Equation of the sphere

The sphere is the three-dimensional analogue of the circle: the locus of points equidistant from the centre.

Property — Equation of the sphere

Sphere with centre $C(x_0,y_0,z_0)$ and radius $R$: $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=R^2.$ In expanded form $x^2+y^2+z^2+ax+by+cz+d=0$ one has $C\left(-\tfrac{a}{2},-\tfrac{b}{2},-\tfrac{c}{2}\right), \qquad R=\sqrt{x_C^2+y_C^2+z_C^2-d}.$

Example

Let $2x^2+2y^2+2z^2+6x+8y-2z+4=0$. Dividing by $2$: $x^2+y^2+z^2+3x+4y-z+2=0$. Then $C\left(-\tfrac32;-2;\tfrac12\right), \qquad R=\sqrt{\tfrac94+4+\tfrac14-2}=\sqrt{4{,}5}=\frac{3\sqrt2}{2}.$

Links

Topics: Analytic geometry in space
Concepts: Sphere
Skills: Analytic geometry · Using formulae