by quickfur » Sat Jul 07, 2012 2:34 pm
This is my approach, which may not be the same as yours, but maybe you might get some ideas from it:
I think of a 4D racing game from an FPS kinda viewpoint, where you're looking in the direction you're going (more or less), parallel to the ground. So the horizon would project to a plane halfway up the projection volume, and, let's take the simplest case of a straight road, the road will appear as a tapered thing, say a square pyramid or a cone, with the tip touching the horizon (that's the point where the road "meets" the horizon), and whose base is the bottom of the projection (where the viewpoint is "standing on").
Before we talk about cars, let's first talk about how wheels will look like, and how tyre rotation will appear, because they are very different from what we might imagine according to our 3D-centric thinking. Let's say we have a cubindrical wheel, and for simplicity, let's consider a detached wheel just rolling along the road (we'll worry about attaching it to a vehicle later). What will it appear like? Our 3D-centric thinking will probably imagine the cubinder resting on one of its cylindrical surfaces, and it will turn along the axis of this cylinder as it rolls forwards towards the 4D viewpoint, which will appear to be getting larger and moving downwards from the tip of the cone to the base. At least, this is the way I thought of it at first. But actually, that's completely wrong. The surface that the cubinder will roll on is actually not any of the 4 cylinders on its surface; those cylinders are flat from the 4D POV, the shape can't roll on them at all! (They are the analogue of the circular lids of a 3D cylinder: place a cylinder on its lid, and it won't roll at all.)
So what you will actually see, if the cubindrical wheel is rolling towards you, is actually a cube-shape volume (slightly bulging around its 4 horizontal sides) doing the "inside-out" rotation as it moves downwards from the tip of the cone to its base. Well, it's the "ana" part of the inside-out rotation, that is, because the wheel is opaque, so you'll only see a cube slightly bulging around the sides, and horizontal squares appearing at the top (they are the tyre treads) and "sliding down" the sides of the square before disappearing when it touches the bottom face of the square. They aren't really sliding in 4D, of course; they are fixed ridges on the curved surface of the cubinder (which is a square torus, that is, the torus formed by sweeping a square in the XY plane along the circle in the WZ plane, assuming Z is the vertical direction). As they rotate with the cubinder, they come into view at the top of the bulging cube image of the cubinder, and slide downwards (actually rotating forwards) and touch the road at the bottom of the cube, then disappear from view. The part of the cubinder that touches the road is not any of its cylindrical cells, but a square-shaped cross section of its square torus.
This, of course, is the simplest case of the wheel farther along the road, rolling directly at the 4D viewpoint. If you were sitting in a vehicle with, say, 4 exposed front wheels, they would appear as the top half of a bulging cube in 4 corners, arranged in a square, at the bottom of the projection volume (touching the base of the conical image of the road, and maybe protruding from the cone a little, if they are tall wheels or if the road is very narrow -- note that the bottom of the wheels are actually within the road surface; the protruding is just a visual artifact).
Now, what if the wheels are turned? First, let's say the 4D viewpoint is standing in the exact center of the road, and a wheel is rolling on the road towards the viewpoint, but along the +X side of the road instead of directly at the 4D viewpoint. How would it appear? It will still project to a bulging cube, except now the cube's volume consists of two distinct volumes, one is like a square tube bent sideways into a C shape (well, not as extreme as that, but just slightly bent) and then a cylinder-shaped volume fitted into the C shape, filling up the rest of the bulging cube's volume. The cylinder appears very flattened sideways, and as the wheel turns, you can now see the cylinder turning too. The tyre treads are still squares that appear at the top of the cube, but now they move down in a C-shaped curve instead of straight down as before. As the wheel approaches the 4D viewpoint, its image will move down the sloping side of the conical image of the road, from the tip to the edge of the circular base.
Now, notice that the cubinder's image is a bulging cube, which means it has 4 horizontal sides. When it's directly facing the 4D viewpoint, all you see is the bulging cube. There are 360° of freedom where it can point away from the 4D viewpoint; so the bent C-shaped image can be along any of the 360° horizontal directions. You can have an entire circle of wheels on the road rolling towards you along the side of the road, and in projection you'll see a circle of bulging cubes with the C-shaped volumes pointing outwards from the axis of the cone, all apparently sliding down the sides of the cone until they sink into the bottom of the image. They won't actually collide with you, of course, even though from your 3D viewpoint one of them looks like it might -- but actually, they're all "off to the side". What will collide with you is the case where the bulging cube is moving directly downwards from the tip of the cone to the center of its base: when it gets there, it will bump you in the face.
Now, let's say you're sitting in a vehicle with four front exposed front wheels (like your typical racing car has uncovered wheels). As I said, they will all appear as the top halves of bulging cubes, in a square formation at the bottom of the projection volume. When you turn +X-ways, they will twist in that direction to make the square tube bent into C-shape shape (well, the top half of that shape). When you turn -X-ways, they will twist in the opposite direction. When you turn +Y or -Y, they will also twist in those directions. In-between directions will also do the same thing. So how do you tell which way the wheels are pointing? Just look at which way the front of the bulge is pointing. Opposite of that will be the image of the cylindrical volumes (which are actually the hub caps of the wheels). There are 360° of directions your wheel can turn in -- this is the XY component of the wheel's actual 4D direction. How distorted the C-shape curve is, is how far in that direction the wheel is actually pointing, so every one of the 360° directions actually has their own 360° direction. In 4D, you have a sphere of possible directions to travel in. The analogous 3D case only has 2 sideway directions, left and right, and you can turn slightly to the left or very sharply to the left; in 4D, you can turn in, say, the 47° direction either very gently or very sharply. If you like, think of this as a sphere with an arrow pointing from the center to the "north pole"; the first 360° sideway directions determines the longitude on of the arrow head when you turn it, and the 2nd 360° sideway directions determines the latitude of the arrow head.
If you want to give a more obvious clue as to which direction the vehicle is moving, you can draw an imaginary arrow in front of the vehicle pointing forwards, which in projection will appear to be an arrow starting at the base of the cone (the road's image) and pointing in some upward direction. When it's pointing directly at the tip of the cone, you're parallel to the road; otherwise, you're misaligned and will drive off the road shortly.
Of course, all this is just the simple case of a single straight road. If the road curves, then its projected image will be a zigzagging cone (or a twisty tentacle, starting from the bottom of the projected volume and winding until its tip touches the horizon plane). Now, notice that there's a lot of volume under the horizon, which is actually the amount of land area (or rather, land volume) that is available. It's very easy to imagine, say, a torus encircling the cone image of the road: that torus would be another road, a circular road, that encircles the road you're on, but isn't connected to it. It could be full of cars, or boulders, or whatever, and you could still drive through without hitting them because your road passes right "through" them (well, not through them literally, of course, technically speaking you're passing "by" them, even though they're all around you when you pass). Navigating on a curving road in 4D is tricky business; you have to be able to steer in 360° of sideways, plus control how much you're veering in that direction at any given time so that you remain on the road. There are just so many more ways to fly off the road if you're not careful!
While driving around in 4D, you may also see other landscape features, like mountains, which will appear as jagged cones above the horizon plane, and rivers, which will appear as winding "tube" shapes below the horizon plane. Note that the road can easily pass beside the river without ever needing to cross it. It can even spiral around the river and you never have to get your wheels wet. Other straight roads can criss-cross the volume below the horizon plane without ever touching the road you're on. You may also see the sun as a sphere somewhere above the horizon plane. Note that it has full 3D freedom as to where to be! Sunrise/sunset will appear as a hemisphere on the horizon plane.
Even more interestingly, if you drive past a city, you can have houses all around the road (in fact, the same building can completely encircle the road, and you can still just drive right through without hitting any walls).
Lots of fun can be had in 4D.
Last edited by
quickfur on Sun Jul 08, 2012 5:22 am, edited 1 time in total.