Here I respond concerning envelopes of surfaces in $\mathbb{R}^3$ (not 
necessarily higher dimensions).

One source (with references) is the CGAL manual, [Envelopes of Surfaces in 3D](http://doc.cgal.org/Manual/3.7/doc_html/cgal_manual/Envelope_3/Chapter_main.html).
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[![TwoTriangles][1]][1]
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<sup>CGAL figure.</sup>

Their software doesn't explicitly handle quadratic surfaces, however (aside from spheres).

These twos papers, and their citations, may help:

> Boissonnat, Jean-Daniel, and Katrin TG Dobrindt. "On-line construction of the upper envelope of triangles and surface patches in three dimensions." *Computational Geometry* 5.6 (1996): 303-320. [Journal link](http://www.sciencedirect.com/science/article/pii/0925772195000070).

> Meyerovitch, Michal. "Robust, generic and efficient construction of envelopes of surfaces in three-dimensional spaces." *ESA: European Symposium on Algorithms*. 2006. [Springer link](https://link.springer.com/book/10.1007/11841036#page=808).
[PDF download](https://www.cs.tau.ac.il/CGAL/Papers/envelope.pdf).

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[![Meyerovitch][2]][2]
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<sup>Meyerovitch figure.</sup>
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  [1]: https://i.sstatic.net/KuHG5.gif
  [2]: https://i.sstatic.net/u6OaB.png