I know that for a torus (with one hole) the parametric equations describing it are $x= (c + a\cos v)\cos u, y= (c + a\cos v)\sin u, z= a\sin v$, where $c$ is the radius from the center of the hole to the center of the torus tube and $a$ is the radius of the tube. My question is what would be the corresponding parametric equations describing a $2$-torus, i.e. a torus with $2$ holes. How about an $n$-torus?
[Math] Parametric Equations for a $2$-torus
differential-geometrygeneral-topologysurfaces
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My understanding:
For simplicity we are viewing finite 2-dimensional surfaces "floating" in 3-dimensional space.
Some, such as a closed disc or an multi-holed anulus, have boundaries and some, such as a sphere or multi-holed tori, do not. A torus has a "donut hole" which is intuitively obvious to see but very difficult to define. The thing is, a hole is can not be viewed locally "from" the surface itself, but must be viewed as how the surface itself is "shaped" in space.
A "hole" is a 3-dimensional hole in the space the surface occupies. If you think of the 2-dimensional surface as the "skin" of a 3-dimensional solid, the solid has a hole in it but the skin is intact.
A "cuff" is a 2-dimensional hole that is in the surface itself. If you cut a hole out of the skin of a sphere and the insides can now leak out.
The subtle part is that a cuff need not appear as a hole in the skin, it could be the border of a surface. Imagine a sphere. Cut a little patch of it and you have a giant bubble with a hole in it. But stretch the hole out and you have a bowl with a rim. Flatten it out and you simply have a closed disc. The cuff has become a border.
Consider a sphere. Cut two cuffs in it. Stretch those cuffs apart so that that remaining surface becomes cylinderical. Then you have a straw. Shrink one cuff and widen the other and flatten and you have an anulus.
Interestingly if you glue the two cuffs of a straw together you get a torus.
Alright, I'm kind of embarrassed at how difficult I made this problem, but I'm glad that the solution is very elegant.
A "toroidal knot patch" can be made simply by using one of the following equations:
$$ \vec r_1 (t, u) = (R+r\cos (p(t+u)))\cos qt \mathbf i + (R+r\cos(p(t+u)))\sin qt \mathbf j + r\sin (p(t+u)) \mathbf k $$
or
$$ \vec r_2 (t, u) = (R+r\cos (pt))\cos(t+u) \mathbf i + (R+r\cos(pt))\sin(t+u)\mathbf j + r\sin(pt)\mathbf k $$
As $u$ increases in the first equation, the patch is "increased" in the poloidal direction. As $u$ increases in the second equation, the patch is "increased" in the toroidal direction.
To find the extents $u$ should range over to correspond to the orthogonal projection from the tube diameter to the torus surface, we project the curve $u$ (at constant $t$) to the tangent plane to the torus at $t$.
This just turns into a problem of projecting a vector to a plane. Namely, for tube diameter $d$, solve
$$ ((\vec r(t,u)-\vec r(t,0))-((\vec r(t,u)-\vec r(t,0))\cdot \hat N(t))\hat N(t)) \cdot \hat B(t) = d/2 $$
for $u$. In the above, $\vec r$ can be either $\vec r_1$ or $\vec r_2$, and $\vec r(t,u)-\vec r(t,0)$ is the vector pointing from the toroidal knot centered at $t$ to $u$ in the direction of $u$ and $\hat N(t)$ and $\hat B(t)$ are the Darboux frame normal and binormal to the torus knot at $t$.
Below are graphs plotting using WinPlot
First Equation (Poloidal Direction):
$$ p = 10, R = 5, r = 2, q = 1 $$ such that $$ \vec r(t) = (5+2\cos (10(t+u)))\cos t \mathbf i + (5+2\cos (10(t+u)))\sin t \mathbf j + 2\sin (10(t+u)) \mathbf k $$
$ 0 \le u \le 0.1 $
$ 0 \le u \le 0.3 $
$ 0 \le u \le 0.5 $
Second Equation (Toroidal Direction):
$$ p = 10, R = 5, r = 2, q = 1 $$ such that $$ \vec r(t) = (5+2\cos (10t))\cos (t+u) \mathbf i + (5+2\cos (10t))\sin (t+u) \mathbf j + 2\sin (10t) \mathbf k $$
$ 0 \le u \le 0.1 $
$ 0 \le u \le 0.3 $
$ 0 \le u \le 0.5 $
Best Answer
A few words concerning nomenclature: The surface of a donut is a $2$-torus, or a closed surface of genus $1$. A donut with $2$ holes is not a $2$-torus but a closed surface of genus $2$. An $n$-torus for arbitrary $n\geq1$ is the manifold obtained from ${\mathbb R}^n$ by identifying points whose coordinates differ by integers.
A $1$-torus is nothing else but the circle $S^1$. The map $$u:\quad\phi\mapsto(\cos\phi,\sin\phi)\qquad (-\infty<\phi<\infty)$$ used to ”parametrize" $S^1$ is actually a covering map: It is locally (i.e., for short $\phi$-intervals) a diffeomorphism, but to each point $z\in S^1$ belong infinitely many $\phi$'s with $u(\phi)=z$.
In the same way the parametrization you gave for the $2$-torus is a covering map: Each point $(x,y,z)$ on the torus is produced infinitely many times. When you want to compute the area of the torus you have to restrict this map to $[0,2\pi]\times[0,2\pi]$, even though it is defined on all of ${\mathbb R}^2$.
Now the closed surfaces of genus $\geq2$: Here no elementary parametrization is possible. In the theory of Riemann surfaces it is shown that such a surface $S$ possesses a "universal" covering map $\pi: \ D\to S$ $\ (D$ is the unit disk in the $z$-plane), where again each point $p\in S$ is produced infinitely many times. Here the different $z$'s that produce the same point $p\in S$ are not related by translations $z\mapsto z+2\pi j +2i\pi k$, as in the case of the $2$-torus, but by a group of Moebius transformations.