Artists don't draw all the way to the vanishing points, we only draw in the part that looks normal. Look at the figure showing off how straight lines behave again, and you'll see that everything starts off actually looking like normal straight lines, and things don't actually go crazy until we start getting close to our vanishing points.
Which is why on paper, you place the vanishing points outside your drawing, so your drawing can stay constrained to the part of exponential space that looks normal. E.g. if you constrain the viewport to the unit area (the triangle between 1/0/0, 0/1/0, and 0/0/1) and scale your scene down to fit inside of that, things will actually look perfectly fine, with a gorgeous perspective (but will also be visually indistinguishable from a wide angle camera positioned closed to your major scene point, so just use that).
However, with computers we can trivially see what will happen if you do try to use the full space, rather than only working inside an (incredibly sensible!) crop. The result is pretty wild.
Artists don't draw all the way to the vanishing points, we only draw in the part that looks normal
I am a professional artist and I would just like to confirm this line. 3-point perspective tends to start looking weird once you get outside of a certain sweet spot; I did so many beginner drawings with the vanishing points too close, which resulted in a weirdly exaggerated set of shapes.
The effect is also pretty dramatic and I think this is a big part of illustration work. Comic books in particular use three point perspective to add drama.
Which is why on paper, you place the vanishing points outside your drawing, so your drawing can stay constrained to the part of exponential space that looks normal. E.g. if you constrain the viewport to the unit area (the triangle between 1/0/0, 0/1/0, and 0/0/1) and scale your scene down to fit inside of that, things will actually look perfectly fine, with a gorgeous perspective (but will also be visually indistinguishable from a wide angle camera positioned closed to your major scene point, so just use that).
However, with computers we can trivially see what will happen if you do try to use the full space, rather than only working inside an (incredibly sensible!) crop. The result is pretty wild.