Fitts' Law intuitively and mathematically explains why pie menus are faster and have lower error rates than linear menus. Fitts's Law says in effect: the bigger and closer a target is, the faster and more reliably you can hit it.

https://en.wikipedia.org/wiki/Fitts%27s_law

>Fitts's law (often cited as Fitts' law) is a predictive model of human movement primarily used in human–computer interaction and ergonomics. The law predicts that the time required to rapidly move to a target area is a function of the ratio between the distance to the target and the width of the target. Fitts's law is used to model the act of pointing, either by physically touching an object with a hand or finger, or virtually, by pointing to an object on a computer monitor using a pointing device. It was initially developed by Paul Fitts.

Pie menus both minimize the target distance, while also maximizing the target size. And the physical directional gesture required to select an item does not demand your continuous visual attention and cognitively taxing hand-eye feedback loop. You don't need to look at the screen to select items from a pie menu, so you can reliably "mouse ahead" or gesture, which is impossible with linear menus.

Another advantage is that they support "rehearsal", by seamlessly training novice users to become experts.

Unlike keyboard shortcuts, the physical action of the novice and expert is the same, only experts can do it faster without looking at the screen.

Unlike traditional "invisible" gesture recognition systems like Palm Graffiti or StrokePlus.net, pie menus are "self revealing" in that the can pop up a menu that shows all the available options and their direction. It's quite difficult to learn which invisible gestures are available and what they all mean, but easy to discover them with pie menus.

Pie menus also support "browsing" and "reselection", which gesture recognition doesn't, allowing users to correct errors or even browse around highlighting every item to preview its effects.

Since pie menus can provide live continuous preview of the effects of the selected item in the application itself (which is especially nice when you use the distance as a parameter, for example a pie menu that lets you create eight objects, with a pull-out size parameter: the more you pull out, the bigger the object previewed at your cursor, which switches to different objects as you browse around the menu, then turns into a real object when you release the button), so you can see the effect and release the button when it's perfect, or go back to the center to cancel. Often the only feedback you need is the live preview of the pie menu selection, and popping up a menu would be a distraction.

Once you learn the directions, you can quickly select items by stroking in the desired direction without even looking at the menu, so pie menus can perform the selection without popping up the menu. There's no need to pop up the "self revealing" menu until you stop dragging the mouse, or you can pop it up instantly by "clicking up" the menu. So a novice can get directions instantly, while an expert can fly ahead without waiting for directions, then pause for directions or confirmation that you have the right selection at any time.

There's a smooth escalator moving you up the learning curve every time you use a pie menu, training your muscle memory through rehearsal.

That is how pie menus "Lead, follow, or get out of the way".

We performed an empirical study in 1988 that measured eight-item pie menus to be 15% faster than linear menus, with less frequent errors.

An Empirical Comparison of Pie vs. Linear Menus (Jack Callahan, Don Hopkins, Mark Weiser, and Ben Shneiderman, Proc. ACM CHI’88):

https://donhopkins.medium.com/an-empirical-comparison-of-pie...

>Pie menus gain over traditional linear menus by reducing target seek time, lowering error rates by fixing the distance factor and increasing the target size in Fitts’s Law, minimizing the drift distance after target selection, and are, in general, subjectively equivalent to the linear style.

[...]

>Pilot study results:
A pilot study of 16 subjects showed that users were approximately 15% faster with the pie menus and that errors were less frequent with pie menus. Statistically significant differences were found for item seek time but not task type. Subjects were split on their subjective preference of pie and linear menus. Some commented that they were able to visually isolate an item easier with linear menus and that it was hard to control the selection in pie menus because of the sensitivity of the pie menu selection mechanism. These subjects tended to be the most mouse naive of all whereas those who had heard of or seen a mouse/cursor controlled system but had not used one extensively tended to prefer pie menus. The most mouse naive users, while finding linear menus easier, tended to be better at pie menus and commented that with practice, they would probably be superior and in fact prefer the pie menus because of their speed and minimization of hand movement with the mouse. Not surprisingly, therefore, most of those preferring linear menus did not have a strong preference on the scaled subjective questionnaire.

More about Fitts's Law and pie menu in this thread from Sept 27, 2022 on: Cairo: Alternative Windows Desktop:

https://news.ycombinator.com/item?id=32992673