special relativity - Why isn't there a limit for a Euclidean rotation, as for a Minkowski rotation?

From invariance of the Minkowski scalar product, we get the Lorentz transformations. In addition, we get a constant $c$ preventing space-like and time-like intervals being rotated into one another.

The Euclidean transformations are derived in the same way from invariance of the Euclidean scalar product, yet there isn't a constant that limits these rotations as above. Why?

Answer

The natural way to parametrize a boost in spacetime (hyperbolic geometry) is by the quantity known as rapidity, just as the natural way to parametrize a rotation in Euclidean space is the angle.

An angle measures distance along a unit circle. That is, the circumference of a unit circle is $2\pi$, so we parametrize angles by the distance the angle subtends on the unit circle.

Rapidity measures invariant interval along the unit hyperbola. It is the natural way of parametrizing distance along the unit hyperbola, and boosts are characterized by interval along the unit hyperbola.

In terms of speed, rapidity is expressed $\mu = \tanh^{-1}{v/c}$. As $v$ approaches $c$, $\mu$ approaches infinity. So with this natural parameterization, there is no "speed limit".

Update

To get an idea why rapidity is more natural, look at this Wikipedia page and see how simple the expression is for $\gamma$ (simple enough to write it here: $\gamma = \cosh \mu$), and how closely the Lorentz transformation equations resemble the equations for rotation in Euclidean space.

Note also the simplicity of the formula for addition of velocity: $\mu_\mathrm{final} = \mu_1 + \mu_2$

Update #2

This answer gives an excellent real explanation for why rapidity is the natural way to parametrize boosts.