A directed edge has a direction associated with it, so it is thought of as coming from one of the vertices and going to the other one. An undirected edge treats both vertices interchangeably. Often, a real number is associated with each edge. These numbers are called weights.
Curves have what is known as an "arc-length". This is the length the curve would have if it were straightened, such that, it became a line. The arc-length, of some curved function, f(x), between points a and b, is equal to the integral of the square root of the quantity, one plus the square derivatived (or squared slope) of f(x) multiplied by the derivative of x -- s = ∫ √ (1 + [df/dx(x)]2dx. The arc-length formula is derived from the distance formula.