rotated hyperbolic cylinder parameterization

Question

A hyperbolic cylinder is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2} = -1$, but thats a hyperbolic cylinder that goes along the Z-axis. How do you parametrize a hyperbolic cylinder that goes instead of symmetric along z=z, symmetric along some direction z=cx+dy+e. (i need to fit data with a hyperbolic cylinder model, but the hyperbolic cylinder is not defined along the z-axis, but along some other direction)

Answer 1

$\newcommand{\R}{\mathbf{R}}\newcommand{\e}{\mathbf{e}}$I'm not entirely sure about your terminology. (For example, $z = cx + dy + e$ defines a plane in $\R^{3}$, not a direction; maybe that's your plane of reflection?) However:

If $\{\e_{1}, \e_{2}, \e_{3}\}$ denotes the standard Cartesian basis of $\R^{3}$, then the cylinder $(x/a)^{2} - (y/b)^{2} = -1$ can be parametrized by $x = a\sinh u$, $y = b\cosh u$, and $z = v$, namely by the vector-valued function $$ \Phi(u, v) = (a\sinh u)\e_{1} + (b\cosh u)\e_{2} + v\e_{3}. $$ Replacing the standard basis by a basis adapted to your needs should parametrize the corresponding surface.

For example, if you merely want the plane $z = cx + dy$ as plane of reflection symmetry, you can take $$ \e_{1} = \frac{(-d, c, 0)}{\sqrt{c^{2} + d^{2}}},\quad \e_{3} = \frac{(c, d, -1)}{\sqrt{c^{2} + d^{2} + 1}},\quad \e_{2} = \e_{3} \times \e_{1}. $$ If the choice of $\e_{1}$ and $\e_{2}$ matters, you can replace the above with the rotated vectors $$ \e_{1}' = (\cos\theta)\e_{1} + (\sin\theta)\e_{2},\quad \e_{2}' = -(\sin\theta)\e_{1} + (\cos\theta)\e_{2},\quad \e_{3}' = \e_{3}. $$

If you want the "center" of your cylinder somewhere other than the origin, simple translation (i.e., adding constants to the component functions) will now do the job.