Tracing Field‐Coherent Quad Layouts

Abstract

Given a cross field over a triangulated surface we present a practical and robust method to compute a field aligned coarse quad layout over the surface. The method works directly on a triangle mesh without requiring any parametrization and it is based on a new technique for tracing field-coherent geodesic paths directly on a triangle mesh, and on a new relaxed formulation of a binary LP problem, which allows us to extract both conforming quad layouts and coarser layouts containing t-junctions. Our method is easy to implement, very robust, and, being directly based on the input cross field, it is able to generate better aligned layouts, even with complicated fields containing many singularities. We show results on a number of datasets and comparisons with state-of-the-art methods

Partitioning Surfaces Into Quadrilateral Patches: A Survey

“…A potential difficulty with this approach is the hardness of constructing a global (quantized) seamless parametrization in the first place [MPZ14, CBK15]. One can consider using a cross field (which also provides two orthogonal directions everywhere on the surface) instead [PPM*16, KLF15], as it is significantly easier to construct for a given set of nodes (i.e. field singularities) [VCD*16].…”

“…This can, for instance, be due to the existence of attractors in the field or due to globally unfit field holonomy [MPZ14, KNP07]. With a suitable formulation one can, however, at least obtain partial or non‐conforming layouts in such problematic cases, that can be post‐processed into quad layouts by refinement operations [PPM*16].…”

“…This hard problem has recently been tackled by pre‐filtering the set of all possible arcs such that firstly it becomes relatively small and, in particular, finite, and second arcs intersect only in specific ways, such that intersections can be taken into account more easily. The simplified problem can then be formulated as a global binary optimization problem and approached using standard solvers [RRP15, ZZY16, PPM*16, RP17]. An objective based on the arcs' geometric quality (e.g.…”

“…Additional criteria are necessary to reduce it to a finite size. For instance, one can use an arc length cut‐off, a threshold on a spiral pitch measure [RRP15], a cardinality threshold [PPM*16] or regional restrictions [ZZY16, RP17]. Figure shows an example candidate set on a simple surface.…”

A New Class of Guided C2 Subdivision Surfaces Combining Good Shape with Nested Refinement

“…[Bar13, PXXZ16, RSAF16, JMPR16]. Quad meshing [BLP*12, VCD*16] has matured over the past decade starting with leveraging directional fields [MPZ14, PPM*16]. Such meshes provide natural input for guided subdivision surfaces.…”