You're right, that's a great thing about them pie menus: they are "self revealing", since they show you how to use them.

There are three phases of using pie menus, and they support "rehearsal" to move you smoothly and seamlessly and unconsciously from novice to expert:

1) Novice: Click to pop the menu up. Look at the screen. Read the menu items. Browse the items (by pointing at them, possibly revealing more information or hiding unselected item labels, see the Unity3D pie menus for an example). Each time you use the menu this way, you're rehearsing the faster mouse gesture to make the same selection.

2) Intermediate: Remember which direction the item you want is in. Press and move in that direction, then look at the screen and wait for the menu to pop up confirming that you have selected the right item. When you know you got the right item, release the mouse button to select the desired item. This increases your confidence that you can (unconsciously) move onto the next stage.

3) Expert: Remember which direction you want, and swipe (mouse ahead) in the appropriate direction, even making multiple selections through nested menus. It's not even necessary to look at the menu (you can keep looking at whatever the menu selection affects, like the object you clicked on to pop it up, and the pie menu gesture tracking can provide real-time feedback of previewing the changes during the gesture, before even showing the menu. At any time you can stop moving the mouse and the menu will pop up and reveal the possible and currently selected items and their directions.

Color selection, font style, or size selection menus are great examples of previewing menu selection effects without needing to show the menu, where it's more obvious to just see the effect on the object directly instead of reading about it in the menu, popped up over and blocking your view of the object itself.

Also, the distance as well as the direction can be used as a parameter, like a 2-dimensional font "pull-out" style (direction) size (distance) pie menu.

Linear menus with keyboard shortcuts suffer from the fact that the keyboard shortcuts are totally different actions than using the menus the slow way with the mouse, so slowly using linear menus with the mouse is NOT rehearsal for quickly using linear menus with the keyboard, and all your time using linear menus the slow way is wasted, instead of rehearsing to use them the fast way every time you use them the slow way, like pie menus.

Pie menus also support "browsing" and "re-selection", as opposed to marking menus or gesture recognition. All possible pie menu gestures are valid easily understandable selections, while with marking menus and gesture recognitions, most possible gestures are syntax errors.

So you can move the mouse into and around the pie menu (even before it's popped up) to browse different items (seeing their effect on the object you clicked on, instead of the menu popping up and blocking your view of it). This enables you to correct mistakes, as well as simply browse and see all the available options by pointing at them (possibly revealing more detailed descriptions, like the Unity3D menus).

https://donhopkins.medium.com/gesture-space-842e3cdc7102

Gesture Space

The space of all possible gestures, between touching the screen / pressing the button, moving along an arbitrary path (or not, in the case of a tap), and lifting your finger / releasing the button. It gets a lot more complex with multi touch gestures, but it’s the same basic idea, just multiple gestures in parallel.

Excerpt About Gesture Space

I think it’s important to trigger pie menus on a mouse click (and control them by the instantaneous direction between clicks, but NOT the path taken, in order to allow re-selection and browsing), and to center them on the exact position of the mouse click. The user should have a crisp consistent mental model of how pie menus work (which is NOT the case for gesture recognition). Pie menus should completely cover all possible “gesture space” with well defined behavior (by basing the selection on the angle between clicks, and not the path taken). In contrast, gesture recognition does NOT cover all gesture space (because most gestures are syntax errors, and gestures should be far apart and distinct in gesture space to prevent errors), and they do not allow in-flight re-selection, and they are not “self revealing” like pie menus.

Pie menus are more predictable, reliable, forgiving, simpler and easier to learn than gesture recognition, because it’s impossible to make a syntax error, always possible to recover from a mistaken direction before releasing the button, they “self reveal” their directions by popping up a window with labels, and they “train” you to mouse ahead by “rehearsal”.

More comments from the HN discussion of "Visualizing Fitts' Law":

https://news.ycombinator.com/item?id=16614889

>Pie menus benefit from Fitts' Law by minimizing the target distance to a small constant (the radius of the inactive region in the menu center where the cursor starts) and maximizing the target area of each item (a wedge shaped slice that extends to the edge of the screen).

>They also have the advantage that you don't need to focus your visual attention on hitting the target (which linear menus require), because you can move in any direction into a big slice without looking at the screen (while parking the cursor in a little rectangle requires visual feedback), and you can learn to use them with muscle memory, with quick "mouse ahead" gestures.

[...]

Updated links from that old message:

An Empirical Comparison of Pie vs. Linear Menus (Published at AMC CHI '88):

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

The Design and Implementation of Pie Menus: They’re Fast, Easy, and Self-Revealing.
(Originally published in Dr. Dobb’s Journal, Dec. 1991, User Interface Issue, cover article.)

https://donhopkins.medium.com/the-design-and-implementation-...

Pie Menus: A 30 Year Retrospective.
By Don Hopkins, Ground Up Software, May 15, 2018.
Take a Look and Feel Free!

https://donhopkins.medium.com/pie-menus-936fed383ff1