Is there an algebraic method for hyperbolic rotations?

Question

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?

Answer 1

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.