# 3-dimensional parametric equations

1. Apr 22, 2007

### JolleJ

3-dimensional parametric equations [Updated]

Look lower for update....
1. The problem statement, all variables and given/known data
Well, my problem is that I need to give some examples on 3-dimensional parametric equations. So far I've found out what parametric equations are, and more specifically what 3-dimensional parametric equations are. But now I am being asked to give some real-world examples of these.

2. Relevant equations
A 3-dimensional parametric equations is an equation of something in a 3d-coordinate system, where each coordinate x,y,z are expressed by the same parameter t: x(t) = f1(t) ^ y(t) = f2(t) ^ z(t) = f3(t)

3. The attempt at a solution
Well, so far I've found out that Solar Winds, Aurorae and the movement of the plasma inside a Tokamak are all 3-dimensional parametric equations. My problem is that while I know that the movements can be expressed by 3-dimensional parametric equations, I have absolutely no idea how these equations look like. I've searched all around the Internet, but I can't find any equations for this - or anything at all that looks like it.

I hope you can help.

Update:
I have now advanced a bit, and acutally found a simulation of the particles moving inside a tokamak, which shows that the particles drift up or down depending on their charge q. So now I have a new problem:

1. The problem statement, all variables and given/known data
My problem is now that I do understand the mathematics / physic equations used in the simulations.
The simulations starts with introducing all the varibles and functions:
Code (Text):
B0:=1
v,m:=1,.01
x,y,z:=3,0,0
vx,vy,vz:=v,v*q,0
t,dt:=0,.01
Integratemethod:=RK4
func det(a,b,c,d)
return a*d - b*c
endfunc
func R(x,y)
return (x^2+y^2)
endfunc
func acc(va,vb,ba,bb)
return (va*bb-vb*ba)/m
endfunc
func Bx(x,y,z)
return y*B0/R(x,y)
endfunc
func By(x,y,z)
return -x*B0/R(x,y)
endfunc
func Bz(x,y,z)
return 0
endfunc
Model tokamak
x':=vx
y':=vy
z':=vz
vx':=q*det(vy,vz,By(x,y,z),Bz(x,y,z))/m
vy':=q*det(vz,vx,Bz(x,y,z),Bx(x,y,z))/m
vz':=q*det(vx,vy,Bx(x,y,z),By(x,y,z))/m
endmodel
After this, it makes a loop which constantly calculates the integrated function of "tokamak" (why this?). And after this adding the timedifference dt to the time variable t:
Loop:
Code (Text):
integrate tokamak(t,dt)
t:=t+dt
2. Relevant equations
I can see the that function det, is finding the determinant, though I do not know why this is relevant.
All of it is something with vectors, but I am not sure how.

3. The attempt at a solution
Tried looking at it so long, but I am not good enough at vectors and integration yet, so I simply cannot see excacly what is going on.

I really hope that one of you can open my eyes.

Last edited: Apr 23, 2007
2. Apr 22, 2007

### Mindscrape

Movement inside a Tokamak will be a complicated example of a 3-dimensional parametric equation. Basically any 3-D motion can be parametrized. A particle moving in a straight line, for example, would follow a motion of $$f(t) = at\mathbf{i} + bt\mathbf{j} + ct\mathbf{k}$$ where a, b, and c are constants and i, j, and k are x, y, and z coordinate directions, respectively. The familiar example of projectile motion could be described as $$f(t) = at\mathbf{i} + bt\mathbf{j} - gt^2 \mathbf{k}$$.

Other, more complicated examples, could be a helix $$s(t) = Rcos(t)\mathbf{i} + Rsin(t)\mathbf{j} + ct\mathbf{k}$$.

If you know Calc 1, you could find a Calc 3 book that will have some good examples of 3-D parametric equations.

3. Apr 23, 2007

### JolleJ

Thank very, very much. In reality I would some advanced examples of 3d parametric equations, like the Tokamak. Any chance that I can find some equations for it somewhere?

When you say Calc 3, do you mean Calculus 3?

4. Apr 23, 2007

### JolleJ

Updated my question...

5. Apr 23, 2007

:rofl:
What do you think, of course he means calc 3. :rofl: I mean, calculus 3. What country are you in? here in the states schools break down Calculus into 3 parts, 1, 2 , 3. 3 is vector Calculus.

6. Apr 23, 2007

### JolleJ

Well I'm from from Denmark:tongue2: And here, we certainly do not split the subjects up like that...:uhh: But now I know. Thanks

7. Apr 23, 2007

### Mindscrape

You don't know what a 3-D parametric equation is but you understand advanced examples? In the United States, we split Calculus into derivatives, integrals and series, and multivariate calculus.

If you want some more "advanced" examples, you might try looking in a book on Electricity and Magnetism.

8. Apr 24, 2007

### JolleJ

I know what 3-d parametric equations are, but having a hard time finding some good examples.

Anyway, I'm still trying to crack trhough the code...

9. Apr 24, 2007

### Mindscrape

Is that a pseudocode or a specific program code? As far as I can tell, it is a code applied to a specific example, with conditions that are predetermined (uniform magnetic field perpindicular to a plane, a given B-field magnitude, and other such things).

The loop at the end probably starts at t=0 (right?) and numerically integrates the functions up top (declared under tokamak) for each small incremental t (known as dt, and probably also defined elsewhere), then loops through all the way until a certain time t=t_final. This will give an approximate function of position, since it will give points of position along each incremental dt.