4D road markings

[Keiji]

Not without turning it upside down

[gonegahgah]

One interesting effect of 4D having 360° of sideways is that a sign pointing to, say, the left, can be rotated so that it points to the right instead, without changing the direction it's facing. So a "curve right" sign can be rotated to be a "curve left" sign or a "curve ana" or "curve kata" sign, and you can still attach it to the pole in the same way as before.

And having the sign point any of the 360° of sideways is also possible for the same sign; though the facets will appear a little different for each degree of turn.

Now the interesting question is, if you paint a 3D left shoe on the sign, can you rotate it so that it becomes a right shoe?

If you mean can a left shoe in 4D also be turned to look like a right shoe by rotation, with both pointing up, then certainly, though I'm still gunning for a <insert-name1> shoe, a <insert-name2> shoe, and a <insert-name3> shoe to go with three legs

[quickfur]

Actually, you can't turn the drawing of a 3D left shoe into a right shoe just by rotation alone. You may try to do it by turning it upside-down, but it's still a left shoe, just upside-down, not a right shoe. Well, you can turn it into a right shoe if you turn the sign over so that its front is now touching the pole, but you wouldn't be able to see the drawing of the shoe unless the sign were transparent.

What I was getting at is that while chiral 3D objects can be turned over in 4D into their mirror images, you can't do this with a drawing on a 4D surface without also inverting the sense (i.e. facing-direction) of the surface. If the sign remains attached to the pole on its back surface, then only 3D rotations are possible for its front face. This is more than you can do with a 3D sign (which can only have a 2D rotation if it remains attached to the pole at the back). But to turn a left shoe into a right shoe requires a 4D rotation: one that will change where the sign is attached to the pole. To perform such a rotation it's no longer possible to keep the sign attached on its back surface.

[gonegahgah]

Actually, you can't turn the drawing of a 3D left shoe into a right shoe just by rotation alone. You may try to do it by turning it upside-down, but it's still a left shoe, just upside-down, not a right shoe. Well, you can turn it into a right shoe if you turn the sign over so that its front is now touching the pole, but you wouldn't be able to see the drawing of the shoe unless the sign were transparent.

What I was getting at is that while chiral 3D objects can be turned over in 4D into their mirror images, you can't do this with a drawing on a 4D surface without also inverting the sense (i.e. facing-direction) of the surface. If the sign remains attached to the pole on its back surface, then only 3D rotations are possible for its front face. This is more than you can do with a 3D sign (which can only have a 2D rotation if it remains attached to the pole at the back). But to turn a left shoe into a right shoe requires a 4D rotation: one that will change where the sign is attached to the pole. To perform such a rotation it's no longer possible to keep the sign attached on its back surface.

Just to add some further clarification to what your saying QuickFur, for other readers now that I understand this myself from help here, a sign in 4D can be rotated sideways in relation to the backing pole and re-attached to the backing pole so that the sign now has a different orientation sideways while still having the same bottom-top orientation. So a left object sign can be rotated to an ana object sign to a right object sign to a kata object sign or any angle in-between while still being the right way up. It doesn't need to be turned upside down which in our 3D world, as QuickFur says, just makes it the same orientation as it was but just upside-down. But, once the backing pole is attached, the pole itself has to remain at the back. In 4D you can't simply rotate the sign, while it is already attached, to make any other direction. First you have to unattach the sign from the backing pole, rotate it sideways, and then reattach it to the backing pole. And you can do this keeping the sign upright at all times in 4D; which we aren't able to do in our 3D world without turning it around to the back which, as QuickFur said, is useless because we can't then see it from our approach direction.

[gonegahgah]

It occurs to me, now as I consider the steps for animating the view of the 'road bends to ???? ahead' sign, that the question exists: 'to which side - of the 360° of roadside - should I put the road sign in the ground. The 4Ders could perhaps choose to have multiple of the same sign around the road to provide enough alert?

It means, in the rotational views, that sometimes the signs will be in the sky.
Wanting to look closer at them would require bringing them down into our 3D plane first.

[quickfur]

It just occurred to me that one very functional road layout in 4D is to have a circle of lanes with an island running down the middle. I've considered this before, but dismissed it for wasting space on the middle island, but I just realized that such an island is the perfect place for putting road signs! Rather than putting signs off to the side where they can be obscured by trees or buildings or whatever, why not put them right in the middle of the road? All lanes will circle around it, so there is a single place to keep an eye out for signs. That way, you cannot possibly miss the sign. (Of course, this presupposes a native 4Der POV; for your empowered-3Der scenario, you will probably still have to rotate one of the views in order to get the sign into view.)

As for what orientation to use for the sign, the answer is simply, just point the arrow wherever the physical direction is. To explain what I mean, consider the usual map projection (i.e., project vertically from above the ground): the road would project to a cylindrical tube in the 3D volume of the map, let's say it's vertical, stretching from the bottom of the map vertically up to the middle, just a little past where our vehicle's position is. A little further above, it curves off in one of the 360° of lateral directions, say it's 123° for no particular reason. Just before this curve, there's a sign standing from the middle island to warn drivers of the upcoming curve. Now, the sign will obviously be oriented vertically, such that it faces the driver, so in the map projection, it gets projected to a polygon -- in this case, a diamond (rotated square). Under this projection, the vertical part of the arrow drawn on the sign collapses into a dot, with the arrowhead projecting to a foreshortened arrow head, but it still retains the direction in which it is bending. So all we have to do is to draw this arrowhead such that it lines up with the 123° of the upcoming curve, that is, its map projection is pointing exactly in the same direction as the road will curve ahead.

So from the driver's POV, the arrowhead is pointing at 123° lateral, exactly in the same direction as the road will curve. So there is no confusion at all -- the meaning will be visually obvious.

And recently I started to be of the opinion that the best way to lay out writing in 4D road signs (e.g. street names, etc.) is vertically, so that there will be no confusion on the part of the drivers since they will read the letters in exactly the same way (top to bottom) regardless of their relative orientation to the sign. This also allows us to minimize the width of the signs, which is important since it allows us to just leave a small island in the middle of the road without fear that tall vehicles might collide with the sign.

[gonegahgah]

That all sounds like a good idea to me QuickFur.
So -: If you don't have a definite side, because of the higher dimensions, then utilise the inside :-
The solution may also help to address the issue of multiple lane confusion so that you only have to look two ways instead of all ways for other traffic.

[gonegahgah]

I'm following the video tutorials and gradually getting to know how to use SketchUp...
The following is a pre-step in drawing the 4D Road; which will eventually be using both rotation and projection. At the moment it only shows projection.

Pre-step


This shows a normal projection - being one possible normal projected offset - of the road and the projection here occurs into the groundwards direction. The solid area - which is the part of the road that is in the current view - won't change between the normal projection and the additional rotated projection; but the shadow - depicted as transparent grey/green and depicting the rest of the road in the 4th directions - will change.

The process of adding rotation will involve turning the road through all its angles around a forward pointing line and, for each angle, taking the horizontal ground slice then rotating that slice - as it now is - into the sky the same corresponding angle that it was off in the 4th direction before rotation; and doing this around the line through your feet or parallel to this just in front of your feet. It is fairly straight forward for this road I'm depiciting as it is flat. The composite of the total rotation will form the present view of the road.

The result will be like a narrow shadow cylinder standing on its rim with the curve forward. It will surround yourself around your feet and its radius will correspond to the length that the road is straight. The section that turns into the 4th direction will be off in the sky (or ground) and will narrow down with the turn into the corner. So it will still head off into the sky (or ground) like the projection version.

The 'road bends ahead' sign will be a little trickier because it has height as well. Each height of the sign stays relative to its lower sections so it has to be sliced up first before rotating each slice and then rotating these up into the sky. It still should look quite good but it will be a bit of work; I'm sure.

But, I'm getting there. I'm ready to start experimenting with drawing the road and then the sign after that...
I look hopefully forward to an engine that one day does all that work for me!