Name for a coordinate system defined on a surface of a [unit] sphere One of the commonly used coordinate systems in 3D is a spherical one. Which would be defined by: * a radius $r$ * polar and azimuthal angles $\theta$ and $\phi$, where "mathematicians and physicists would argue about which one is which". Now, I am working in a coordinate system (?) in 2D which is effectively a set of points on the surface of the sphere ($R\ne 0$), which can be taken as a unit one ($R=1$), since this does not really change anything. Now, a point is uniquely defined by only $\theta$ and $\phi$. * Is there a specific name for this particular coordinate system? * Is there a specific term which would describe such a mapping from 3D to 2D? (say for spheroidal, cylindrical, and other coordinate systems) Motivation: do not want to invent a bicycle in terms of naming conventions.

For your first question, I don't think there's a very standard terminology. You could call them "latitude longitude coordinates" and no-one would object or be confused. You could perhaps even call them "spherical coordinates" since they are just a restriction to the sphere of the usual spherical coordinates in 3-space.

For your second question, your use of the term "mapping" is confusing to me. Perhaps you simply mean the process of restricting a 3-d coordinate system to a 2-d coordinate system after setting one of the original 3 coordinates equal to a constant? If so, you could call that "restriction of coordinates".