@article{9652,
  abstract     = {In 1998 Burago and Kleiner and (independently) McMullen gave examples of separated nets in Euclidean space which are non-bilipschitz equivalent to the integer lattice. We study weaker notions of equivalence of separated nets and demonstrate that such notions also give rise to distinct equivalence classes. Put differently, we find occurrences of particularly strong divergence of separated nets from the integer lattice. Our approach generalises that of Burago and Kleiner and McMullen which takes place largely in a continuous setting. Existence of irregular separated nets is verified via the existence of non-realisable density functions ρ:[0,1]d→(0,∞). In the present work we obtain stronger types of non-realisable densities.},
  author       = {Dymond, Michael and Kaluza, Vojtech},
  issn         = {1565-8511},
  journal      = {Israel Journal of Mathematics},
  keywords     = {Lipschitz, bilipschitz, bounded displacement, modulus of continuity, separated net, non-realisable density, Burago--Kleiner construction},
  pages        = {501--554},
  publisher    = {Springer Nature},
  title        = {{Highly irregular separated nets}},
  doi          = {10.1007/s11856-022-2448-6},
  volume       = {253},
  year         = {2023},
}

@article{12972,
  abstract     = {Embroidery is a long-standing and high-quality approach to making logos and images on textiles. Nowadays, it can also be performed via automated machines that weave threads with high spatial accuracy. A characteristic feature of the appearance of the threads is a high degree of anisotropy. The anisotropic behavior is caused by depositing thin but long strings of thread. As a result, the stitched patterns convey both color and direction. Artists leverage this anisotropic behavior to enhance pure color images with textures, illusions of motion, or depth cues. However, designing colorful embroidery patterns with prescribed directionality is a challenging task, one usually requiring an expert designer. In this work, we propose an interactive algorithm that generates machine-fabricable embroidery patterns from multi-chromatic images equipped with user-specified directionality fields.We cast the problem of finding a stitching pattern into vector theory. To find a suitable stitching pattern, we extract sources and sinks from the divergence field of the vector field extracted from the input and use them to trace streamlines. We further optimize the streamlines to guarantee a smooth and connected stitching pattern. The generated patterns approximate the color distribution constrained by the directionality field. To allow for further artistic control, the trade-off between color match and directionality match can be interactively explored via an intuitive slider. We showcase our approach by fabricating several embroidery paths.},
  author       = {Liu, Zhenyuan and Piovarci, Michael and Hafner, Christian and Charrondiere, Raphael and Bickel, Bernd},
  issn         = {1467-8659},
  journal      = {Computer Graphics Forum},
  keywords     = {embroidery, design, directionality, density, image},
  location     = {Saarbrucken, Germany},
  number       = {2},
  pages        = {397--409},
  publisher    = {Wiley},
  title        = {{Directionality-aware design of embroidery patterns}},
  doi          = {10.1111/cgf.14770 },
  volume       = {42},
  year         = {2023},
}