The hyperboloid given by:
$$ x^2 + y^2 - z^2 = -1 $$
can be parameterized as:
$$ \begin{align} x &= \sinh(r)\ \cos(\theta)\\ y &= \sinh(r)\ \sin(\theta)\\ z &= \cosh(r)\\ \end{align} $$
Conversely, given $(x, y, z)$ we can find $(r, \theta)$:
$$ \begin{align} r &= \cosh^{-1}(z)\\ \theta &= \arctan(y/x)\\ \end{align} $$
Translating an $(x, y, z)$ point $a$ units along the $x$ axis is accomplished by a hyperbolic rotation.
$$ \begin{bmatrix} \cosh(a) & 0 & \sinh(a) \\ 0 & 1 & 0 \\ \sinh(a) & 0 & \cosh(a) \\ \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} $$
Questions
- How is an $(r, \theta)$ point translated?
- Suppose I have the three points $A$ at $(0, 0°)$, $B$ at $(1, 0°)$, and $C$ at $(1, 45°)$. If these three points are translated 1 unit to the right, will $C$ still be at $45°$ relative to $A$?
- Will the alternate angle to $45°$ be $180°-45° = 135°$?
- Will $C$ still be $1$ unit away from $A$?
I've drawn a little picture to help illustrate the problem.
$$ \begin{array}{c|ll} point & before & after \\ \hline A & (0, 0°) & (1, 0°) \\ B & (1, 0°) & (2, 0°) \\ C & (1, 45°) & (?, ?) \\ \end{array} $$