how to calculate control points on a bezier curve?

Question

I do have a bezier curve, and at a certain point, I want a second bezier curve "branching off" the first curve in a smooth manner. Together with calculating the intersection point (with a percentage following the Bezier curve), I need also the control point (the tangent and weight). The intersection point is calculated with the following piece of javascript:

getBezier = function getBez(percent,p1,cp1,cp2,p2) {
    function b1(t) { return t*t*t }
    function b2(t) { return 3*t*t*(1-t) }
    function b3(t) { return 3*t*(1-t)*(1-t) }
    function b4(t) { return (1-t)*(1-t)*(1-t) }
    var pos = {x:0,y:0};
    pos.x = p1.x*b1(percent) + cp1.x*b2(percent) + cp2.x*b3(percent) + p2.x*b4(percent);
    pos.y = p1.y*b1(percent) + cp1.y*b2(percent) + cp2.y*b3(percent) + p2.y*b4(percent);
    return pos;
}

(Non IE browsers can see it in action at http://www.iscriptdesign.com -> Tutorial -> Groups & Paths). All I need now is the controlpoint or (tangent and weight) for the branchpoint ( I don't have a clue where to start, and I hope somebody can point to some code, or mathematical equation, if possible as function from the same parameters as the getBezier function above).