I would like to know how to do neat illustration for geometric concept such as this one : 
This image come from "DoCarmo - Differential geometry of curve and surfaces, p41" and is an illustration of the covariant derivative defined on a regular surface S.
I would like answer to such questions :
- Do I need to use Geogebra or a software such as Photoshop to reach such a result ?
- Do I need to find parametrization to all of these object or instead, draw them like on a sheet of paper but on a specific software ?
- How much time do it usually take to a mathematician to create its illustration ?
- ...
I will be also very grateful of some tips. Thank you a lot.