# 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. – bubba Oct 13 '13 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. – David Oct 14 '13 at 2:42

## 1 Answer

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.

• Hi Muphrid. I recognize your answer as coming from geometric algebra/clifford algebra. Looking at some of your other answers, you seem really knowledgeable in this area. Could you recommend some books or resources? Many thanks. – David Oct 17 '13 at 4:20
• I'll speak a bit about a few books I have personal experience with. Alan MacDonald's books are introductory level and well-suited to connecting concepts to traditional teachings. Doran and Lasenby gives a good overview of physics applications. Dorst, Fontijne, and Mann is oriented for computer science applications. Both are intermediate level in terms of necessary background and sophistication. Hestenes and Sobczyk is a rigorous, in-depth text; it's a little older, and newer texts have refined some things, but it's worth it for rigorous linear algebra and differential geometry applications. – Muphrid Oct 17 '13 at 4:31