Subdivision Directional Fields

Abstract

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the mixed finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure preserving: it reproduces curl-free fields precisely and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover’s distance computation, for efficient and robust computation.

Metric Optimization in Penner Coordinates

“…If 𝜋 ′ • 𝑃 𝜏 𝑐 (𝑥) = 𝜋 ′ • 𝑃 𝜏 𝑐 (𝑦) for any two points 𝑥, 𝑦 ∈ 𝑀 𝑐 such that 𝜋 (𝑥) = 𝜋 (𝑦), then 𝜋 ′ • 𝑃 𝜏 𝑐 extends to a unique continuous map 𝑃 𝜏 between the two hyperbolic surfaces (𝑀, 𝜎) and (𝑀, 𝜎 ′ ). This continuity constraint for a given edge 𝑒 𝑖 𝑗 can be expressed succinctly and concretely by 𝜏 𝑘 𝑖 𝑗 + 𝜏 𝑙 𝑗𝑖 = 𝜎 ′ 𝑖 𝑗 − 𝜎 𝑖 𝑗 We also note that this construction closely resembles the halfedge forms defined in [Custers and Vaxman 2020]. We hope to explore this connection more deeply in future work.…”

Constrained willmore surfaces

“…Therefore it is convenient to consider all objects as living on a modied cell complex, whose discrete dierential forms naturally contain the discrete forms of M as well as the piecewise constant 1-forms ⌦ 1 F (M; R 3 ). The idea is closely related to the halfedge forms introduced in [Custers and Vaxman 2020].…”

“…Aside from making our approach compatible with the many existing geometry processing algorithms which use dierential coordinates, it also enables us to nd conformal immersions for abstract metric surfaces, valuable in mathematical visualization. For triangle meshes such dierential coordinates are piecewise maps, constant per triangle, and we allow them to vary independently [Custers and Vaxman 2020]. Surprisingly, optimization over such triangle elds is well dened even if triangles do not "glue" together and provides a richer search space for the optimization, often allowing more rapid progress towards convergence.…”

A Simple Discretization of the Vector Dirichlet Energy

“…Knöppel et al [KCPS15] and Sharp et al [SSC19] use a finite‐difference‐like method that evolves the approach of Knöppel et al [KCPS13]: the same vertex flattening preprocessing is performed, but then a finite‐differencelike approach is used to construct discrete operators. Custers and Vaxman [CV18] present a subdivision scheme for per‐face tangent vectors using a data structure of scalar quantities on halfedges, and apply it to vector design and optimal transport. Their basis functions are constructed using mesh subdivision.…”