Definition:Sphere
Contents |
Definition
Geometry
A sphere is a surface in solid geometry such that every straight line falling upon it from one particular point inside it are equal.
As Euclid defined it:
- When, the diameter of a semicircle remaining fixed, the semicircle is carried round and restored again to the same position from which it began to be moved, the figure so comprehended is a sphere.
(The Elements: Book XI: Definition $14$)
Center
That point is called the center of the sphere.
Radius
A radius (plural radii, pronounced ray-dee-eye) of a sphere is a straight line segment whose endpoints are the center and the surface of the sphere.
The radius of a sphere is the length of one such radius.
Thus it is the three-dimensional version of the circle.
Every point on the sphere is at the same distance from its center.
Topology
In the field of topology the concept is generalized.
The $n$-dimensional sphere, or $n$-sphere, is the set:
- $\Bbb S^n = \left\{{x \in \R^{n+1} : \left|{x - y}\right| = r}\right\}$
where $r \in \R_+$ is called the radius of the sphere and $y \in \R^{n+1}$ is called the center of the sphere.
Frequently, the radius is taken as $1$ and the center as the origin.
Note
As the sphere is defined here, it is specified as being the surface only, that is, not the inside.
