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"quote": " $u$ 与 $-u$ 对 $\\vec v$ 的作用相同，因此 $\\mathrm{SO}(3)$ 中每个元素在 $\\mathrm{Sp}(1)$ 中恰有两个原像。映射 $\\Phi$ 是满射，所以它是次数为二的[覆盖映射(covering map)](../../../../../concepts/topology/covering-map)。因此，$\\mathrm{Sp}(1)$ 二重覆盖 $\\mathrm{SO}(3)$，这个群记作 $\\operatorname{Spin}(3)$。从几何上看，$\\mathrm{SO}(3)$ 是将三球面 $S^3=\\mathrm{Sp}(1)=\\operatorname{Spin}(3)$ 上的对径点等同后得到的空间。该空间即[实射影 3-空间(real projective 3-space)](../../../../../concepts/topology/real-projective-space)，也可视为 $\\mathbb R^4$ 中过原点的直线所成的空间。",
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"quote": "Both $u$ and $-u$ act identically on $\\vec v$, so each element of $\\mathrm{SO}(3)$ has two preimages in $\\mathrm{Sp}(1)$. The map $\\Phi$ is surjective, hence it is a [covering map](../../../../../concepts/topology/covering-map) of degree two. Thus $\\mathrm{Sp}(1)$ double-covers $\\mathrm{SO}(3)$, and we denote this group by $\\operatorname{Spin}(3)$. Geometrically, $\\mathrm{SO}(3)$ is $S^3=\\mathrm{Sp}(1)=\\operatorname{Spin}(3)$ with antipodal points identified. The resulting space is [real projective 3-space](../../../../../concepts/topology/real-projective-space), also the space of lines through the origin in $\\mathbb R^4$.",
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