toruksen
The torus is a doughnut-shaped surface that appears frequently in geometry and topology. In its standard geometric construction, it is formed by revolving a circle of radius r about an axis in the same plane at a distance R from the circle’s center, with R > r. The result is a compact, orientable surface of genus 1, which is topologically equivalent to the cartesian product S^1 × S^1.
A common coordinate description uses two angles u and v:
with 0 ≤ u < 2π and 0 ≤ v < 2π. Equivalently, it satisfies the implicit equation (√(x^2 + y^2)
Topologically, the torus is the product S^1 × S^1, has Euler characteristic 0, and has a fundamental
Variants and notable examples include the flat torus, obtained as the quotient of the Euclidean plane by
Applications span mathematics, physics, computer graphics, and architecture, where toroidal shapes model periodic structures, magnetic confinement