Reconstruction of tensor product spline surfaces to integrate surface-surface intersection geometry and topology while maintaining inter-surface continuity
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In the field of Computer-Aided Geometric Modeling (CAGD) stands one critical problem that plagues not only users of Computer-Aided Design (CAD) applications, but all downstream uses of a geometric model, namely, computational analysis, animation, graphics, rendering, metrology, manufacturing, and 3D printing. When complicated real world models are created using the core Boolean operations of a CAD program, massive numbers of approximations set off a cascade of consequences that requires extensive additional CAD infrastructure, creates models unfit for direct downstream use, and necessitates entire industries to mitigate the crippling side effects. Termed the surface-surface intersection (SSI) issue, or trim problem, intersecting spline surfaces are known to create complex nonlinear exact solutions of a very high degree. CAD applications have addressed this by approximating these intersections and introducing gaps between the discrete surfaces, using an independent topological structure to then manage model integrity through solid modeling processes. This document accepts the intersection approximation as an unavoidable computational circumstance and uses this to update the otherwise independent surfaces to create a single conformal, watertight model. This is accomplished using a novel approach that first analyzes the trim geometry and topology, and then uses this information to form an admissible global parameter space. The attention to parameter space management is often an underappreciated and neglected process that is shown to be critical in any sound solution to the trim problem. Finally, the model space geometry is updated to embed the trim information while maintaining the existing trajectories and continuities of the original objects. This is accomplished through solving a system of isoparametric curve equations, under the right constraints, in reverse, a surface creation strategy without precedent in CAGD literature. The result is a watertight model representing the users' deign intent and ready for downstream processing, e.g. computational analysis, without the need of the current immense framework of amelioration.
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