Question

How to visualize a torus as a cartesian product of two circles?

Answer 1

torus resembles a big circle, but it has a circular cross section for every point on that original circle. (Just as the Euclidean plane R2 has a separate copy of R for every point along the x-axis.)

Let's say you take a circle in the xy plane and create a torus around it. Then suppose you want to make a coordinate system to describe it.

To find a point on the torus, first you move a certain distance along the outer edge of the torus, staying in the xy-plane. Describe that distance with an angle theta1. This narrows it down to a circular cross-section. Then, move around that cross-section a certain distance until you find the point you're looking for; call that distance theta2. It should be clear that holding either theta1 or theta2 constant and letting the other one vary, you get a circle.