cartesian equations for toratopes

Discussion of shapes with curves and holes in various dimensions.

cartesian equations for toratopes

Postby PWrong » Tue Jan 31, 2006 11:51 am

I've found a simple method for finding the cartesian equation for a toratope. I'm using Marek14's notation without the plusses.

First, note that (11) = (x,y) means that x^2 + y^2 = r^2.
That is, sqrt(x^2 + y^2) - r = 0
Now let A and B be toratopes with equations A=0 and B=0 respectively. Then the equation of (A, B) is sqrt(A^2 + B^2) - r = 0

Let's look at the 3D torus ((11)1) = ((x,y),z)
We have
(x,y): sqrt(x^2 + y^2) - r_1 = 0
so,
((x,y),z): sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_2 = 0

Similarly, (A,B,C) means sqrt(A^2 + B^2 + C^2) - r = 0
and AB means (A=0 and B=0)

Here's a more complicated example. (321)211 = ((xyz)(wv)u)(ts)pq

3: sqrt(x^2 + y^2 + z^2) - r_1 = 0
(321): sqrt( (sqrt(x^2 + y^2 + z^2) - r_1)^2 + (sqrt(w^2+v^2) - r_2)^2 + u^2) - r_2 = 0

(311)11: sqrt( (sqrt(x^2 + y^2 + z^2) - r_1)^2 + (sqrt(w^2+v^2) - r_2)^2 + u^2) - r_3 = 0 and sqrt(t^2 + s^2) - r_4 = 0 and |p| - r_5 = 0 and |q| - r_5 = 0

Does this all make sense to everyone? I'll try to implement this into my mathematica program somehow.
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Postby moonlord » Thu Feb 09, 2006 4:12 pm

Yes, it makes sense... except... your 'more complicated example'...
So, let's see if I got it...

(11) is a circle, because x**2+y**2==1
(111) is a sphere, because x**2+y**2+z**2==1
(1..1) is a n-sphere, because sum[i=1,i<n,i++](x_i**2)==1

How about the square, the cubinder and all other polytopes that also have |x_i|=1? I don't think I understood the notation for those...

EDIT: Is sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_2 = 0 the cartesian equation for the torus? Why so? Maybe you can explain a little...
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Postby PWrong » Fri Feb 10, 2006 12:27 pm

(11) is a circle, because x**2+y**2==1
(111) is a sphere, because x**2+y**2+z**2==1
(1..1) is a n-sphere, because sum[i=1,i<n,i++](x_i**2)==1


That's right.

The square has notation x,y = 11. It has the equation |x| =1 and |y| =1
the cubinder has a circle and two extra dimensions, so it's notation is (x,y),z,w = (11)11.

Some rotopes (exactly half in fact) have no brackets around them. You could call these "open rotopes". These include all the rotatopes except the spheres. They also include things like the torinder, ((x,y),z),w = ((11)1)1
If you like, you can put square brackets around them. Each open rotope corresponds to a closed rotope. In general, the closed rotope is much harder to visualise.

1111 = tesseract -> (1111) = glome
(11)11 = cubinder -> ((11)11) = circle with a sphere at every point
(111)1 = spherinder -> ((111)1) = sphere with a circle at every point
((11)1)1 = torinder -> (((11)1)1) = circle^3
(11)(11) = duocylinder -> ((11)(11)) = tiger

EDIT: Is sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_2 = 0 the cartesian equation for the torus? Why so? Maybe you can explain a little...

Ok. This is the parametric equation:

x = (R + r cos(u) ) cos(v)
y = (R + r cos(u) ) sin(v)
z = r sin (u)

Now we use the identity, cos(v)^2 + sin(v)^2 = 1
x^2 + y^2 = (R + r cos(u)) ^ 2
sqrt(x^2 + y^2) = R + r cos(u)
(sqrt(x^2 + y^2) - R) ^ 2 = r^2 cos(u)^2

add z^2 to both sides:
(sqrt(x^2 + y^2) - R) ^ 2 + z^2 = r^2 cos(u)^2 + r^2 sin(u)^2

and use the same identity again.
(sqrt(x^2 + y^2) - R) ^ 2 + z^2 = r^2
sqrt( (sqrt(x^2 + y^2) - R) ^ 2 + z^2) - r = 0
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Postby moonlord » Fri Feb 10, 2006 5:23 pm

Now I see... Thanks a lot!

So, the misterious tiger has the cartesians:

Code: Select all
x**2 + y**2 = 1
z**2 + w**2 = 1
(sqrt(x**2 + y**2) -1)**2 + (sqrt(z**2 + w**2) -1)**2 = 1


I think I've understood the torus now. Generalising:

Code: Select all
sqrt((sqrt(( ... sqrt((sqrt(x_1**2 + x_2**2) - r_1)**2 + x_3**2) - r_2)**2 +x_4**2) ... x_n)**2) - r_(n-1) = 0


I can't derive the parametric from this, however. Maybe you can help.

Are these correct, anyway?[/code]
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Postby PWrong » Thu Feb 16, 2006 3:50 pm

So, the misterious tiger has the cartesians:

Code:
x**2 + y**2 = 1
z**2 + w**2 = 1
(sqrt(x**2 + y**2) -1)**2 + (sqrt(z**2 + w**2) -1)**2 = 1

Yep, that's correct. Although more generally it would have three different radii instead of 1.

I think I've understood the torus now. Generalising:
Code:
sqrt((sqrt(( ... sqrt((sqrt(x_1**2 + x_2**2) - r_1)**2 + x_3**2) - r_2)**2 +x_4**2) ... x_n)**2) - r_(n-1) = 0

That would be the topological (n-1)-torus. I can't think of a simple way to derive the parametric equations directly from the cartesian equation. But I can tell you the equations for the 3-torus (((21)1)1).

x = r_1 cos(t_1) + r_2 cos(t_1) cos(t_2) + r_3 cos(t_1) cos(t_2) cos(t_3)
y = r_1 sin(t_1) + r_2 sin(t_1) cos(t_2) + r_3 sin(t_1) cos(t_2) cos(t_3)
z = r_2 sin(t_2) + r_3 sin(t_2) cos(t_3)
w = r_3 sin(t_3)
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Postby moonlord » Thu Feb 16, 2006 5:18 pm

I'll think about it...

Maybe you or someone will make a animation of a 3-torus passing a hyperplane... Maybe from different angles... :lol: Thanks in advance :wink:
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