TITLE: Finding upper segments of intersecting parabolas
QUESTION [9 upvotes]: I have multiple parabolas ($y = ax^2 + bx + c$) which may intersect with each other (or some of them may not intersect). I am trying to find upper segments of these parabolas, e.g. bold part in the picture:

I need to find it in $O(n\log n)$. There's a solution for line version of this problem: https://stackoverflow.com/questions/7420193/how-to-find-upper-envelopes-of-intersected-lines-in-onlogn 
But I cannot find a way to apply this solution to quadratic version since there are $a$, $b$ and $c$ variables to consider. Any help will be appreciated.

REPLY [7 votes]: You are seeking what is called in the literature the upper envelope of a collection of $x$-monotone curves.  Actually, searching for "lower envelope" will give you more hits; obviously they are equivalent.  You might look at the CGAL implementation, e.g., via this link:

Update. To address joriki's comment, let me quote Theorem 6.5 (p. 137) from the Sharir-Agarwal book, Davenport-Schinzel Sequences and Their Geometric Applications:

Given a collection $\Gamma$ of $n$ $x$-monotone Jordan arcs, each pair of which intersect in at most $s$ points, the lower envelope of $\Gamma$ can be computed in $O(\lambda_{s+1}(n) \log n)$ time.

Because $s=2$ in the case of parabolas, the relevant term is $\lambda_3(n)$.
This $\lambda$-function is a combinatorial quantity that has been proved to
have certain bounds.  A tight bound is known for $\lambda_3(n)$: it is $\Theta( n \alpha(n) )$, where $\alpha(n)$ is the inverse of Ackermann's function, i.e. $\le 5$ in this universe. So the complexity in the theorem is just a hair above $O( n \log n)$.