5
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – bubba Oct 13 '13 at 9:38
  • $\begingroup$ 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. $\endgroup$ – David Oct 14 '13 at 2:42
4
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – David Oct 17 '13 at 4:20
  • $\begingroup$ 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. $\endgroup$ – Muphrid Oct 17 '13 at 4:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.