Hyperbolic Coordinate Transformation in n-Sphere $x= r Cosh\theta$
$y= r Sinh\theta$
In 2D, the radius of hyperbolic circle is given by:
$\sqrt{x^2-y^2}$, which is r.
What about in 3D, 4D and higher dimensions.
In 3D, is the radius
$\sqrt{x^2-y^2-z^2}$?
Does one call them hyperbolic n-Sphere? How is the radius defined in these dimensions. In addition these coordinates doesn't seem equivalent. For Cosh only span positive space, while Sinh span both positive and negative space. How can we resolve this? What another hyperbolic coordinate system is good that resolves this problem? I am mainly interested in changing my coordinates x,y,z, etc. to hyperbolic coordinate system.
 A: With $x=r\cosh\theta$ and $y=r\sinh\theta$ you get a hyperbola, or at least one branch of a hyperbola. Usually I'd interpret “hyperbolic circle” as “circle in hyperbolic geometry”, and unless you are dealing with some rare instances of the Beltrami-Klein model, the hyperbola you describe is not a circle of hyperbolic geometry.
The three-dimensional generalization of a hyperbola is called a  hyperboloid. For the $x^2-y^2-z^2=r^2$ you mention, you'd get the two-sheeted hyperboloid of revolution. That's the surface you get by rotating your hyperbola around the $x$ axis. You might parametrize one of its sheets (corresponding to one branch of the hyperbola) as
\begin{align*}
x&=r\cosh\theta\\
y&=r\sinh\theta\cos\varphi\\
z&=r\sinh\theta\sin\varphi
\end{align*}
But the one-sheeted hyperboloid $x^2-y^2+z^2=r^2$ is just as valid a generalization. It's what you get if you rotate your hyperbola around the $y$ axis. You might parametrize it as
\begin{align*}
x&=r\cosh\theta\cos\varphi\\
y&=r\sinh\theta\\
z&=r\cosh\theta\sin\varphi
\end{align*}
