Sphere Cavalieri

A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. 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 centre of the sphere, and 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 object 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.

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 (=1). 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 Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere.
If a particular point on a sphere is (arbitrarily) designated as its ''north pole'', its antipodal point is called the ''south pole''. The great circle equidistant to each is then the ''equator''. Great circles through the poles are called lines of longitude or meridians. A line connecting the two poles may be called the axis of rotation. Small circles on the sphere that are parallel to the equator are lines 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 a ''sphere'' and a '' ball'', which is 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 (closed or open) 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 are sometimes called spherules, e.g. in Martian spherules.

Equations

In analytic geometry, a sphere with center and radius is the locus of all points such that
:$(x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2.$
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
:$x_0 = \frac, \quad y_0 = \frac, \quad z_0 = \frac, \quad \rho = \frac.$
Then the equation
:$f(x,y,z) = a(x^2 + y^2 +z^2) + 2(bx + cy + dz) + e = 0$
has no real points as solutions if $\rho < 0$ and is called the equation of an imaginary sphere. If $\rho = 0$, the only solution of $f(x,y,z) = 0$ is the point $P_0 = (x_0,y_0,z_0)$ and the equation is said to be the equation of a point sphere. Finally, in the case $\rho > 0$, $f(x,y,z) = 0$ is an equation of a sphere whose center is $P_0$ and whose radius is $\sqrt \rho$.

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