The dotted red circle and dotted purple circle are there to give a stronger scent of a 3D space and the arrangement of the 2D images(like a fan). It also indicates that each slice is about 1o apart and a legend that'd give you an idea where the z-axis should be.
For peeps who are good at math would immediately recognize what i am trying to do and see the solution. For a slow learner like me, it took me a while to flash back to basic geometry.
The solution is indeed very simple: Spherical Coordinate System.
Since, formally speaking, what i am trying to do is to convert spherical coordinates to Cartesian coordinates. The correspoding (x,y,z) can then be expressed by:
I should have known better!