Sphere


A sphere is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the same distance from a given point in three-dimensional space. That given point is the center of the sphere, and the distance is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.
The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings.

Basic terminology

As mentioned earlier is the sphere's radius; any line from the center to a point on the sphere is also called a radius. 'Radius' is used in two senses: as a line segment and also as its length.
If a radius is extended through the center to the opposite side of the sphere, it creates a diameter. Like the radius, the length of a diameter is also called the diameter, and denoted. Diameters are the longest line segments that can be drawn between two points on the sphere: their length is twice the radius,. Two points on the sphere connected by a diameter are antipodal points of each other.
A unit sphere is a sphere with unit radius. For convenience, spheres are often taken to have their center at the origin of the coordinate system, and spheres in this article have their center at the origin unless a center is mentioned.
A great circle on the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.
Although the figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere.
A particular line passing through its center defines an axis.
The sphere-axis intersection defines two antipodal poles. The great circle equidistant to the poles is called the equator. Great circles through the poles are called lines of longitude or meridians. Small circles on the sphere that are parallel to the equator are circles of latitude. In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding.
Mathematicians consider a sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space. They draw a distinction between a sphere and a ball, which is a solid figure, a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and a sphere is the boundary of a ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between "circle" and "disk" in the plane is similar.
Small spheres or balls are sometimes called spherules.

Equations

In analytic geometry, a sphere with center and radius is the locus of all points such that
Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface, a type of algebraic surface.
Let be real numbers with and put
Then the equation
has no real points as solutions if and is called the equation of an imaginary sphere. If, the only solution of is the point and the equation is said to be the equation of a point sphere. Finally, in the case, is an equation of a sphere whose center is and whose radius is.
If in the above equation is zero then is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity.

Parametric

A parametric equation for the sphere with radius and center can be parameterized using trigonometric functions.
The symbols used here are the same as those used in spherical coordinates. is constant, while varies from 0 to and varies from 0 to 2.

Properties

Enclosed volume

In three dimensions, the volume inside a sphere is
where is the radius and is the diameter of the sphere. Archimedes first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere. This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle. This formula can also be derived using integral calculus to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along the -axis from to, assuming the sphere of radius is centered at the origin.
At any given, the incremental volume equals the product of the cross-sectional area of the disk at and its thickness :
The total volume is the summation of all incremental volumes:
In the limit as approaches zero, this equation becomes:
At any given, a right-angled triangle connects, and to the origin; hence, applying the Pythagorean theorem yields:
Using this substitution gives
which can be evaluated to give the result
An alternative formula is found using spherical coordinates, with volume element
so
For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since, where is the diameter of the sphere and also the length of a side of the cube and ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1m, or about 0.524 m3.

Surface area

The surface area of a sphere of radius is:
Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to because the total volume inside a sphere of radius can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius is simply the product of the surface area at radius and the infinitesimal thickness.
At any given radius, the incremental volume equals the product of the surface area at radius and the thickness of a shell :
The total volume is the summation of all shell volumes:
In the limit as approaches zero this equation becomes:
Substitute :
Differentiating both sides of this equation with respect to yields as a function of :
This is generally abbreviated as:
where is now considered to be the fixed radius of the sphere.
Alternatively, the area element on the sphere is given in spherical coordinates by. The total area can thus be obtained by integration:
The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area.
The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as
where is the density.

Other geometric properties

A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters; this is very similar to the traditional definition of a sphere as given in Euclid's Elements. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. Replacing the circle with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor axis, an oblate spheroid.
A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine a unique circle in a plane.
Consequently, a sphere is uniquely determined by a circle and a point not in the plane of that circle.
By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane is a real plane, the circle may be imaginary or consist of a single point.
The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.

Pencil of spheres

If and are the equations of two distinct spheres then
is also the equation of a sphere for arbitrary values of the parameters and. The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane in the pencil.