[Kosh]

Is that using 4d equations?

[Mika]

Um, to clear one thing up, this hasn't been yet truely four dimensional stuff.

One can parameterize a location vector in Cartesian coordinate system as:
r(t) = x(t)i + y(t)j + z(t)k

where x(t), y(t) and z(t) are some (preferably continuous) functions of time and i, j and k are the Cartesian unit vectors.

When it comes to the motion of particles, elementarily they are defined by the second time derivative of the location vector:
d²r(t) / dt²

This is the acceleration of a particle.

The actual difference between four dimensional and three dimensional stuff is that in Galilean transformations the t parameter is assumed the same for everyone, while in relativistic stuff t is not at all the same for everybody.

Now when that is said, Aardwolf, it is perfectly explainable that things infinite distance away work well while things closer to the "warp cylinder" do not work so well. This is simply because:

1) A cylindrical "refraction" (or how you call it in this case) system cannot image point as a point to the observer, but it will work as a point to line transformer. For example, you have to think about a star, which is imaged by a cylindrical lens. Now the rays starting from the star are arriving on the cylinder lens think about the local curvature that the ray sees. In other direction, there is no curvature (hence cylinder) and along the perpedicular axis there is normal spherical curvature. So now rays say "Crickey, there is refractional power along the other axis, but no power at all along the other axis!" So, the bastards focus only in the direction of the other axis, while continuing freely and unrivaled along the other axis. Thus the point (star is pretty good approximation of a point) is spread as a line.

2) When imaging a system infinite distance away, imaging is close to perfect since the rays are pretty much arriving parallel to the observer's imaging system's first surface, which allows the imaging system to work as ideally as possible. When the object is closer, there is always divergence added, and thus the image of those objects will appear as blurred (out of focus).

This is not to undermine your work efforts, the effect is quite good. As a side note one could try applying the refraction laws of rays in a cylindrical lens which would have a varial radius and a varial index (several Snell's law calculation required depending on the number of index steps) along the rotation axis on the problem, the effect would be approximately the same for the observer. I'm not sure which is computationally faster though.

I should shut up about this since this is starting to parallel my actual daily work as optical engineer...

Mika

EDIT: Typos

[Mika]

One more thing, you might want to consider using the surface that has minimalized surface area (in rotationally symmetric objects), as the tendency of nature is to arrive in the situation with minimalized potential energy.

Such a curve is defined by y(x)=a*cosh ((x-b)/a), where a and b are constants.

A picture of such surface can be found (photographed) in:
http://www.funsci.com/fun3_en/exper2/exper2.htm

Look at Figure 19.

Mika

[Aardwolf]

You're right about the bent ray being parameterizable in 3 dimensions as

r(t) = x(t)i + y(t)j + z(t)k

for the most part, but as you reach the other side, your parameterization has to squeeze an entire side of the universe into a region the size of the worm hole or less.

[Mika]

I'm not sure if I understood what you meant.

Take a look at:
http://www.cartage.org.lb/en/themes/sciences/physics/Optics/Optical/Lens/Lens2.htm

There are some common lens types included in the page. What I wanted to show is the wide-angle or fisheye lens. There are lenses that can image half of a hemisphere on to a circular detector (actually rectangular but no matter). If you look at the image, there is a location where all red, blue and green rays (fields) cross each other. This is called the aperture stop of the system, and it controls the rays incoming from the half hemisphere (in infinity) that are allowed to hit the detector on the right hand side. The image forms where all the rays from each respective field are brought to a common focus, i.e. the location where all the rays of equal color are brought to a same point.

This is actually a mapping from half-hemisphere to a limited area. This is possible because the photon has actually locational coordinates (call it the location space) and the angular coordinates (call it the angle space). Locational coordinates define where the ray is, while angular coordinates define where the ray is going to go, and there is a relation between them. Note that the above holds well in Euclidian space, while in general situation it might not be so in curved coordinates. But there the rays will still obey the minimized path distance principle.

Mika

[Dark RevenantX]

The game actually calculating this will **** up framerates.  Just do the simplest thing and have a vector-based system that uses four dimensions rather than three.  Just use a simple geometric model for the funnel.  Approximate where the light will hit you.  End of story.

You WILL commit suicide if you actually try to make this realistic.  Or, more specifically, the computer that actually has to calculate this **** in real time will.

[zonination]

You should find out how to use portals

[Dark Hunter]

Now you're thinking with portals! 

[zonination]

Okay, I've come up with a very rudimentary 3d equation for a vortex, in case you're interested. It's only 3d, but it represents the bending of space-time around a mass, so you can apply it to 4d if you ever get that far.

z= -r^2/(x^2+y^2+1/m) where 'm' is a mass. A mass of 1 is 'large'
Also, make sure that x^2+y^2 > 1/m if m < 0.

Oh, and r is the radius of... something...

If you want a worm hole in real physics, you use objects with negative mass. In this equation, if you want a wormhole, set m<0 (but be sure to obey the bounds i gave in the second line above)...

...I'm not a master of 4d, but 3d graphing is what I do in my free time...
...sometimes...

That's a very sexy equation by the way... I wrote that equation on the board today in one of my classes and got a few smiles from some females of the Terran species. I rule.

[Mika]

The game actually calculating this will **** up framerates.  Just do the simplest thing and have a vector-based system that uses four dimensions rather than three.  Just use a simple geometric model for the funnel.  Approximate where the light will hit you.  End of story.

You WILL commit suicide if you actually try to make this realistic.  Or, more specifically, the computer that actually has to calculate this **** in real time will.

One question: how is four dimensional parameterization easier than three dimensional? I'm not sure what do you actually mean by four dimensional at this point? Like physical three dimensions plus time dimension [x,y,z,t] ? Note that even in classical mechanics there are problems that should be called then four dimensional, since they also are parameterized as a function of time.

This is not to say that the computational cost of the real physics stuff is huge. But this information presented in the thread can be used to find out the easiest way of creating a visual that looks approximately the same. Personally this is quite interesting for me at least, since I have been wondering how is the stuff actually rendered in games since there is not time to calculate many reversed ray paths from camera to source. Do the graphics engines actually use rays at all?

As a side note to Aardwolf, considering observer inside the funnel, there are several interesting effects, depending which way do you want the rays to behave. If all the rays coming from the half hemisphere are parallel in one location inside the funnel, then observer in that location will only see a bright spot limited by his field of vision, but no image of the the universe. If the image of the universe is required, then the rays cannot travel parallel in any place of the funnel, there must be some angular divergence left. Then the observer will see the image of the universe, limited by his field of vision. The image is magnified by some factor, depending on the observer's location in the funnel and his field of vision.

If then, a ship would travel in front of the funnel, but not entering it, the ship could be magnified by a massive amount for the observer. However, the image would not be sharp. Though I'm not sure if you could make such an effect with the current graphics engine tools.

I should draw some sketches about this to check if my deductions are right.

Mika

[Mika]

And the most troublesome topic is how you would like to render a ship that is inside the funnel? Would the rays reflected/emitted by that ship obey the same laws as the rays entering the funnel?

Mika

[Aardwolf]

Well, no, the fourth dimension in this is not time (as I have said more than once), it is just an arbitrary spacial dimension, such that only three orthogonal axes of the four-dimensional space are used at any point. Sure, you could parameterize everything with three variables, but think what that parameterization would look like.

You can't write a formula in terms of the axial vectors for a point in one of two distinct spaces. The genus of this space is increased by one by adding a worm hole to it, which makes parameterization much much harder and much less linear for one or both sides, when you consider that both sides of the worm hole are approximately infinite.

Unfortunately,
although this idea for a worm hole would be awesome, it's not practical, as to render it would require a ton of math to get it accurately, and an approximation using render-to-texture would be terribly flawed (as I have explained elsewhere in this thread).

What would work, however, is a worm hole like this: