Looking for the equations of hypertori

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benorin
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I'm looking for the equations of hypertori (e.g. n-dimensional tori). By equations I'm mean explicit, implicit, or parametric equations that represent hypertori (please, no :devil: topological glue-ing :devil: in the construction!), and by hypertori I mean the family of surfaces obtained by generalizing the usual, doughnut-looking torus (e.g., a 3-d torus, a 2-torus embedded in 3-space) to [tex]\mathbb{R}^{n}[/tex] . Example:

Hyperellipsoid: set of all points [tex]\left( x_{1},x_{2},...,x_{n} \right) \in\mathbb{R}^{n}[/tex] such that [tex]\sum_{k=1}^{n} \left( \frac{x_{k}}{a_{k}}\right)^{2}=1, a_{K}\in\mathbb{R}[/tex].
 
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The standard n-torus is the set of points of the form (x1, ..., xn,y1,...,yn) that satisfy the equation (x12 + y12 - 1, ..., xn2 + yn2 - 1) = 0.
Note that the 2-torus is properly a submanifold of R4, but is commonly parameterized as a submanifold of R3.
 

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