But, ok. We're talking about schemes to rotate a single object rather than the camera.
The turntable controls they talk about are a special case of gimbal controls where we lock rotation to one or two axes. But in my personal experience, when it's two or three axes people still call it gimbal controls whereas turntable controls is rotation in a single axis (as if on a turntable, hence the name). But then again, 3d terminology is so mixed up across different fields that maybe some people have only heard the two axes version called a turntable. Not a big deal, but why no mention at all of gimbal controls?
Then, trackballs. In my experience, when it's limited to a single hemisphere of rotation, these are called arcball controls. Trackballs are supposed to emulate a trackball mouse which don't have this limitation.
And finally, no mention at all of the dreaded gimbal lock (where two of axes end up overlaid on each other and the controls loses a degree of freedom), which is a major reason for choosing one type over another.
Overall, not an amazing article. I checked it again to see if I missed anything and realized it's a blog post for some app - so, basically an ad - which probably explains the lack of effort.
It's also missing the absolute most natural user interface for rotating objects, which is when you are in VR or with hand tracking. If you have only one hand controller you attach the object to it so you can simply inspect it under any angle, and if you have two-hand tracking you can have translation, scale and rotation based on a virtual segment between the hands. The little Leap motion device worked like this it was very natural.
Some systems implement a trackball UI that behaves like a real trackball, i.e. it also has inertia, so you can give an impulse towards a direction and the 3D model will continue to rotate after you no longer click the mouse button, until you stop the rotation.
I do not remember using any trackball UI where the rotation was limited. When the trackball UI did not have inertia, than you would need to do 3 pushes for a rotation over 360 degrees or 2 pushes for a rotation over 180 degrees, but you could still reach any angles.
I guess lots of people (somehow) got used to it without knowing there's an alternative?
Honestly I don't see that supposed failing as much of a downside really, because A) most of those extra rotations don't change the camera view in any meaningful way, they just roll it, B) you can typically rotate the object on top of the camera if you really need these rotations, and C) it makes control much more accurate.
My personal preference, and how I've set up my visualization engine (For chemistry vis projects and similar): Move the camera; not the object. FPV controls, plus a roll axis, with crouch/jump for up/down. Works best with a videogame controller, but FPS standard + Q/E is fine too! Full control, and intuitive if you've played those sorts of games. This is full 6 axis.
Turntable makes most sense when there is well defined and easily identifiable up direction like when you are working with 3d environments or humanoid models. With mechanical parts that you make in CAD software up direction isn't always as clear, you can easily forget where the up was if you don't pay attention to axis gizmo. Plenty of parts can have critical features on all sides, and the preferred "up" direction in most useful views for the part doesn't necessarily match with the up direction of whole assembly.
Turntable view has one downside which the article didn't mention - lack of "state independence". The behavior of mouse movement depends on the angle from which you looked at the start. This can result in situations where you meant to rotate around one axis but it rotates around different one or the rotation happens in opposite directions. Tumble and trackball don't have this problem as rotation isn't affected by global axis direction.
One more potential factor is use of 3d mice. I haven't used one myself, but I would assume that it somewhat avoids the most confusing aspects of non turntable modes. Thus in the best case reducing the need to improve default behavior for regular mice, in the worst case hacky 3d mice integration depending on the chosen rotation style which might brake if something other than tumbler is chosen (I really hope that no 3d software does this, but I lack the experience to confirm this).
This makes me wonder has any software tried dynamic turntable. That is turntable where the role of z axis is replaced by whichever axis was closer to up at the start of mouse drag. Always one of the 3 axis, not the local rotation up direction.
Any decent CAD system will support both turntable and tumble, defaulting to a user-selected choice, with easy options to override the behavior and switch to the other.
Actually, when I accidentally tumble models with that kind of UI, I just drag it in a circle until it's right side up.
But turntable is clearly far superior to the others. Unless you really need to yaw the model then stick with that.
But hey, most implementations talk about using quaternions, which scare people.
And so the author of the article (...and a lot of software) is completely ignorant of how e.g. 3DS Max, Maya, etc do it.
FWIW, here's a proper arc ball implementation. Try it vs. your favorite 3D software, and by God, implement it in yours:
(Demo requires mouse)
I don't understand this part. I could easily see all 6 faces with turntable. Does it really matters to be able to make arbitrary rotations?
Also, Matt Keeter has some serious skills.
I always thought his Antimony CAD program[1] was neat, and sad that it seems to have died. I've yet to figure out how to get it to run on newer versions of linux (The last time I tried was about 3 years ago, and I was just using a raspberry pi.)