TIGHTENING AND BLENDING SUBJECT TO SET-THEORETIC CONSTRAINTS

Our work applies techniques for blending and tightening solid shapes represented by sets. We require that the output contain one set and exclude a second set, and then we optimize the boundary separating the two sets. Working within that framework, we present mason, tightening, tight hulls, tight blends, and the medial cover, with details for implementation. Mason uses opening and closing techniques from mathematical morphology to smooth small features. By contrast, tightening uses mean curvature flow to minimize the measure of the boundary separating the opening of the interior of the closed input set from the opening of its complement, guaranteeing a mean curvature bound. The tight hull offers a significant generalization of the convex hull subject to volumetric constraints, introducing developable boundary patches connecting the constraints. Tight blends then use opening to replicate some of the behaviors from tightenings by applying tight hulls. The medial cover provides a means for adjusting the topology of a tight hull or tight blend, and it provides an implementation technique for two-dimensional polygonal inputs. Collectively, we offer applications for boundary estimation, three-dimensional solid design, blending, normal field simplification, and polygonal repair. We consequently establish the value of blending and tightening as tools for solid modeling.

DISSERTATION

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ERRATA
The sentence
“By contrast, tightening uses mean curvature flow to minimize the measure of the boundary separating the opening of the interior of the closed input set from the opening of its complement, guaranteeing a mean curvature bound.”
In the Summary of my dissertation on page xvii should instead be
“By contrast, tightening uses mean curvature flow to minimize the measure of the boundary separating the opening of the input set from the opening of its complement, guaranteeing a mean curvature bound.”
To verify the correction, see Subsection 4.2.1 Definition of r-tightening on p. 26. For background on operators involving topology and closure, see Subsection 8.3.1 Topological operators p. 87-88 and Section 8.6 Closure structures p. 96-99 in my earlier draft. For context concerning morphological opening, see Subsection 8.3.2 Morphological operators p. 88-89 in the same draft.

ADDENDA