I am having some difficulty lining up the definition and my intuition for rational equivalence of cycles. My intuition is based off of the idea that two cycles being rationally equivalent is analogous to the two cycles being homotopic.

If it is requested of me to state the definition of rational equivalence, I will; for the moment I will refrain and simply link to the wikipedia page for the chow ring and adequate equivalence relation.

Here are two examples that I have thought of that are giving me trouble.

In $\mathbb{k}[x,y]$, consider the line given by $x = 0$ and the unit circle. By taking the rational function $\frac{x^2+y^2 - 1}{x}$ on $\mathbb{k}[x,y]$, we get that these two are rationally equivalent. These two being homotopic is only semi-okay with me. I can imagine making the circle larger and larger, making it look more and more like a line. But it seems to me that the final step of taking it to the line is not allowed by homotopy.

Consider the 2-torus $\mathbb{T}$, thought of as sitting in $\mathbb{A}^3$ with center at the origin. Let $S_1$ be (one of the) circles obtained by intersecting $\mathbb{T}$ with the $xy$-plane, and let $a$ be the regular function defining it. Let $S_2$ be one of the circles obtained by intersecting with the $xz$-plane, and $b$ the regular function defining it. Then the rational function $\frac{a}{b}$ gives a rational equivalence of $S_1$ and $S_2$. This goes very much against my intuition.

Hopefully somebody will tell me where I am going wrong in my thinking.