Need 3-ball to 3-sphere projection equation

[mikef]

I am trying to find the equation that will let me do an inverse azimuthal equidistant projection from the values contained within a 3-ball onto the surface of a 3-sphere. I have found plenty of information concerning 3-spheres, but nothing having to do with this kind of projection. I have also found the inverse azimuthal projection equation from a 2-ball onto a 2-sphere on Mathworld (http://mathworld.wolfram.com/AzimuthalEquidistantProjection.html), but nothing dealing with higher dimensions.

 

[jvicens]

Hello,

Thank you for your inquiry. The inverse azimuthal equidistant projection from a 3-ball onto a 3-sphere is a complex mathematical problem that has not yet been extensively studied or solved. However, there are some potential approaches that may be helpful in finding a solution.

Firstly, it may be useful to consider the properties of the 3-sphere and the 3-ball in relation to each other. The 3-sphere can be thought of as a higher dimensional version of the 2-sphere, and the 3-ball as a higher dimensional version of the 2-ball. This means that the equations for the inverse azimuthal equidistant projection from a 2-ball onto a 2-sphere may provide some insight into the problem.

Additionally, it may be helpful to look at the equations for other types of projections from a 3-ball onto a 3-sphere, such as the stereographic projection or the gnomonic projection. These equations may provide a starting point for developing an equation for the inverse azimuthal equidistant projection.

It is also worth considering the properties of the inverse azimuthal equidistant projection itself. This projection preserves distances from the center of the projection to any point on the 2-sphere, and it may be possible to adapt this property to the 3-sphere.

Finally, it may be beneficial to collaborate with other mathematicians and researchers who have expertise in higher dimensional geometry and projections. Their insights and perspectives may be valuable in developing an equation for this specific projection.

I hope this information is helpful in your search for the equation for the inverse azimuthal equidistant projection from a 3-ball onto a 3-sphere. Best of luck in your research!