One of the most essential ingredients in the theory of motivic integration are the space of arcs (in german: the "formalen Schleifen": literally that translates as "formal loops") of a given $k$-variety
$X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.

In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "space of formal loops".

My question is if there any geometrical reason & motivation to call these objects "arcs" or "formal loops"? Does there exist any analogy to the topological loops and loop spaces which motivates the choice of the name "arcs" or "formal loops" for these objects occuring in motivic integration?