Can derive the velocity parameter using Wick rotation

The \(x^2 + y^2\) invariant implies the Euclidean coordinate transformation is in terms of cosines and sines. In relativity the invariant is \(x^2 - t^2\). We can relate the coordinate transformation in relativity to that of Euclidean geometry by defining \(t \equiv iT\). The transformation of \(x\) and \(T\) will be simply sines and cosines like for \(x\) and \(y\), however when we substitute back \(T \rightarrow -it\), we find that the transformation formula only makes sense if \(\cos \theta\) is real and \(\sin \theta\) is imaginary, or if \(\theta\) is imaginary: \(\theta = i \eta\) for \(\eta\) real. We can then write the transformation in terms of \(\sinh \eta\) and \(\cosh \eta\).