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.

 

[tacman]

The equations for hypertori can be quite complex and vary depending on the specific type of hypertorus being described. However, here are a few examples of equations for different types of hypertori:

1. n-dimensional torus: This can be represented by the following parametric equations:

x = (a + b cos(t_1)) cos(t_2) ... cos(t_n)
y = (a + b cos(t_1)) cos(t_2) ... sin(t_n)
z = (a + b cos(t_1)) sin(t_2)

where a and b are constants and t_1, t_2, ..., t_n are the parameters.

2. Hyperellipsoid: This is represented by the equation given in the content, where a_k represents the semi-axis lengths of the hyperellipsoid.

3. Hypersphere: This can be represented by the following implicit equation:

x_1^2 + x_2^2 + ... + x_n^2 = r^2

where r is the radius of the hypersphere.

4. Hypercube: This can be represented by the following parametric equations:

x = a cos(t_1) cos(t_2) ... cos(t_n)
y = a cos(t_1) cos(t_2) ... sin(t_n)
z = a cos(t_1) sin(t_2) ... cos(t_n)
...
w = a sin(t_1)

where a is the length of each side of the hypercube and t_1, t_2, ..., t_n are the parameters.

These are just a few examples of equations for different types of hypertori. There are many more variations and types of hypertori that can be represented by different equations. It is important to note that the equations for hypertori can become quite complex and may require advanced mathematical knowledge to understand and work with.