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I want write $x^2+y^2-z^2-t^2$ in mixture of polar and hyperbolic coordinates.

In $3$D, it is possible as follows: $$x^2-y^2-z^2=r^2 ,$$ where \begin{align} x&=r \sinh \theta \\ y&=r \cosh\theta \cos\phi \\ z&=r \cosh\theta \sin\phi \end{align}

How does one implement the coordinate transformation in 4D?

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Introduce the coordinate $w$ and coordinate $\alpha$, given by $$x = w \cos \alpha, \qquad y = w \sin \alpha .$$ (These are essentially polar coordinates on the $xy$-plane.) Then, in these coordinates, the equation $$r^2 = x^2 + y^2 - z^2 - t^2$$ becomes $$r^2 = w^2 - z^2 - t^2 .$$ Now, just apply the formula you gave for the $3$-dimensional case. This same method can be generalized to general dimension and signature of the quadratic form.

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