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:''Za druge pomene [[Krogla (razločitev)]].''
<div style="float:right;margin:0 0 0 1em">[[slika:Krogla.png|Krogla]]</div>
'''Krogla''' je v [[matematiki]] okroglo [[simetrija|simetrično]] [[telo]]. Točke krogle so od središča oddaljene največ za [[polmer]].
'''Krogla''' v [[metrični prostor|metričnem prostoru]] pomeni [[množica|množico]] [[točka|točk]], ki niso oddaljene več kot za določeno razdaljo od neke določene točke. Njeno [[površina|površino]] imenujemo ''[[sfera]]''.
[[Enačba]] krogle s polmerom ''R'' in središčem v izhodišču [[koordinatni sistem|koordinatnega sistema]] je:
x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> + <= R<sup>2</sup>
[[en:Ball (mathematics)]]
Njeno [[površina|površino]] imenujemo ''[[sfera]]''.
== V topologiji ==
In [[solid geometry|three-dimensional]] [[Euclidean geometry|Euclidean]] [[geometry]], a '''sphere''' is the set of points in '''R'''<sup>3</sup> which are at distance ''r'' from a fixed point of that space, where ''r'' is a positive [[real number]] called the '''radius''' of the sphere. The fixed point is called the '''center''' or '''centre''', and is not part of the sphere itself. The special case of ''r'' = 1 is called a '''unit sphere'''.
[[Image:jade.png|thumb|A jade sphere with luminosity effects and blended layers.]]
In [[analytic geometry]], a sphere with center (''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>) and radius ''r'' is the set of all points (''x'', ''y'', ''z'') such that
:<math>(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2 \,</math>
The points on the sphere with radius ''r'' can be parametrized via
:<math> x = x_0 + r \sin \theta \; \cos \phi </math>
:<math> y = y_0 + r \sin \theta \; \sin \phi \qquad (0 \leq \theta \leq \pi \mbox{ and } -\pi < \phi \leq \pi) \,</math>
:<math> z = z_0 + r \cos \theta \,</math>
(see also [[trigonometric function]]s and [[spherical coordinates]]).
A sphere of any radius centered at the origin is described by the following [[differential equation]]:
:<math> x \, dx + y \, dy + z \, dz = 0. </math>
This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other.
The [[surface area]] of a sphere of radius ''r'' is:
:<math>A = 4 \pi r^2 \,</math>
and its enclosed [[volume]] is:
:<math>V = \frac{4 \pi r^3}{3}</math>
The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are
roughly spherical, because the [[surface tension]] minimizes surface area.
The circumscribed [[cylinder]] for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to [[Archimedes]].
A sphere can also be defined as the surface formed by rotating a [[circle]] about its [[diameter]]. If the circle is replaced by an [[ellipse]], the shape becomes a [[spheroid]].
===Generalization to higher dimensions===
Spheres can be generalized to higher [[dimension]]s. For any [[natural number]] ''n'', an ''n''-sphere is the set of points in (''n''+1)-dimensional Euclidean space which are at distance ''r'' from a fixed point of that space, where ''r'' is, as before, a positive real number.
* a 0-sphere is a pair of points <math>(-r, r)</math>
* a 1-sphere is a [[circle]] of radius ''r''
* a 2-sphere is an ordinary sphere
* a [[3-sphere]] is a sphere in 4-dimensional Euclidean space
Spheres for ''n'' > 2 are sometimes called [[hypersphere]]s. The ''n''-sphere of unit radius centred at the origin is denoted ''S''<sup>''n''</sup> and is often referred to as "the" ''n''-sphere.
===See also===
*[[en:Ball (mathematics)]]
*[[Circle]], [[3-sphere]], [[hypersphere]]
*[[Metric space]]
*[[Riemann sphere]]
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