# Is there an algebraic method for hyperbolic rotations?

Given a 2d vector, how do you rotate it in space? You could use a rotation matrix,

$$\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta &\cos\theta \end{bmatrix} \begin{bmatrix} x\\y\end{bmatrix}$$

or you could represent it as a complex number, and multiply it by a complex exponential.

$$x'+iy'=e^{i\theta}(x+iy) =(\cos\theta+i\sin\theta)(x+iy)$$

For a hyperbolic rotation

$$\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix} \cosh\theta & \sinh\theta \\ \sinh\theta &\cosh\theta \end{bmatrix} \begin{bmatrix} x\\y\end{bmatrix}$$ is there an algebraic method of representing this transformation, in analogy to the complex numbers?

• You could substitute $\cosh\theta = \cos i\theta$ and $\sinh\theta = -i\sin i\theta$, but I don't know if the results would be useful to you. Commented Oct 13, 2013 at 9:38
• Just tried that substitution, and I get a formula like $x'+iy'=(\cos i\theta+\sin i\theta)x+(\cos i\theta-\sin i\theta)iy$. It's a good idea, but isn't really a useful result. Commented Oct 14, 2013 at 2:42

You can posit the existence of a number $\epsilon \neq \pm 1$ such that $\epsilon^2 = 1$. Then the exponential takes the form $e^{\epsilon \theta} = \cosh \theta + \epsilon \sinh \theta$. Such numbers are called split-complex numbers.