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.
[![TwoTriangles][1]][1]
CGAL figure.
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.
Meyerovitch, Michal. "Robust, generic and efficient construction of envelopes of surfaces in three-dimensional spaces." ESA: European Symposium on Algorithms. 2006. Springer link. PDF download.
[![Meyerovitch][2]][2]
Meyerovitch figure.