Singular Foliations for Knit Graph Design

摘要

We build upon the stripes-based knit planning framework of [Mitra et al. 2023], and view the resultant stripe pattern through the lens of singular foliations. This perspective views the stripes, and thus the candidate course rows or wale columns, as integral curves of a vector field specified by the spinning form of [Knöppel et al. 2015]. We show how to tightly control the topological structure of this vector field with linear level set constraints, preventing helicing of any integral curve. Practically speaking, this obviates the stripe placement constraints of [Mitra et al. 2023] and allows for shifting and variation of the stripe frequency without introducing additional helices. En route, we make the first explicit algebraic characterization of spinning form level set structure within singular triangles, and replace the standard interpolant with an “effective” one that improves the robustness of knit graph generation. We also extend the model of [Mitra et al. 2023] to surfaces with genus, via a Morse-based cylindrical decomposition, and implement automatic singularity pairing on the resulting components.