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 or balls 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 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 (that is, the volume of a ball, but classically referred to as the volume of 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