {\frac {1}{6}}\pi ^{3}}$	5.168	${\displaystyle \pi ^{3}}$	31.01
7th	${\displaystyle {\frac {16}{105}}\pi ^{3}}$	4,725	${\displaystyle {\frac {16}{15}}\pi ^{3}}$	33.07
8th	${\displaystyle {\frac {1}{24}}\pi ^{4}}$	4.059	${\displaystyle {\frac {1}{3}}\pi ^{4}}$	32.47
9	${\displaystyle {\frac {32}{945}}\pi ^{4}}$	3,299	${\displaystyle {\frac {32}{105}}\pi ^{4}}$	29.69
10	${\displaystyle {\frac {1}{120}}\pi ^{5}}$	2,550	${\displaystyle {\frac {1}{12}}\pi ^{5}}$	25.50
12	${\displaystyle {\frac {1}{720}}\pi ^{6}}$	1,335	${\displaystyle {\frac {1}{60}}\pi ^{6}}$	16.02
20th	${\displaystyle {\frac {1}{3628800}}\pi ^{10}}$	0.0258	${\displaystyle {\frac {1}{181440}}\pi ^{10}}$	0.516
25th	${\displaystyle {\frac {8192}{7905853580625}}\pi ^{12}}$	0.000958	${\displaystyle {\frac {8192}{316234143225}}\pi ^{12}}$	0.0239

The unit sphere with respect to the sum norm is geometrically a cross polytope , its volume is . ${\displaystyle {\tfrac {2^{n}}{n!}}}$

The unit sphere with respect to the maximum norm is a hypercube with edge length 2, so it has the volume . ${\displaystyle 2^{n}}$

Remarks

• The unit sphere forms the edge of the unit sphere. Correspondingly in two-dimensional the unit sphere is not the circle , but the circular disk.

• More generally, a unit sphere can be defined in any metric space . It should be noted that a point does not have to be marked as the zero point from the outset and therefore one can only speak of the unit sphere of a metric space to a limited extent . Furthermore, especially in the case of metrics that are not norm-induced, the unit spheres are even further removed from view. Specially applicable in a vector space with the discrete metric : , and ${\displaystyle X}$ ${\displaystyle B_{X}=\{0\}}$ ${\displaystyle {\overline {B_{X}}}=X}$ ${\displaystyle \partial B_{X}=X\backslash \{0\}.}$

• When looking at surroundings , the unit ball is also referred to as a 1-ball or 1- ball .

properties

• The closed unit sphere is convex . (The convexity follows from the triangle inequality .) ${\displaystyle {\overline {B_{X}}}}$
• It is point-symmetric to the origin 0: . ${\displaystyle x\in {\overline {B_{X}}}\implies -x\in {\overline {B_{X}}}}$
• Conversely, in a finite-dimensional vector space , a norm is defined by every closed convex set , which is point-symmetrical to the origin and contains the origin inside , which has this set as a unit sphere:, for (see Minkowski functional ). ${\displaystyle B}$ ${\displaystyle \lVert x\rVert _{B}=\min\{\,t>0:{\tfrac {x}{t}}\in B\,\}}$ ${\displaystyle x\neq 0}$
• The closed unit sphere is compact if and only if is finite dimensional . ${\displaystyle {\overline {B_{X}}}}$ ${\displaystyle X}$
• The closed unit sphere is weakly compact if and only if is reflexive . ${\displaystyle {\overline {B_{X}}}}$ ${\displaystyle X}$
• The closed unit sphere in the topological dual space of is always weak - * - compact ( Banach-Alaoglu theorem ). ${\displaystyle {\overline {B_{X^{\prime }}}}}$ ${\displaystyle X^{\prime }}$ ${\displaystyle X}$

Applications in the natural sciences

The unit sphere is used in various ways in the geosciences , especially for calculations on the globe . They are done with so-called spherical triangles and the formulas of spherical trigonometry , if an accuracy of about 0.1% is sufficient, for example in geography and cartography , globe calculations and navigation . The true distances are obtained from the spherical arcs by multiplying by the earth's radius .

For greater accuracy - especially in geodesy - the earth ellipsoid should be used instead of the earth globe . With the leveling method , triangle calculations are also possible spherically.

Geologists use a unit sphere, which they call a layer sphere, to analyze the direction of rock strata or fissures . The normal vectors of the respective planes are entered in it and then displayed in true-to-area azimuthal projection .

Also astronomical calculations are carried out on the unit sphere around the observer ever since. It corresponds to the free-eyed view of the sky and is called the celestial sphere , on which spherical astronomy has defined its own coordinate systems for angle measurements and star locations . It is irrelevant whether the spherical radius is assumed to be 1 or ∞.

literature

• Dirk Werner : Functional Analysis . 6th, corrected edition. Springer-Verlag, Berlin 2007, ISBN 978-3-540-72533-6
• Ivan Mueller: Spherical Astronomy as applied to Geodesy . Frederic Ungar Publ., New York 1969