In mathematics, an n-sphere or hypersphere is an ${\displaystyle n}$-dimensional generalization of the ${\displaystyle 1}$-dimensional circle and ${\displaystyle 2}$-dimensional sphere to any non-negative integer ${\displaystyle n}$. The ${\displaystyle n}$-sphere is the setting for ${\displaystyle n}$-dimensional spherical geometry.

Considered extrinsically, as a hypersurface embedded in ${\displaystyle (n+1)}$-dimensional Euclidean space, an ${\displaystyle n}$-sphere is the locus of points at equal distance (the radius) from a given center point. Its interior, consisting of all points closer to the center than the radius, is an ${\displaystyle (n+1)}$-dimensional ball. In particular:

• The ${\displaystyle 0}$-sphere is the pair of points at the ends of a line segment (${\displaystyle 1}$-ball).
• The ${\displaystyle 1}$-sphere is a circle, the circumference of a disk (${\displaystyle 2}$-ball) in the two-dimensional plane.
• The ${\displaystyle 2}$-sphere, often simply called a sphere, is the boundary of a ${\displaystyle 3}$-ball in three-dimensional space.
• The 3-sphere is the boundary of a ${\displaystyle 4}$-ball in four-dimensional space.
• The ${\displaystyle (n-1)}$-sphere is the boundary of an ${\displaystyle n}$-ball.

Given a Cartesian coordinate system, the unit ${\displaystyle n}$-sphere of radius ${\displaystyle 1}$ can be defined as:

${\displaystyle S^{n}=\left\{x\in \mathbb {R} ^{n+1}:\left\|x\right\|=1\right\}.}$

Considered intrinsically, when ${\displaystyle n\geq 1}$, the ${\displaystyle n}$-sphere is a Riemannian manifold of positive constant curvature, and is orientable. The geodesics of the ${\displaystyle n}$-sphere are called great circles.

The stereographic projection maps the ${\displaystyle n}$-sphere onto ${\displaystyle n}$-space with a single adjoined point at infinity; under the metric thereby defined, ${\displaystyle \mathbb {R} ^{n}\cup \{\infty \}}$ is a model for the ${\displaystyle n}$-sphere.

In the more general setting of topology, any topological space that is homeomorphic to the unit ${\displaystyle n}$-sphere is called an ${\displaystyle n}$-sphere. Under inverse stereographic projection, the ${\displaystyle n}$-sphere is the one-point compactification of ${\displaystyle n}$-space. The ${\displaystyle n}$-spheres admit several other topological descriptions: for example, they can be constructed by gluing two ${\displaystyle n}$-dimensional spaces together, by identifying the boundary of an ${\displaystyle n}$-cube with a point, or (inductively) by forming the suspension of an ${\displaystyle (n-1)}$-sphere. When ${\displaystyle n\geq 2}$ it is simply connected; the ${\displaystyle 1}$-sphere (circle) is not simply connected; the ${\displaystyle 0}$-sphere is not even connected, consisting of two discrete points.

Description

For any natural number ${\displaystyle n}$, an ${\displaystyle n}$-sphere of radius ${\displaystyle r}$ is defined as the set of points in ${\displaystyle (n+1)}$-dimensional Euclidean space that are at distance ${\displaystyle r}$ from some fixed point ${\displaystyle \mathbf {c} }$, where ${\displaystyle r}$ may be any positive real number and where ${\displaystyle \mathbf {c} }$ may be any point in ${\displaystyle (n+1)}$-dimensional space. In particular:

• a 0-sphere is a pair of points ${\displaystyle \{c-r,c+r\}}$, and is the boundary of a line segment (${\displaystyle 1}$-ball).
• a 1-sphere is a circle of radius ${\displaystyle r}$ centered at ${\displaystyle \mathbf {c} }$, and is the boundary of a disk (${\displaystyle 2}$-ball).
• a 2-sphere is an ordinary ${\displaystyle 2}$-dimensional sphere in ${\displaystyle 3}$-dimensional Euclidean space, and is the boundary of an ordinary ball (${\displaystyle 3}$-ball).
• a 3-sphere is a ${\displaystyle 3}$-dimensional sphere in ${\displaystyle 4}$-dimensional Euclidean space.

Cartesian coordinates

The set of points in ${\displaystyle (n+1)}$-space, ${\displaystyle (x_1,x_2}$Zdroj:https://en.wikipedia.org?pojem=Hypersphere
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.