field of view - When do straight lines become curved when talking about projection?

This in not a question directly pertaining to photography, but the theory of projection in general.

Imagine that I'm drawing my surroundings, say, in a city with straight lines everywhere. I start by drawing what is directly in front of me, and going further and further out. I intend to draw everything around me. When will I be forced to draw curved lines rather than straight ones? Or are straight lines actually unrealistic? I'm just having a hard time wrapping my head around how we can make "flat" images from a "spherical" field of view, I'm not even sure if I'm using the right words to describe the questions.

There is undoubtedly extensive research and papers made on the subject, I'd appreciate if someone can point out some of that for me to get learned on the subject.

Here follows a painting which inspired me to ask the question:

The Courtyard by Irvine Peacock