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does anyone know an explicit imbedding $h\colon T^2 \to \mathbb{R}^3$ of the torus $T^2=\mathbb{S}^1 \times \mathbb{S}^1$ into $\mathbb{R}^3$ ?

Thanks in advance !!!


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2 Answers

The standard parameterization is the one given on the top of the Wikipedia webpage for "torus".


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I see, so that using this parameterization and the fact that in this case $T^2=\mathbb{S}^1 \times \mathbb{S}^1$, then $R=r=1$, and if $x_u= \sin u, y_u=\cos u, x_v=\sin v, y_v=\cos v$ for $(x_u,y_u,x_v,y_v)\in T^2$, then the imbedding $h$ is $h(x_u,y_u,x_v,y_v) = (y_u + y_v * y_u, x_u + y_v * x_u, x_v)$. Thanks a lot for the tip !! –  onebengaltiger Mar 8 '12 at 4:58
Be careful with this. If you choose the parameters $R=r=1$, then your torus will not have a real hole in the middle (see the "horn torus" picture). Look on the wikipedia page, it says that $R$ is the distance between the center of the "hole" and the center of the "tube" around it, and $r$ is the radius of the tube. If $r \geq R$ then you won't get a real hole, better to choose values like $R=2$ and $r=1$. These radii do not need to be both $1$ since stretching the torus does not change its topology. –  Vhailor Mar 8 '12 at 15:04
I see, you are right, i did not analyze the parameters $R,r$ to fully understand them and i thought that they should correspond to the radii of the two circles $\mathbb{S}^1×\mathbb{S}^1$ Thanks a lot again for your advice !!!! –  onebengaltiger Mar 8 '12 at 19:50
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$(x,y)$ maps to $((3+\sin(y))\cos(x),(3+\sin(y))\sin(x),\cos(y))$

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Thanks for the tip !!!! –  onebengaltiger Mar 8 '12 at 4:59
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