From what I understand, an elliptic contains a set points satisfying the equation
$y^2=x^3 + ax + b$
together with the point at infity.
It seems clear how multiplication with a scalar and a point works, and how point addition works. Discussions about cryptographic applications usually jump right to curves defined over finite fields at this point. I think I understand how that works as well.
What doesn't make sense is whether a curve defined over the field of real numbers could also work as a trapdoor.
If I know that point Q results from multiplication of some integer with the generator point on a curve over real numbers
$Q = dG$
what feasible methods exist to compute d given only a, b, and Q, and G?
In other words, aside from convenience in implementation, what exactly does the finite field add to elliptic curves as trapdoors?
I've seen this discussion, but it doesn't seem to directly address the question here.