When \(e_1=e_2=e_3=R\) we can solve \(E(t)=0\) analytically. \[y_1^2+y_2^2+y_3^2-(t+R^2)^2=0\] \[t = - R^2 + \sqrt{y_1^2+y_2^2+y_3^2}\] Note: We consider the positive \(t\). Negative \(t\) corresponds to the point \(\tilde{x}\) on the other side of the ellipsoid.