Question on Chapter12

[JF-Hu]

I want to ask a question. In Chapter12 ("Lowering and elevating curve order"), the picture shows:

.

It is identical to:

.

And we have:

.

Because B(n+1,t) and B(n,t) represent the same curve, so:

which is just the inverse form to your conclusion. So I want to know where is wrong before. Wishing for your reply, thank you very much!

[Pomax]

you don't have enough weights (they go up to w_k rather than w_n) but the crux is this: "Because B(n+1,t) and B(n,t) represent the same curve". Because this is not true: they look like the same curve (and that's good, because that's what we're trying to achieve) with each point on the lower order curve having a corresponding point on the other, but those points will have different t values, and their first, second, etc. derivatives are completely different. They are not the same curves except in the very narrow sense that they cover each other.

[JF-Hu]

At the beginning of Chapter 12, you add ((1-t)+t) to the low order Bézier function, then after transformation, you get the higher order Bézier function. But when elevating curve order, we will get higher Bézier function and the exact same curve at the same time. So i don't understand the meaning of "they look like the same curve". In generally, when we want to lower curve order, the new lower Bézier function just looks like the original one.