Example 3. Find the translation trajectory of $(1,2)$ with respect to $\left(t, t^{2}\right)$.
Let the translation point be $(x, y)$, then
$$
\left\{\begin{array}{l}
x=1+t \\
y=2+t^{2} .
\end{array}\right.
$$

Eliminating $t$, we get $y=(x-1)^{2}+2$.
When a figure undergoes a translation, the translation increment for all points is the same. Therefore, once the translation increment is found, the translation trajectory of certain points can be easily determined.