Visualizing $\mathbb{S}^{3}$ as consisting of two "polar" great circles together with a family of nested Clifford tori interpolating between them. The following text describes a construction of the unit sphere $\mathbb{S}^{3}$.

To consider the next example, it is helpful to practice visualizing $\mathbb{S}^{3}$ as consisting of two "polar" great circles $\{(z,0): z \in \mathbb{S}^{1} \subset \mathbb{C}\}$, $\{(0,w): w \in \mathbb{S}^{1} \subset \mathbb{C}\}$, together with a family of nested Clifford tori interpolating between them. Thinking of $\mathbb{S}^{3}$ as the unit sphere in $\mathbb{C}^{2}$, these Clifford tori are
$$
T_{\alpha}:=\left\{(z \cos \alpha, w \sin \alpha): z, w \in \mathbb{S}^{1} \subset \mathbb{C}\right\}
$$
for $\alpha \in(0, \pi / 2)$. (Note that $T_{\alpha}$ is the set of points at spherical distance $\alpha$ from the circle $\{(z, 0)\} .$ ) If we stereographically project to $\mathbb{R}^{3}$, then the two circles become the $z$ -axis and the unit circle in the $x y$ -plane, while the nested tori $T_{\alpha}$ grow around the circle and shrink around the line.

I’ve been trying but could not visualize how the two circles and the Clifford tori describe $\mathbb{S}^3$. Is there an equivalent construction for $\mathbb{S}^{2}$?
The picture provided with the text looks similar to the last one in this answer. However the process described there is also not clear to me. What is the relationship between $\mathbb{S}^{3}$ and two solid tori?
 A: Here's a capsule summary. Via stereographic projection, the three-sphere may be identified with $\mathbf{R}^{3}$ with one point at infinity. The left-hand figure shows a closed half-plane. The blue line at left is a circle (with one point at infinity). When this half-plane is revolved using the blue line as axis, each circle sweeps out a Clifford torus, and the blue spot where the circles acculumate sweeps out a circle that links the axis.
(Tangential fun facts: If $p = (1, 0)$ and $q = (-1, 0)$, then each circle is the solution set of $|x - q|/|x - p| = k$ for some real $k > 1$. The blue axis is the solution set for $k = 1$, and the point $p = (1, 0)$ is in a sense the limit as $k \to \infty$. These circles are orthogonal to the family of circles through $p$ and $q$.)
The tori partition the complement of the two circles in $S^{3}$ because the circles partition the open punctured right half-plane, and this partition is a product foliation (essentially by the fun facts above).
Now let's focus on one of the circles sweeping out a Clifford torus. The animation on the right shows the fibation of a typical Clifford torus into Hopf circles, which are images of great circles on $S^{3}$ under stereographic projection.
One remaining detail you may want to ponder is that as $k \to 1^{+}$, the Hopf circles' radii grow without bound and the planes they lie in approach longitudinal planes, while as $k \to \infty$ the Clifford torus collapses to the unit circle swept out by $p$.


