That last image is not how artists draw rotated cubes in three-point perspective. So, I have to conclude that the author's formalization only works for lines parallel to the axes.
Indeed. As you stated, mutual vanishing points, as used in illustration, are when parallel lines of the scene are extended to infinity and converge. Animation of a rotating object requires animation of a set of vanishing points for each frame.
This is nearly deja vu for me, as a conversation that I've seen several times at the intersection of arts and computer graphics. I can only conclude that a sub-population have received instruction on "vanishing points" that were overly specific and missed the footnote on how it generalizes. Specifically: the artist and draftsman idioms of 2 or 3-point perspective or vanishing points are short cuts for perspective rendering of subjects which have cartesian layouts, such as rectilinear buildings and street scenes built on a cartesian grid, with lots of parallel or perpendicular edges. The missing footnote is that this does not work for arbitrary subjects.
Real perspective rendering has infinite potential vanishing points in one scene. Each point represents nothing more than the infinite continuation of any line segment in the scene, as it would be rendered by a true perspective rendering. The vanishing point of any one line is when the viewing ray converges with that line, to the limit of angular resolution in your rendering. All more distant segments project into the same small picture element, whether formed by raster graphics or your finest pencil, pen, brush, or engraving tool.
For a complex scene, the skilled artist would choose [edited to delete typo "three"] different vanishing points appropriate for each set of parallel lines. E.g. a vineyard with parallel rows of plantings might use different points than a road passing by at an odd angle, and a set of high-tension powerlines crossing the scene would have its own vanishing points as well. If the power lines follow a ragged course, each segment between two towers would need its own vanishing points. Furthermore, the artist would have to approximate the parabolic sag of the wire below these projected line segments, perhaps using parallel lines to locate the envelope within which the line sags.
Artist is my day job and this is a decent approximation of what a complex perspective drawing entails. And, really, it probably entails a lot of eyeballing it, too - sketch in something that feels roughly right against whatever guidelines you may have constructed, and leave it at that.
"Two-point perspective" and "three-point perspective" are good enough approximations for the common problem of "drawing a scene set within a place built around a lot of right angles".
I feel like the computing analogy to make here is "a very opinionated framework": it makes life a lot easier, if you want to do things the same way it's built to do. If you want to do stuff outside of what it's built to do you're gonna have to do a lot of the work yourself.
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.
I think there's two issues making it this way. The big one is the use of 'exponential space' based on each distance being halved along an axis, which is not accurate to how artists perspective works if drawn properly (someone smarter than me may do the math to figure out what the real formula is here).
The second point is the vanishing point for verticals is ridiculously close to the cube which makes it an incredibly tortured perspective for the eye to understand to start with.
If you're a professional artist, and you draw on actual paper, you will almost never work outside of the very first grid space. This article shows you what happens if you were to use the full exponential space, which (and let me stress this yet again) *no one should ever do*.
Any tutorial on perspective drawing teaches you to keep your vanishing points out of your picture, and to stay the hell away from them. That's solid advice, and in this article we investigate why that's solid advice.
This is just not the case. You can have three orthogonal vanishing points all inside the bounds of an image and it can look just fine. It just corresponds to having a field of view wider than 90 degrees (which looks a bit unnatural but can be a dramatic way of presenting an image).
Imagine standing on the corner of a city block, such that without turning your head you can look down both streets and up at the sky. You can see the two streets recede off into the distance and the skyscraper above you foreshorten towards a vanishing point far overhead.
Of course a vanishing point can be inside a perspective drawing, after all where do you put the vanishing point for one-point perspective?
