Editing Sphere (section)

==Equations==
In [[analytic geometry]], a sphere with center {{math|(''x''<sub>0</sub>, ''y''<sub>0</sub>, ''z''<sub>0</sub>)}} and radius {{mvar|r}} is the [[Locus (mathematics)|locus]] of all points {{math|(''x'', ''y'', ''z'')}} such that
:<math> (x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2.</math>
Since it can be expressed as a quadratic polynomial, a sphere is a [[quadric surface]], a type of [[algebraic surface]].<ref name="EB" />

Let {{mvar|a, b, c, d, e}} be real numbers with {{math|''a'' ≠ 0}} and put
:<math>x_0 = \frac{-b}{a}, \quad y_0 = \frac{-c}{a}, \quad z_0 = \frac{-d}{a}, \quad \rho = \frac{b^2 +c^2+d^2 - ae}{a^2}.</math>
Then the equation
:<math>f(x,y,z) = a(x^2 + y^2 +z^2) + 2(bx + cy + dz) + e = 0</math>
has no real points as solutions if <math>\rho < 0</math> and is called the equation of an '''imaginary sphere'''. If <math>\rho = 0</math>, the only solution of <math>f(x,y,z) = 0</math> is the point <math>P_0 = (x_0,y_0,z_0)</math> and the equation is said to be the equation of a '''point sphere'''. Finally, in the case <math>\rho > 0</math>, <math>f(x,y,z) = 0</math> is an equation of a sphere whose center is <math>P_0</math> and whose radius is <math>\sqrt \rho</math>.<ref name=Albert54 />

If {{mvar|a}} in the above equation is zero then {{math|1=''f''(''x'', ''y'', ''z'') = 0}} 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]].<ref name=Woods266>{{harvnb|Woods|1961|loc=p. 266}}.</ref>

===Parametric===
A [[parametric equation]] for the sphere with radius <math>r > 0</math> and center <math>(x_0,y_0,z_0)</math> can be parameterized using [[trigonometric function]]s.
:<math>\begin{align}
x &= x_0 + r \sin \theta \; \cos\varphi \\
y &= y_0 + r \sin \theta \; \sin\varphi \\
z &= z_0 + r \cos \theta \,\end{align}</math><ref>{{harvtxt|Kreyszig|1972|p=342}}.</ref>

The symbols used here are the same as those used in [[spherical coordinates]]. {{mvar|r}} is constant, while {{mvar|θ}} varies from 0 to {{mvar|π}} and <math>\varphi</math> varies from 0 to 2{{mvar|π}}.