information:filtering, information:extraction
Sphere with charts

A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of R 3 {\displaystyle \mathbb {R} ^{3}}: S = { (x, y, z) ∈ R 3 ∣ x 2 + y 2 + z 2 = 1 }. {\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.}

The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of R 2 {\displaystyle \mathbb {R} ^{2}}. Consider the northern hemisphere, which is the part with positive z coordinate (coloured red in the picture on the right). The function χ defined by χ (x, y, z) = (x, y), {\displaystyle \chi (x,y,z)=(x,y),\ }

maps the northern hemisphere to the open unit disc by projecting it on the (x, y) plane. A similar chart exists for the southern hemisphere. Together with two charts projecting on the (x, z) plane and two charts projecting on the (y, z) plane, an atlas of six charts is obtained which covers the entire sphere.

This can be easily generalized to higher-dimensional spheres.