Suppose \(\mathbb{S}^{d}_R\) denotes the \(d\)-sphere with radius \(R>0\). Then for \(p\in \mathbb{S}^{d-1}\) and \(r>0\)
\begin{equation*} K(p,r)=\{q \in \mathbb{S}^{d}_R \mid d_{R}(p,q)\le \pi r\} \end{equation*}defines the spherical cap at \(p\) with radius \(r\), where \(d_{R}\) denotes the spherical distance .