Hyperbolic Coordinate Transformation in n-Sphere

[junt]

$x= r Cosh\theta$

$y= r Sinh\theta$

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.

 

[jvicens]

In higher dimensions, the radius of a hyperbolic n-sphere is defined by the equation:

$\sqrt{x_1^2 + x_2^2 + ... + x_n^2 - y_1^2 - y_2^2 - ... - y_n^2}$,

where x and y are the hyperbolic coordinates. This can also be written as:

$\sqrt{x^T x - y^T y}$,

where x and y are column vectors.

In terms of the specific coordinates given in the forum post, for 3D the radius would be:

$\sqrt{x^2-y^2-z^2} = r$

And in 4D, it would be:

$\sqrt{x^2-y^2-z^2-w^2} = r$

This can also be thought of as the distance from the origin to a point on the hyperbolic n-sphere.

In higher dimensions, the hyperbolic coordinates do not necessarily span the entire space. This is similar to spherical coordinates in 3D, where the radius only spans the positive space and the angles only span a portion of the space. To resolve this, you can use a different set of hyperbolic coordinates that cover the entire space, such as the Poincaré coordinates or the Beltrami coordinates.

In summary, the radius of a hyperbolic n-sphere in higher dimensions is defined by the equation above and can be thought of as the distance from the origin to a point on the hyperbolic n-sphere. Different sets of hyperbolic coordinates can be used to cover the entire space.