Re-mesh of any given tri-mesh into a quad-only mesh. This is achieved by integrating two conjugated vector fields.

The family of conjugated vector fields comes from a non-linear interpolation of two vector fields: one aligned to the principal curvatures and another with the principal stress direction.

This results in a tradeoff which brings edges aligned to the principal stress curves and faces being almost planar.

All the algorithms implemented for this research have been collected into Capybara, a grasshopper plugin, presented at IASS 2018 at MIT

###### WORKFLOW

###### INITIAL MESH

###### PRINCIPAL STRESS (BLACK) PRINCIPAL CURVATURE(RED)

###### SUBSET OF THE VECTOR FIELDS

###### INTERPOLATED FIELDS

###### CONJUGATED FIELDS

###### FRAME FIELD MESH

###### FRAME FIELD MESH WITH ISOCURVES

###### GRIDSHELL

Initial Vector fields and final quad-only mesh. The rectangles show the areas where vector fields were sampled.

## BIBLIOGRAPHY

###### OVERVIEW ON TOP-DOWN APPROACH

Structuring free-form building envelopes

Thesis (M.Phil.) - University of Bath, 2012. 2012

###### MESH CURVATURE

Szymon Rusinkiewicz. Estimating Curvatures and Their Derivatives on Triangle Meshes. Symposium on 3D Data Processing, Visualization, and Transmission, September 2004.

###### N-POLYVECTOR FIELD

Olga Diamanti, Amir Vaxman, Daniele Panozzo, Olga Sorkine-Hornung. Designing N-PolyVector Fields with Complex Polynomials, 2014

###### CONJUGATED FIELDS FOR QUAD MESH PLANARITY

Yang Liu, Weiwei Xu, Jun Wang, Lifeng Zhu, Baining Guo, Falai Chen, Guoping Wang. General Planar Quadrilateral Mesh Design Using Conjugate Direction Field, 2008.

###### QUAD REMESHING

Daniele Panozzo, Enrico Puppo, Marco Tarini, Olga Sorkine-Hornung. Frame Fields: Anisotropic and Non-Orthogonal Cross Fields, 2014.

###### QUAD PLANARIZATION

Sofien Bouaziz, Mario Deuss, Yuliy Schwartzburg, Thibaut Weise, Mark Pauly Shape-Up: Shaping Discrete Geometry with Projections, 2012