How to convert Theta-Phi Array to UV or Az-El Array

I have a matrix which includes the r(magnitude in my case) values for each degree. An example matrix is like thi:

Theta Phi R

-180 0 0.1

-180 1 0.2

....

+180 180 0.5

I previously created a rectangular plot using those variables on theta-phi window. Now, I want to create the same plots by using UV and Az-El coordinate systems. My intention is to see the differences of the plot bendings in different coordinate systems (like the world map projections in different coordinate systems). The problem is, when I convert these using ordinary conversion method using Spherical Corrdinate Conversion , the UV values would not be integers as expected, i.e. usign coordinate transformation makes the bending on the coordinate values, not on the magnitude locations.

Is there any method I can apply to change the corrdinate system in orger to see the bendings using different spherical coordinates?