Torus
Definition
A torus is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit.
Mathematical Description
In terms of Cartesian coordinate system, a torus can be defined as the set of points (x, y, z) in three-dimensional space that satisfy the equation:
(x² + y² + z² + R² - r²)² = 4R²(x² + y²)
where R is the distance from the center of the tube to the center of the torus, r is the radius of the tube, and R > r. This equation corresponds to a torus centered at the origin and oriented in the xy plane.
Topology
In topological terms, a torus is a closed surface defined as the product of two circles: S¹ × S¹. This can be visualized as a rectangle in the plane with opposite edges identified: if one moves off the right-hand edge, one reappears at the corresponding point on the left-hand edge (and vice versa), and similarly for the top and bottom edges.
Geometry
In geometric terms, a torus can be described as a quadric surface in three dimensions, or more generally, a hypersurface in higher dimensions. A torus can also be defined parametrically by the following equations:
x = (R + r cos v) cos u y = (R + r cos v) sin u z = r sin v
where u and v are parameters which take on real values. In this equation, R and r represent the radii of the torus and tube, respectively.
Applications
Tori are frequently used in a variety of fields including computer graphics, algebraic topology, and physics. In computer graphics, toroidal surfaces are commonly used in 3D modeling and animation due to their simple mathematical properties and ease of rendering. In algebraic topology, the torus is a fundamental example of a compact manifold without boundary, and plays a key role in the classification of such manifolds. In physics, toroidal shapes are used in the design of certain types of nuclear fusion reactors, such as the tokamak.
See Also
