Since points on a euclidean plane can be represented by one coordinate on a space-filling curve, is there any curve such that if two vectors $(x_0,y_0)$ and $(x_1,y_1)$ were represented by $a$ and $b$, the sum $a+b$ would represent the vector $(x_0+x_1,y_0+y_1)$? Could this curve be generalized to three dimensions?

EDIT: Even one with $ab$ representing $(x_0+x_1,y_0+y_1)$ would be a start, I can't find anything.

EDIT2: Never mind $a+b$, is there any way to do some sort of simple operation between $a$ and $b$ to represent vector addition?