In one dimension, you could try to parametrize the curve $C(t)=\{x(t),y(t)\}$ in such a way that the "velocity" is constant, and then quantize the paramenter $t$ - that is feasible, but that would lead you to equidistant points along the curve, and that's not apparently what you are after.
Elsewhere, the problems seems awkard, only tractable by some iterative algorithm.
In more dimensions, it's worse, it even becomes bad defined: what would "n equidistant points" mean?
There are many iterative algorithms, some related to vector quantization or clustering, some inspired by physical systems. For example, you could throw N random initial points over your domain, and move them according to some "energy" function, that increases at short distances: in that way, the points would try to go as far as the others as they can, and that "low energy" configuraton would correspond -more or less, conceptually- to the "equidistant points" you are envisioning.
If you, additionally, want to impose some organization/ordering/topology to the points (for example, in 2D you could want them to conform some distorted mesh, with each points having four -north-east-south-west 'neighbours') then you should take a look at Self Organizing Maps (Kohonen).