Editing Sphere (section)

===Other geometric properties===
A sphere can be constructed as the surface formed by rotating a [[circle]] one half revolution about any of its [[diameter]]s; 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.<ref>{{harvnb|Albert|2016|loc=p. 60}}.</ref>

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.<ref>{{harvnb|Albert|2016|loc=p. 55}}.</ref> This property is analogous to the property that three [[collinear|non-collinear]] points determine a unique circle in a plane.

Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle.

By examining the [[Circle of a sphere#Sphere-sphere intersection|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.<ref>{{harvnb|Albert|2016|loc=p. 57}}.</ref> Although the radical plane is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point).<ref name=Woods267>{{harvnb|Woods|1961|loc=p. 267}}.</ref>

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.<ref>{{harvnb|Albert|2016|loc=p. 58}}.</ref> They intersect at right angles (are [[Orthogonality|orthogonal]]) if and only if the square of the distance between their centers is equal to the sum of the squares of their radii.<ref name=Woods266 />

====Pencil of spheres====
{{Main|Pencil (mathematics)#Pencil of spheres}}
If {{math|1=''f''(''x'', ''y'', ''z'') = 0}} and {{math|1=''g''(''x'', ''y'', ''z'') = 0}} are the equations of two distinct spheres then
:<math>s f(x,y,z) + t g(x,y,z) = 0</math>
is also the equation of a sphere for arbitrary values of the parameters {{mvar|s}} and {{mvar|t}}. 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 (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil.<ref name=Woods266 />