Looking for the equations of hypertori

[benorin]

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 topological glue-ing 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 $$\mathbb{R}^{n}$$ . Example:

Hyperellipsoid: set of all points $$\left( x_{1},x_{2},...,x_{n} \right) \in\mathbb{R}^{n}$$ such that $$\sum_{k=1}^{n} \left( \frac{x_{k}}{a_{k}}\right)^{2}=1, a_{K}\in\mathbb{R}$$.

 

[hypermorphism]

The standard n-torus is the set of points of the form (x₁, ..., x_n,y₁,...,y_n) that satisfy the equation (x₁² + y₁² - 1, ..., x_n² + y_n² - 1) = 0.
Note that the 2-torus is properly a submanifold of R⁴, but is commonly parameterized as a submanifold of R³.