Inspect a paper on 4D rotation, anyone ?

Vlion · 2002-02-22T04:03:12

Heya everybody- I`ve just gotten done writing about 4D rotation. But, before I put it on my website, I want to make sure everything is accurate. So... Take a read, please. ---------------------- Rotation in 4 Spatial Dimensions. Paul "V''lion" Nathan. So you have decided to make a 4D engine, have you ? Good for you ! I`m going to discuss 4D rotation, extrapolating the 4D equations from the 3D equations, as well as using 2D rotation for various purposes. I`m not going to go into the actual mathematical proof for rotation; thats well documented on the internet in 2D and 3D contexts and it does not change for 4D. Lets start by rotations in the plane- Call it the XY plane, and we will be using 2-space. I will use C++ code. math equations for 2D are: x'' = y * sin(theta) + x * cos(theta) y'' = y * sin(theta) - y * cos(theta) code. void c2DPoint::2DRotate(float theta) { float x0, y0; x0 = y * sin(theta) + x * cos(theta) y0 = y * sin(theta) - y * cos(theta) //This just assigns the results to itself, assuming c2DPoint has a x and a y. *this.x = x0; *this.y = y0; } Now, you are wondering, why just not use x and y, instead of x0 and y0 ? Because if you did, you would screw up the equation. It needs the original, unaltered x and y to operate with. So it requires 1 angle and 1 equation set to do a 2-space rotation. Lets go into the 3-space rotation. The maths hairly, but don`t let it confuse you ! Just remember that y'' is simply a different variable than y. x'' = y * sin(theta) + x * cos(theta) y'' = y * sin(theta) - y * cos(theta) z'' = z x'''' = x'' * sin(phi) + x'' * cos(phi) y'''' = y'' z'''' = z'' * sin(phi) - z'' * cos(phi) x'''''' = x'''' y'''''' = y'' * sin(chi) + y'' * cos(chi) z'''''' = z'' * sin(chi) - z'' * cos(chi) Heres the code; void c3DPoint::Rotate(float theta = 0, float phi = 0, float chi = 0) { c3DPoint temp0, temp1, temp2, temp3; temp0 = temp1 = temp2 = *this; //XY plane temp1.x = temp0.y * sin(theta) + temp0.x * cos(theta); temp1.y = temp0.y * cos(theta) - temp0.x * sin(theta); temp1.z = temp0.z; //XZ plane temp2.x = temp1.z * sin(phi) + temp1.x * cos(phi); temp2.y = temp1.y; temp2.z = temp1.z * cos(phi) - temp1.x * sin(phi); //YZ temp3.x = temp2.x; temp3.y = temp2.z * cos(chi) + temp2.y * sin(chi); temp3.z = temp2.z * cos(chi) - temp2.y * sin(chi); *this = temp3; } We now have 3 angles, and 3 equation sets. I won`t do the math for 4 dimensions, but it works _exactly_ as does the 2D math and the 3D math. What that means for you is you have to do- heres the funky part- 6 sets of equations, and 6 angles. That really shocked me when I found it out- I just didn`t believe it. But, heres how it works. In 2D, the plane is XY. All angles are expressed from a initial and a terminal side in the plane. In 3D, we have 1 more axis. But- 2 more planes: YZ and XZ. So in 4d, we have 4 axes, and 3 more planes. XQ, YQ, and ZQ. Add all the planes up and you get 6 planes, with 4 axes. If you have 1 rotation a plane, that means- 6 angles. Here is the relevant code from my own engine, bugle4d: //Works similar to the OGL function, but operates against the point that calls it. //6 angles. XY plane, XZ plane, YZ plane, XQ plane, YQ plane, ZQ plane. void c4Point::Rotate(float theta = 0, float phi = 0, float chi = 0, float rho = 0, float mu = 0, float lambda = 0) { //I know I could use like only 3 of these, but having all of these makes it more clear c4Point temp0, temp1, temp2, temp3, temp4, temp5, temp6; temp0 = *this; //XY plane temp1.x = (temp0.y * sin(theta)) + (temp0.x * cos(theta)); temp1.y = (temp0.y * cos(theta)) - (temp0.x * sin(theta)); temp1.z = temp0.z; temp1.q = temp0.q; //XZ plane temp2.x = (temp1.z * sin(phi)) + (temp1.x * cos(phi)); temp2.y = temp1.y; temp2.z = (temp1.z * cos(phi)) - (temp1.x * sin(phi)); temp2.q = temp1.q; //YZ temp3.x = temp2.x; temp3.y = (temp2.z * cos(chi)) + (temp2.y * sin(chi)); temp3.z = (temp2.z * cos(chi)) - (temp2.y * sin(chi)); temp3.q = temp2.q; //XQ temp4.x = (temp3.q * cos(rho)) + (temp3.x * sin(rho)); temp4.y = temp3.y; temp4.z = temp3.z; temp4.q = (temp3.q * cos(rho)) - (temp3.x * sin(rho)); //YQ temp5.x = temp4.x; temp5.y = (temp4.q * cos(mu)) + (temp4.y * sin(mu)); temp5.z = temp4.z; temp5.q = (temp4.q * cos(mu)) - (temp4.y * sin(mu)); //ZQ temp6.x = temp5.x; temp6.y = temp5.y; temp6.z = (temp5.q * cos(lambda)) + (temp5.z * sin(lambda)); temp6.q = (temp5.q * cos(lambda)) - (temp5.z * sin(lambda)); //After the cumulative effect for the rotations, we assign it to *this. *this = temp6; } I hope this makes 4D rotation a little easier for you. For a full software implementation with matrices, look at: http://research.microsoft.com/~hollasch/thesis/ Steven Richard Hollasch`s 4D engine. It is research and a 4D engine. It is also the only other one in a "real" language. Its written in C for a old API, but his reasearch still alooks sound. Note on notation: For the 4th dimension, I used Q, because when you have homogneous coords, you use W as the extra coord. So to ease confusion, Q. Bugle4d

Math and Physics Programming

Started by Vlion February 18, 2002 11:05 PM

10 comments, last by Vlion 22 years, 1 month ago

Vlion

151

Author

February 21, 2002 09:41 PM

Quite frankly, I don`t expect a great many people to hit my site right now unless I send a update message to a major coding site.

[dream]
However, if I get everything working, I have a innotative demo in my long-range plans, and I suspect that IF I ever finish it, a great deal of people would chck out my site and read my articles, and maybe even work on a 4D engine.
[/dream]

On a less visionary note, I do release my engine source to the public, so that people can use that as a base if they want to write a modern 4D engine. The design is made for portability.

I am well aware that modern games require hordes of math.
Whether many people will make a 4D engine and/or read my artcle, I don`t know.
But I expect that as chaos theory and other strange things with "unreal" dimensions become more in peoples knowledge, more people will make hyper-d engines.
So I figure demand will slowly grow.

Johnathan- what game company ?

~V'lion

Bugle4d

Edited by - Vlion on February 21, 2002 10:44:06 PM

~V'lionBugle4d

Anonymous

February 22, 2002 04:03 AM

Oh, I am an independent contractor right now. I work with various different companies on whatever projects they''re working on.

The most recent game I worked on to be published was Munch''s Oddysee for the Xbox. Right now I am working on a Gamecube game that''s not scheduled to be out for a long time, and also a PC game that''s more of an independent effort.

- Jonathan.

Inspect a paper on 4D rotation, anyone ?

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Inspect a paper on 4D rotation, anyone ?

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