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A sphere is, roughly speaking, a ball-shaped object. In non-mathematical usage a sphere is often considered to be solid (which mathematicians call ball). But in mathematics, a sphere is the boundary of a ball, and is therefore "hollow". This article deals with the mathematical concept of sphere.
Note that topologists and many geometers have different meanings for n-sphere, differing by one dimension. Thus, for example, 2-sphere means circle to many geometers but "ordinary" sphere to topologists. Some geometers use topologists' nomenclature, adding to the confusion.
In three-dimensional Euclidean geometry, a sphere is the set of points in R3 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.
In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the set of all points (x, y, z) such that
The points on the sphere with radius r can be parametrized via
(see also trigonometric functions and spherical coordinates).
A sphere of any radius centered at the origin is described by the following differential equation:
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:
and its enclosed volume is:
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 spheres, because the surface tensionIn physics, surface tension is an effect within the surface layer of a liquid that causes the layer to behave as an elastic sheet. It is the effect that allows insects (such as the water strider) to walk on water, and causes capillary action, for example. tries to minimize surface area.
The circumscribed cylinderThe word cylinder has several meanings. For the geometric object, see Cylinder (geometry . For the engine component, see Cylinder (engine . In firearms the cylinder is the rotating device that contains the firing chambers of a revolver. The phonograph cyl 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 ArchimedesSee also Archimedes computer, Archimedes (disambiguation). Archimedes of Syracuse (circa 287 BC 212 BC), was a Greek mathematician, astronomer, philosopher, physicist and engineer. He was killed by a Roman soldier during the sack of the city, despite orde.
A sphere can also be defined as the surface formed by rotating a circleSee The Circle for the distributed file storage system, and see Ring (diacritic) for the diacritic mark. In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius from a fixed point, called the centre . about its diameterIn geometry, a diameter of a circle is any straight line segment that passes through the center and whose endpoints are on the circular boundary, or, in more modern usage, the length of such a line segment. When using the word in the more modern sense, on. If the circle is replaced by an ellipseThe Ellipse is also an elliptical street immediately in front of the White House. In mathematics, an ellipse is a figure corresponding to a circle which has been stretched in one direction. This is an example of a conic section and can be defined as the l, the shape becomes a spheroidA spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball). If the mi.