I'm looking for a nice geometric interpretation for flat morphisms between schemes. In wiki's article
https://en.wikipedia.org/wiki/Morphism_of_schemes#Intuition
is stated that flat families correspond to families of varieties which vary "continuously".
Futhermore as an example is given by
$${\displaystyle \operatorname {Spec} \left({\frac {\mathbb {C} [x,y,t]}{(xy-t))}}\right)\to \operatorname {Spec} (\mathbb {C} [t])}$$
What I don't understand here what is concretely meant by "continuously"?
The continuity of parametrisation $V_t = xy -t$?
Does anybody know another nice geometric interpretation for flatness? Is every morphism flat which provides such "varying family" of subvarieties by some parameters?