I've never taken a projective geometry course, and I'm trying to understand the real projective plane $\mathbb{P}^2$ and its description using homogeneous coordinates, and how these relate to rational Bézier curves.
I understand that the elements of $\mathbb{P}^2$ can be identified with lines through the origin in $\mathbb{R}^3$. Also (equivalently) elements of $\mathbb{P}^2$ can be described by homogeneous coordinates of the form $[x:y:z]$ where two coordinate triples are regarded as equal if their elements are proportional (a nice simple equivalence relationship). All good so far.
Now let's look at how homogeneous coordinates are typically used with 2D rational Bézier curves. Given $n+1$ 2D points $P_0, \ldots, P_n$ and $n+1$ weights $w_0, \ldots, w_n$, we can define a rational Bezier curve $$ C(t) = \frac{\sum \phi_i(t)w_iP_i}{\sum \phi_i(t)w_i} $$ where the $\phi_i$ are Bernstein polynomials of degree $n$. Fine, no problem.
But then people suggest that we should take the point $P_i = (x_i, y_i)$ and the weight $w_i$ and combine them to get an element of $\mathbb{P}^2$, like this: $\overline{P}_i = [w_ix_i:w_iy_i: w_i]$. Having done this, we can now think about a somehow related Bézier curve: $$ \overline{C}(t) = \sum \phi_i(t)\overline{P}_i $$ This makes us happy because $\overline{C}(t)$ looks like a polynomial curve, rather than a rational one, so we're back to more familiar territory.
But this doesn't make sense to me. The right-hand side of the last equation does additions and scalar multiplications of elements of $\mathbb{P}^2$, but what do these mean?
So, in short, my questions are what do addition and scalar multiplication mean in $\mathbb{P}^2$ (if anything). Is it a vector space? If not, what kind of algebraic structure is it?