Ok how about this.
A cubic bezier curve is determined by four control points, P0(x0,y0), P1(x1,y1), P2(x2,y2), and P3(x3,y3), and is defined by these parametric equations:
x = x0(1 - t)^3 + 3x1*t(1 - t)^2 + 3x2*t^2(1 - t) + x3*t^3
y = y0(1 - t)^3 + 3y1*t(1 - t)^2 + 3y2*t^2(1 - t) + y3*t^3
Where 0 <= t <= 1
Now notice that when t = 0 we have (x,y) = (x0,y0) and when t = 1 we have (x,y) = (x3,y3) so the curve starts at P0 and ends at P3. Go through the algebra and you will see that this is true.
Now here is an example.
Graph the bezier curve with the control points P0(4,1), P1(28,48), P2(50,42), and P3(40,5). Now graph the line segments P0P1, P1P2, and P2P3. (to do that use the parametric equations x = x1 + (x2 - x1)t and y = y1 + (y2 - y1)t. These equations will describe a line segment for some points P1(x1,y1) and P2(x2,y2). Ok now notice that the middle control points P1 and P2 dont lie on the curve, the curve starts at P0, heads for P1 and P2 without reaching them and ends at P3. See that the tangent at P0 passes through P1 and the tangent at P3 passes through P2.
Ok thats a lot of stuff there, just try out the example and do any algebra you need to understand it. Im sure you will need to work these things out on paper first to see it visually to get a good understanding of it. I didnt show how to derive the bezier parametric equations above but with a bit of thinking you should figure out how they were derived. If you dont want to figure that out yourself then im sure somewhere on the net shows how they were derived. If you want to know that is.
Also, if you understand all of that (or when you do) you should easily see how to extend this into 3d. Remember my examples and stuff are for 2d to keep things as simple as can be.
Hope this helps. And i hope i didnt make any stupid mistakes that will confuse you more than need be.
-SirKnight
