Higher-dimensional flight controls...

[pat]

For the following, assume you're piloting a space craft. Assume a Euclidean coordinate system local to your space craft. Have that coordinate system oriented so that the center-of-mass of the ship is at the origin, the x-axis points directly out the nose of your craft, the y-axis points directly through the roof, the z-axis points directly out the right side of the craft, the w-axis points directly out

Typical video games where you're (mostly) locked on a 2-d surface (like car racing games, tank games, first-person-shooters, etc.) usually give you motion controls like, forward, reverse, turn left, turn right. (Some give you strafing, too... but not so much in the car and tank games.) So, you can move forward and back or rotate your coordinate system in the xz plane. With this one rotation, you can achieve any 2-d orientation.

Typical video games where you're (entirely) locked in a 3-d universe (like space flight games and flight sims and such) usually give you motion controls like: forward, roll left, roll right, pitch downward, pitch upward. So, you can move forward or rotate your coordinate system in either the xy plane or the yz plane. With these two rotations, you can achieve any 3-d orientation. Note: you'd still be able to achieve any orientation with xz and yz roations or with xy and xz.

So, we already see that moving from two- to three- dimensions, the controls from the previous dimensions are ignored. I suspect, however, that's a result of gravity being perpendicular to most 2-D games. I suspect that the fact that the 'y' axis is involved in both of the 3-D moves is because of the fact that gravity acts in that direction. And, space-flight games still play on the same idea that we have an intuitive feel for "down" (or at least for airplane controls).

So, if we were to move up to 4-d, what would be the most intuitive set of controls? I'm thinking that we'd keep the 3-d controls: forward, yz-rotations, and xy-rotations. Then, we'd add: yw-rotations. But, maybe xw-rotations would be more intuitive. I'm pretty sure that you'd be able to achieve any 4-D orientation with either set. But, what do y'all think... should we roll into the next dimension or pitch into it?

Another possible explanation for the 3-D flight controls is to think of an airplane with two wings, each with a flap. Rolling in the yz plane amounts to tipping the flaps on the wings in opposite directions (one up, one down). Pitching forward or backward amounts to tipping the flaps on the wings in the same direction (both up or both down). I can see two ways to try to extend this to four dimensions.

One is to say that the flaps can not only move up and down, but also ana and kata. Thus, you could do all kinds of things like: move one flap ana and the other down, move both flaps kata, move one flap up and the other flap up-ana, etc. This would probably result in controls very similar to tank driving games that give you two joysticks... one for the left-tread and one for the right-tread. In this, however, both joysticks would have a full 2-D range of motion. This would be easily implemented on modern joysticks which have two independent analog sticks. The only problem here is that to only have two flaps would mean that they're in the same plane. They'd be offset from each other in z, but not so much in w. Thus, it's somewhat like having flaps in xyz and only a rudder in xw, I think.

The other way to extend this is to assume, instead, that you've got one set of wings with flaps which are offset in +z and -z and can tip in a rotation parallel to the xy plane (these are normal airplane flaps) and another set of wings with flaps which are offset in +w and -w and can tip in a rotation parallel to the xy plane. Again, this would work decently with two analog joysticks. One would be the xyz airplane controls and the other would be the xyw airplane controls.

Thoughts? questions? comments? ideas? alternatives? cookies?

[RQ]

If you are talking about the 4th dimension being out towards us and being perpendicular to the screen where the game is actually 3D, then the screen is only a representation, it is not 3D on 2D, which of course defies logic.

[pat]

If you are talking about the 4th dimension being out towards us and being perpendicular to the screen where the game is actually 3D, then the screen is only a representation, it is not 3D on 2D, which of course defies logic.

No... I'm not at all concerned at this point about visualizing things on the screen. I am asking about what sort of flight controls one would expect in a 4-D space craft. All of the stuff about video-games and joysticks and such is much later. Right now, I want to understand what controls one would want beyond those which a normal flight-stick give to you.

[pat]

Actually, maybe this will be more clear.... The x-axis goes straight into the screen. The y-axis goes straight up on the screen. The z-axis goes straight to the right. The w-axis goes another direction. Would you be most comfortable if the flight controls let you rotate in yw, xw, or maybe zw?

[PWrong]

What shape is the plane? An aeroplane in 3D is roughly a cylinder with wings. The wings are complicated shapes, but basically they cause the plane to move upward when it moves forward. I'll just assume that 4D wings do the same thing. Is the 4D plane a cubinder, spherinder or duocylinder?

I'm not very experienced with polychora, but I'll have a go. The aeroplane can only move in one direction, so it should only have one linear extention. A cubinder is won't work because it has two linear extentions, and a duocylinder won't work because it has no linear extensions. So it must be a spherinder. you might have assumed this already, but I want to know if I have this right before I start working on the problem.

This question seems really complicated

[pat]

The aeroplane can only move in one direction, so it should only have one linear extention. .... So it must be a spherinder. you might have assumed this already, but I want to know if I have this right before I start working on the problem.

My logic exactly. (In fact, it's similar logic to the Tetranail shape.)

As for wings, I was assuming that gravity is acting in the y-direction. As such, there must exist some planes parallel to the x-y plane whose intersection with the ship is an airfoil cross-section (and, further, I suppose, that there is sort of a minimal profile of the ship in the y-direction... I don't want to fly a van around).

And, I'm thinking the steering has to be something like flaps and rudders. I'm defining flap-type control to be some control which acts like some panel diverting air that's flowing along the x-axis (direction of motion) either with or against gravity. I'm defining rudder-type control to be some control which acts like some panel diverting air that's flowing along the x-axis (direction of motion) in some direction perpendicular to gravity.

My experience is that most video games which involve flight omit rudder-type control. There is only flap-type control. Flight-simulators (by necessity) cannot ignore this control.

With flap-type control, the placement of the flap determines what sort of torque can be applied to the craft. If the flap sits on the x-axis, then it can only effect pitching. However, if the flap is offset a bit in the z-direction and another sits offset in the negative z-direction, then the combination of flaps can be used to effect pitching or rolling (when they're moved in tandem or in opposition). The same is true if you have flaps offset in the w-direction---they can be used to effect pitching or rolling (I believe).

I think that if you had flaps in +/- z and +/- w, that will likely be the most intuitive set of controls. And, if fact, I believe there is never any time where you would want to move the flap in +z without moving the one in -z (either in the same direction or the opposite one). I think anything you can do by just moving the +z and +w flap is equivalent to something you can do by moving the +z a little less and the -z in the opposite direction while also moving the +w a little less and the -w in the opposite direction.

But, I'm both babbling and biasing at this point. So... I'll cut out now...

[pat]

The aeroplane can only move in one direction, so it should only have one linear extention. .... So it must be a spherinder. you might have assumed this already, but I want to know if I have this right before I start working on the problem.

My logic exactly.

Actually, this is a bit over-zealous. There are some aeroplanes like the B2-bomber which aren't very cylindrical. The real key is keeping a fairly low profile in the y-direction and keeping the shape aerodynamic in the other directions. But, the spherinder+wings is a good way to start. You can then blur the spherinder into the wings.

Maybe a good way to think about the aerodynamics though is an attempt to facetize the surface of the aeroplane. Then, having a low wind-resistance is roughly equivalent trying to keep to a minimum the sum of the absolute values of the dot-product of the facet normal with the x-axis. And, more lift is possible with higher sums of the absolute values of the dot-product of the facet normal with the y-axis.