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11.2 Circles, Spheres, and Hyperspheres

An \(n\)-sphere, written \(S^{n}\), is defined as the set of \((n + 1)\)-dimensional unit vectors, \[ S^{n} = \{ x \in \mathbb{R}^{n+1} \, : \, \Vert x \Vert = 1 \}. \]

Even though \(S^n\) is made up of points in \((n+1)\) dimensions, it is only an \(n\)-dimensional manifold. For example, \(S^2\) is defined as a set of points in \(\mathbb{R}^3\), but each such point may be described uniquely by a latitude and longitude. Geometrically, the surface defined by \(S^2\) in \(\mathbb{R}^3\) behaves locally like a plane, i.e., \(\mathbb{R}^2\). However, the overall shape of \(S^2\) is not like a plane in that is compact (i.e., there is a maximum distance between points). If you set off around the globe in a “straight line” (i.e., a geodesic), you wind up back where you started eventually; that is why the geodesics on the sphere (\(S^2\)) are called “great circles,” and why we need to use some clever representations to do circular or spherical statistics.

Even though \(S^{n-1}\) behaves locally like \(\mathbb{R}^{n-1}\), there is no way to smoothly map between them. For example, because latitude and longitude work on a modular basis (wrapping at \(2\pi\) radians in natural units), they do not produce a smooth map.

Like a bounded interval \((a, b)\), in geometric terms, a sphere is compact in that the distance between any two points is bounded.