preprocessing:
input: pts, an array of N points indicating a piecewise
            linear path.

distances[0] = 0.0;
for i = 1 to N-1
   distances[i] = distances[i-1] + dist(pts[i], pts[i-1]);

L = lengths[N-1]; // total length of the path.

main program:
input:
input: pts, an array of N points indicating a piecewise
            linear path.
       f, a fraction between 0 and 1.
       L, the total length of the path
       distances, an array of distances along the path.

output: q, a point along the path that's f of the way from
       pts[0] to pts[N-1].

target = f * L; // the distance we want to go.
i = 0;
while (lengths[i] < target) i = i + 1;

// Now lengths[i-1] < target <= lengths[i]
u = pts[i-1];
v = pts[i];
h = lengths[i] - lengths[i-1]; // distance between the pts
r = target - lengths[i-1]; // distance still to be covered.
t = r/h; // fraction of the distance from u to v we have to go.
q = (1-t) * u + t * v;