Those variables on Bezier curve equations are not fixed

I wrote something on reverse engineering Bezier curves about… *goes to check* woah, 2 years ago! I don’t remember it being that long… (You might want to read that article before proceeding…)

Anyway, I’ve received a few comments and emails about its usefulness. Basically, what I did was to find the 4 control points of a cubic Bezier curve from 4 known points which lie on that Bezier curve. 2 of the known points are to be the end points of the Bezier curve (which automatically makes them control points too). The other 2 known points lie somewhere on the Bezier curve.

Here’s where the confusion sets in. Commenter Yonatan pointed out that there is an infinite number of Bezier curves based on how those 2 known points are defined. And he’s right.

Now, I formed that solution based on the “natural” implicit decision that the 4 points are evenly spread out on the Bezier curve. There is no reason for them to be, and the math never assumed they are. The solution arose from the assumption that the control points were evenly spread out, but in the end, it worked for the general case as well. So long as 0 < u,v < 1 and u not equal to v (and logically speaking, u < v), everything worked fine.

So the whole point of this article is this: u doesn’t have to be 1/3, and v doesn’t have to be 2/3. You are supposed to know or decide what value they take. Once you’ve decided, the other control points will be uniquely determined. Let me illustrate:

Same Bezier curve with different control points

Now the 2 Bezier curves are exactly the same (I would know, I copied and pasted them…). Suppose I define u and v such that f and g lie on certain points on one Bezier curve, and they lie on different other points on the other Bezier curve. What happens is that the control points p1 and p2 are different for the 2 Bezier curves, even though the curves are exactly the same!

Disclaimer: I haven’t worked out an example such that it is true (other than the trivial case of a straight line), that a Bezier curve can be drawn with 2 different sets of control points. As in exactly the same. However, based on the math, I can say that 4 points lying on a Bezier curve can be drawn with 2 different sets of control points. The resulting curves might (actually they should) differ slightly, a twitch of a pixel here, a slight upward gradient there. But the 4 points would be exactly positioned as calculated. It’s meant to be a, what’s the word, sensational example. So there.

I can’t tell you what u and v are, although 1/3 and 2/3 should work fine. I gave you the theory and the solution. It’s up to you to decide how to use it. Depending on your context, you might decide on different values for u and v, which will influence how your control points are calculated.

My original intent was to produce a camera path flying in 3D. I didn’t care about the “correctness” of its path, only that there is one. As such, u=1/3 and v=2/3 worked excellently for me.

You might find that 1/3 and 2/3 don’t work for you. That’s fine. u and v are variables. By definition, they’re not fixed. Choose whatever value works for you. Depending on context, you might even want to come up with a simple formula (based on your situation) to calculate u and v dynamically.