Q:

Geometric Operations in GIS Systems

I have worked with 2D geometry for CAD - My questions are general and relate to understanding operations on geometric entities (lines).
Should geometric operations such as line-line intersection be done in the projected spaces (such as EPSG  3857) ?
I've had a brief look at the geos library, but it is unclear whether the inputs should be in a linear projected space.
If operations are done in projected space - does this introduce significant error in the result when the output is un-projected?

A:

Possible error of your geometric operation depends on:

overall size of the objects - bigger size increase errors, 
projection that you use,
datum that you use (each datum suits some parts of the Earth more than the others)
quality of your data.

Generally you don't want to work with unprojected data at all unless there is some specific reasons like finding an orthodrome on a global scale (there is a Gnomonic projection for a regional scale) or you are working with global data in general (in this case you store data unprojected but project it for specific operations you need to perform: distance calculation, etc.). Note that there is no universal projection and for each task and the region of the world an appropriate projection (more precisely - CRS) have to be used for achieving the best results.
If your project demands to work with more than one projection you should pay a lot of attention to your data quality and integrity. Here a question: will a parallel and a meridian cross in any projection? The picture below is an unprojected image of the countries and a parallel and a meridian.

'Yes' would you say - they will cross in any projection. But I say - 'Nope if your data sucks ass'. Lets project our data into the Bonne projection:

Both parallel and meridian were defined only by 2 points each (start and end). That leaded to a disaster in specific projection. But if we know that we will use our lines in specific projection we can adapt our data to it. Lets add some nodes to our lines an project them again: much better result.

So when you are working with GIS, especially if you are going to modify your data - you have to understand the pros and cons of CRS. Don't be afraid to use projections - be afraid to use wrong one.

