Function B(t) giving point at ratio 't' along curve.

We're going to subdivide ratios [0,1] into N uniform slices (larger N gives more accurate lookup) and evaluate the cumulative distance along the segments.

var distances = [];
var accumulated = 0;
var currentPoint = B(0);
for (i in 1...N+1) {
   var nextPoint = B(i/N);
   var distance = ... // some approximation to distance along curve between
                      // B((i-1)/N) and B(i/N) . If N is large, even straight-line distance
                      // would be fine.
   distances.push(accumulated += distance);
}

We now have a table, that gives us the distance along the curve at ratios 1/N, 2/N, 3/N ... 1

We want to generate point B(x) parameterised by distance x instead of ratio t.
We need to lookup the ratio 't' corresponding to an 'x' using our table.

// disatnces = [10, 20, 30];
// x = 5 -> i = 0
// x = 15 -> i = 1

function ratioAtDistance(x) {
   if (x < 0) throw "Cannot eval before curve";
   var i = 0;
   while (i < distances.length && distances[i] < x) i++;
   if (i >= distances.length) throw "Cannot eval after curve";

   var pDist = (i == 0) ? 0 : distances[i-1];
   return (i/N) + (1/N)*(x - distances[i])/(distances[i] - pDist);
}

We can then find point along curve at 'distance' x as B(ratioAtDistance(x))