When I did dragon curves, I’d use U->L->D->R, so I don’t get confused when I am moving down and turning left becomes turning right.
My list of Ls and Rs can be more verbosely written like, “Draw forward, then turn left, then draw forward (now that you’re facing left), then turn right, then draw forward, then turn right again,” and so on. Alex’s directions would go more like, “Draw a line upwards (relative to the page), then draw to the left, then draw a line down, then draw to the right,” etc. Alex’s instructions recognize the fact that, while you’re busy drawing the dragon curve, your view of the page is not actually changing. Up, down, left, and right always point in the same direction, regardless of the last move. In my set of instructions, L and R don’t mean the same thing: you have to keep track of the direction your line is “facing,” the last direction you drew in, in order to turn left or right. The ULDR alphabet is probably a lot better for telling someone how to draw a given iteration of the dragon curve (although it doesn’t occur to me how I’d go about generating those instructions).
This difference in these kinds of instructions show up whenever we are describing an object moving around in space. We probably see it the most often when figuring out directions. The game everyone has to play when looking at a map is in connecting the fixed, objective orientation of a map to their own personal situation. In order to start figuring out which way to go, you have to figure out where you are, as well as ask, “which direction am I facing?” You then have to translate a set of directions on the objective map to a set of subjective directions, ones that you will have to execute from your point of view.
In the drawing, you might start describing your car’s path with, “drive two blocks east, then three blocks south, then two blocks east, then two blocks north.” You’d need to use the fixed system. But when executing your drive, you need to think in terms of your subjective viewpoint, “drive two blocks forward, then turn right, then drive three blocks, then turn left, drive two blocks, turn left again and drive two blocks.”
The directions we attach to the rotating, traveling car, Forward, Backwards, Left, and Right, are what you’d call in physics a body-fixed frame of reference, or a fixed-body coordinate system. The system we attached to the Earth, including North, South, East, and West, is an example of a space-fixed frame of reference, or an inertial coordinate system.
The difference between the objective and the body-fixed coordinate systems is pretty important in physics. The game plan in most situations involves first finding an objective coordinate system in which the laws of physics are the most simple. There might also be plenty of cases when someone would be interested in the alternative, a body fixed system, for instance, to verify that a person driving around still sees the same universe as the rest of us standing on the side of the road.
The car example is not too hard to wrap our heads around, since the car can only rotate around a vertical axis. It’s the same direction whether you’re using the compass rose or the car-fixed system: Up. When dealing with objects that can rotate in all three dimensions, though, things get mucky. Describing rotations in 3D can get complicated when making a distinction between body-fixed coordinates and space-fixed coordinates.
You might be thinking, “I don’t plan my car trips. I ask the GPS on my phone.” Well, sure, okay. FINE. The phone can detect where you are and which direction you’re traveling, and give you a set of instructions having translated NSEW to FBLR.
Google Maps on my Android can show my car directions either from the car’s point of view (with a neat 3D perspective camera following the car) or from overhead, with north fixed in a certain direction. I kind of prefer the latter, which like the dragon fractal instructions, means I would rather put that extra step on myself. I’m not sure what that means.
As a kid, I’d occasionally get the change to go to my uncle’s house and play Sonic the Hedgehog. I loved it. One day, another uncle of mine, his brother-in-law, traded Sonic for a copy of Electronic Arts’ Desert Strike.
I was pretty annoyed as a young kid. This slow, Gulf War-themed, objective-based helicopter mission simulator had nothing like Sonic whipping across colorful, futuristic, robot populated levels. But I did play it a bit.
The helicopter was controlled from an isometric viewpoint above, using a body-fixed system. No matter which direction you were facing, pressing Up on the directional pad made the chopper move forward. It could still look like the chopper was flying to the left on the screen. Pressing Left and Right made it rotate. This can be a bit confusing, since the controls act as if the player were in the cockpit, while the view is fixed from above. (The arcade game Asteroids is another example with a body-fixed control system, and probably a better example since it was way more popular and came first.)
Some years later, as an adult, I got a copy of Desert Strike of my own and did enjoy it. There was a way to change the directional controls, so that Up was always Up, etc. But I preferred the body-fixed controls.
This is a third example, after the dragon curve instructions and the Google Maps orientation, in which I prefer what seems to be the more difficult way of viewing the system. I must be MESSED UP.