Convert a quadratic curve points to polynomial representation?

Question

I have the X,Y of 2 end points and 1 bezier point, of a Quadratic Bezier curve.

Using this data, how can I derive the polynomial representation of the curve?

_{(source: euclidraw.com)}

Answer 1

B(t) = (1-t) * (1-t) * B0 + 2 * (1-t) * t * B1 + t * t * B2

Answer 2

Oog. That would be tricky. Beziers are parametrized curves, namely:

x = f(t)
y = g(t)

where t=0 yields one endpoint and t=1 yields the other.

You could technically figure out how to eliminate "t" and get an equation in x and y, but it would not be a polynomial like y = a + bx + cx² ...; it would be an equation h(x,y) = 0 where h is probably somewhat ugly.

Answer 3

Wikipedia has a section about this. Perhaps this helps.