I'd like to prove or disprove the following:
given a compact arc of a conic section $\mathcal{T} \subset \mathbb{R}^2$ such that for any $P \neq Q \in \mathcal{T}$ the tangent vectors $\tau_P, \tau_Q$ aren't parallel, then i can parametrize it using a quadratic Rational Bézier curve.
First of all I can find the three points: the endpoints of the arc and the intersection of the tangent lines at the endpoints (here i use the hypotesis making this intersection nonempty, as a counterexample we can pick a semicircumference). Then I can, without loss of generality, normalize the weights $w_0=w_2=1$. Now, how can I find $w_1$?
I searched on NURBS,Farin, online and even on Stack Exchange. I didn't find a general case (only the one with circular arcs) and moreover it seems that you can only approximate an arc of conic section (why?).
Edit1: I found these slides https://public.vrac.iastate.edu/~oliver/courses/me625/week11.pdf
but I don't understand nor the direct approach (because who grants me that of the three equations at least two are linearly independent) nor the geometric one (that one I just don't get it at all).
Best Answer
Choose a point $P$ on the conic and a line parameterized by $L(t)= A + tB$. For any given value of $t$, construct a line through $P$ and the line point $L(t)$. Since the conic is a curve of degree 2, this line must intersect the conic in two points. By construction, one of these points is $P$; call the other one $C(t)$. If you work through the algebra, you’ll find that $C(t)$ is a rational quadratic function of $t$.
Alternatively: use the geometry of the given conic to figure out its “shape factor”, $s$. You’ll have to find the intersection of the conic with the line joining its tangent intersection point $P_1$ and the mid-point $M$ of the chord $P_0P_2$. In other words, you have to find $s$ such that this intersection occurs at $(1-s)M + sP_1$. Then compute the weight of the middle control point from this shape factor: $w_1 = s/(1-s)$, as explained in the slides you found.