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to . This forms the basis for stereographic projection.

Volume and area

Let be the surface area of the unit -sphere of radius embedded in -dimensional Euclidean space, and let be the volume of its interior, the unit -ball. The surface area of an arbitrary -sphere is proportional to the st power of the radius, and the volume of an arbitrary -ball is proportional to the th power of the radius.

The -ball is sometimes defined as a single point. The -dimensional Hausdorff measure

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