Running time of computing upper and lower envelope

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What is the running time of computing the upper and lower envelope of a set of piecewise functions (curves)? I am looking for an O(n log n) algorithm for this problem, but couldn't find the paper.

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Chapter 6 of this book describes a simple divide-and-conquer algorithm:

Micha Sharir, and Pankaj K. Agarwal. Davenport-Schinzel sequences and their geometric applications. Cambridge university press, 1995.

The algorithm's exact complexity depends on the curves. For example (p.136): "The lower envelope of a collection of $m$ piecewise-linear functions, whose graphs are composed of a total of $n$ segments, can be computed in $O(n \alpha(m) \log m)$ time."

$\alpha$ is the inverse-Ackermann function.