4D translation to/from null-space.

Vlion · 2002-03-25T22:45:26

This isn`t strictly a math question, but its rather mathematical. I`m working on a 4D engine right now, and my end goal is to have a demo where you can do null-space jumps. Now, we know that in Euclid, d=sqrt(x^2+y^2+z^2+q^2) So if you are movil along a vector in 3d, at point P move in the q component, and move back onto the original vector at point Q, you will have moved a greater distance relative to you in Euclid than if you had kept on the original vector. This defeats the purpose of having a null-space jump, which is to get somewhere very fast. So, I`m going to try and work out a relationship between null-space and real-space. I have a web page with a 143kb pic on it illustrating this. http://thefivelions.bugle4d/distance/index.html Your translation point into null-space is P. Your translation point out of null-space is Q. Your distance traveled in null-space must be less than the distance traveled in realspace from those 2 coords. Find a relationship that hold true in interval [0, inf] between each point in null-space and each point in real-space. Currently, I think that the relationship should probably be non-linear; furthermore, I think that it should take longer for small jumps than for large. So given that my func that describes distance/time in realspace is d(t)=t And my null-space distance/time describing func is d(t)=t^2(t-1)+.2 At a given time t, Each ordered pair has the form (t,d(t)) So: (t,t) (t,t^2(t-1)+.2) So in my null-space point structure, I need to have a counter for distance, which will determine real d Lets try it out. Given that we have a (x,y,z,q) tetra. When q changes from 0, we move into null-space. When it moves back to 0, we move into real-space. Moving around in null-space is moving around in 4-space. To translate back into 3-space with q=0, we use this formula x(d)=d^2(d-1)+.2 y(d)=d^2(d-1)+.2 z(d)=d^2(d-1)+.2 we set q=0, and we make sure no accidental translations occur with our code in case q happens to hit 0 when we are in null-space. How does this look, people ? Bugle4d

Math and Physics Programming

Started by Vlion March 18, 2002 06:28 PM

15 comments, last by Vlion 22 years ago

Vlion

151

Author

March 21, 2002 02:05 PM

Well, to be quite honest, thats exactly what I want to know.
Hence my engine. Hopefully during spring break I can finish the basic code and get a viewer running.
I think I am almost ready to do that.
This thread is my thinking ahead to my goal.
yipeee !
V''lion

Bugle4d

~V'lionBugle4d

Timkin

864

March 21, 2002 07:34 PM

quote:Original post by Vlion
So in real-space I select the point I want to move to, i come up with the point in null-space I need to go to at a given speed, then I proceed to translate to nulls-space, go to that point at the given speed, then translate to real-space...

Well, yes, that would work. There are other ways, but that would suffice.

Timkin

sQuid

149

March 25, 2002 04:08 AM

One interesting possibility might be the 2-d surface of a torus embedded in a 4-d Euclidean space. The torus can be "twisted" to give its surface a Euclidean metric.

Timkin

864

March 25, 2002 06:31 PM

I''m going to need 5 minutes to get my brain around the tangent field for that manifold... come back next year!

While I can easily visualise a torus embedded in 3-d space, embedding in 4-d space is a little harder to picture (but I can use sub-manifold projections to make it easier...). I cannot though see how the twisting of the torus will give it a Euclidean metric on it''s surface... how do you get around the periodicity of the torus'' surface?

I am genuinely interested in hearing how this is done!

Thanks,

Timkin

Vlion

151

Author

March 25, 2002 10:04 PM

oi.
torus in 3-space, yes.
torus surface is a 2-space.

3-space torus in 4-space.
roger that; all coords have a W componet the same.

Twist ?
Like taking two ends of a bike tube and twisting them like a rope ?
How would that change the dimemnsion of it ?
If I am not mistaken, the topological dimension is the same.

I really need to get my 4-space view going.

V''lion

Bugle4d

~V'lionBugle4d

sQuid

149

March 25, 2002 10:31 PM

First up I''d better apologize - I''m just confused too

What I was thinking of was this
this which is about curvature.

Imagine rendering a plane which represents the view of 2-d being living in a 2-d toroidal space where light moves on the surface of this torus. Looking in two directions the being would eventually see itself as the surface is periodic, and in one of those directions space would appear squashed together. As the torus is twisted into the 4th dimension the plane starts to look euclidean, albeit with the periodicity mentioned above.

I think this would be a neat think to code up, but sorry if i was misleading before.

Vlion

151

Author

March 25, 2002 10:45 PM

coool
a periodic eulidean metric; very interesting.
Hey-
If you`d like to see some code for 4d rendering, email me.

You should be able to hack up some kind of 4d parametric function viewer without a terrible abount of pain.

Bugle4d

~V'lionBugle4d

4D translation to/from null-space.

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4D translation to/from null-space.

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