Summary: Let \(p_i\) and \(q_i\) belong to a hyperquadric \(Q\) and \((e_{1_i},\ldots,e_{n_i})\) and \((f_{1_i},\ldots,f_{n_i})\) be orthonormal frames in \(T_{p_i}Q\) and \(T_{q_i}Q\), respectively, where \(1\leq i \leq m\). We study sufficient and necessary conditions for existence of an isometry \(\varphi: \mathbb{R}_\nu^{n+1}\to\mathbb{R}_\nu^{n+1}\) such that \(\varphi(Q)\subset Q\), \(\varphi(p_i)=q_i\) and \(d\varphi(e_{j_i})=f_{j_i}\).