Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta)\end{array}\right)$$ This can then be expanded to representation of ellipses by Hadamard multiplication with one scaling $$M_{S} = \left(\begin{array}{cc} S_x & S_y \\ S_y & S_x\end{array}\right)$$ I have grown curious of how to represent points on hyperbolas. I know that the hyperbolic sin and cos: $$\sinh(\theta) = -i\sin(i\theta)$$ $$\cosh(\theta) = \cos(i\theta)$$ And the corresponding hyperbolic identity: $$\cosh(\theta)^2 - \sinh(\theta)^2 = 1$$

Seeing the same rules for sinh(2x) and cosh(2x) as for their sin and cos counterparts, what I've tried is to just replace $\cos(\theta) \rightarrow \cosh(\theta), \sin(\theta) \rightarrow \sinh(\theta)$ in the matrix above, but it does not seem to work.

• $\pmatrix{\cosh(\theta) & \sinh(\theta) \\ \sinh(\theta) & \cosh(\theta)}$ and the identity is: $\cosh^2-\sinh^2=1$. – Berci Jun 27 '15 at 20:47
• @Berci could you please write an answer out of it so I can accept it? – mathreadler Oct 24 '17 at 6:10