In computer graphics, simplifying a polygonal mesh surface \(\mathcal {M}\) into a geometric proxy that maintains close conformity to \(\mathcal {M}\) is crucial, as it can significantly reduce computational demands in various applications. In this paper, we introduce the implicit shell (ImS), a concept designed to implicitly represent the sandwich-walled space surrounding \(\mathcal {M}\) , defined as \(\{{\textbf {x}}\in \mathbb {R}^3|\epsilon _1\le f({\textbf {x}}) \le \epsilon _2, \epsilon _1< 0, \epsilon _2>0\}\) . Here, f is an approximation of the signed distance function (SDF) of \(\mathcal {M}\) , and we aim to minimize the thickness \(\epsilon _2-\epsilon _1\) . To achieve a balance between mathematical simplicity and expressive capability in f, we employ a first-degree tri-variate tensor-product B-spline to represent f. This representation is coupled with adaptive knot grids that adapt to the inherent shape variations of \(\mathcal {M}\) . In this manner, the analytical form of f can be rapidly determined by solving a sparse linear system. Moreover, the process of identifying the extreme values of f among the infinitely many points on \(\mathcal {M}\) can be simplified to seeking extremes among a finite set of candidate points. By exhausting the candidate points, we find the extreme values \(\epsilon _1<0\) and \(\epsilon _2>0\) that define the thickness. The constructed ImS is guaranteed to wrap \(\mathcal {M}\) strictly, without any intersections between the bounding surfaces and \(\mathcal {M}\) . ImS offers numerous potential applications thanks to its rigorousness, tightness, expressiveness, and computational efficiency. We demonstrate the efficacy of ImS in mesh simplification through the control of global error.
