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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, $S^{n} = \left\{ x \in \mathbb{R}^{n+1} \: : \: \Vert x \Vert = 1 \right\}.$

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.