How far can one see before the earth curves away?

A one-dimensional line or curve can be represented by the coordinates as a function of one parameter. Each value of the parameter corresponds to a point in spacetime and varying the parameter traces out a line. So in mathematical terms a curve is defined by four coordinate functions (where x0 usually denotes the time coordinate) depending on one parameter τ. A coordinate grid in spacetime is the set of curves one obtains if three out of four coordinate functions are set to a constant.