A point has a position
A point in three-dimensional space has three coordinates: x, y, and z. Instead of describing a surface with a single equation, a parametric model gives each coordinate as a function of two variables, u and v.
Walk around two circles
For a torus, u moves around the large ring and v moves around the small circular tube. Both range from 0 to 2π. Every pair of values identifies a point on the surface.
Change the proportions
R is the distance from the center of the torus to the center of its tube. r is the tube radius. Increase the tube radius to make a fuller shape. The sliders keep r below R, so the surface stays a ring torus.
A model becomes a picture
The visualization samples these functions, rotates the coordinates, and projects them onto a flat canvas. Its linework is a view of the surface, rather than a physical simulation. Move your cursor over the shape to change its orientation.