on level2">
<h2><span class="header-section-number">11.2</span> Circles, Spheres, and Hyperspheres</h2>
<p>An <span class="math inline">\(n\)</span>-sphere, written <span class="math inline">\(S^{n}\)</span>, is defined as the set of <span class="math inline">\((n + 1)\)</span>-dimensional unit vectors,
<span class="math display">\[
S^{n} = \{ x \in \mathbb{R}^{n+1} \, : \, \Vert x \Vert = 1 \}.
\]</span></p>
<p>Even though <span class="math inline">\(S^n\)</span> is made up of points in <span class="math i