publications
2023
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Differential Operators on Sketches via Alpha ContoursMariia Myronova, William Neveu, and Mikhail BessmeltsevACM Trans. Graph., Jul 2023A vector sketch is a popular and natural geometry representation depicting a 2D shape. When viewed from afar, the disconnected vector strokes of a sketch and the empty space around them visually merge into positive space and negative space, respectively. Positive and negative spaces are the key elements in the composition of a sketch and define what we perceive as the shape. Nevertheless, the notion of positive or negative space is mathematically ambiguous: While the strokes unambiguously indicate the interior or boundary of a 2D shape, the empty space may or may not belong to the shape’s exterior.
For standard discrete geometry representations, such as meshes or point clouds, some of the most robust pipelines rely on discretizations of differential operators, such as Laplace-Beltrami. Such discretizations are not available for vector sketches; defining them may enable numerous applications of classical methods on vector sketches. However, to do so, one needs to define the positive space of a vector sketch, or the sketch shape.
Even though extracting this 2D sketch shape is mathematically ambiguous, we propose a robust algorithm, Alpha Contours, constructing its conservative estimate: a 2D shape containing all the input strokes, which lie in its interior or on its boundary, and aligning tightly to a sketch. This allows us to define popular differential operators on vector sketches, such as Laplacian and Steklov operators.
We demonstrate that our construction enables robust tools for vector sketches, such as As-Rigid-As-Possible sketch deformation and functional maps between sketches, as well as solving partial differential equations on a vector sketch.@article{10.1145/3592420, author = {Myronova, Mariia and Neveu, William and Bessmeltsev, Mikhail}, title = {Differential Operators on Sketches via Alpha Contours}, year = {2023}, issue_date = {August 2023}, publisher = {Association for Computing Machinery}, address = {New York, NY, USA}, volume = {42}, number = {4}, issn = {0730-0301}, url = {https://doi.org/10.1145/3592420}, doi = {10.1145/3592420}, journal = {ACM Trans. Graph.}, month = jul, articleno = {69}, numpages = {15}, keywords = {vector graphics, differential operators, sketch processing}, }
2021
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On symmetry breaking of dual polyhedra of non-crystallographic group H₃Mariia MyronovaActa Crystallographica Section A, Jul 2021The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes \cal V_H_3(λ) constructed for each type of dominant point λ. Here a polytope \cal V_H_3(λ) is considered as a dual of a \cal D_H_3(λ) polytope obtained from the action of the Coxeter group H_3 on a single point λ∈\bb R^3. The H_3 symmetry is reduced to the symmetry of its two-dimensional subgroups H_2, A_1 \times A_1 and A_2 that are used to examine the geometric structure of \cal V_H_3(λ) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the ‘pancake’ structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the \cal V_H_3(λ) structure into various nanotubes. Moreover, since a \cal V_H_3(λ) polytope may contain the orbits obtained by the action of H_3 on the seed points (a,0,0), (0,b,0) and (0,0,c) within its structure, the stellations of flat-faced \cal V_H_3(λ)-polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C_20 has the dodecahedral structure of \cal V_H_3(a,0,0), the construction of the smallest fullerenes C_24, C_26, C_28, C_30 together with the nanotubes C_20+6N, C_20+10N is presented.
@article{Myronova:ug5018, author = {Myronova, Mariia}, title = {{On symmetry breaking of dual polyhedra of non-crystallographic group~H₃}}, journal = {Acta Crystallographica Section A}, year = {2021}, volume = {77}, number = {4}, pages = {296--316}, month = jul, doi = {10.1107/S2053273321002254}, url = {https://doi.org/10.1107/S2053273321002254}, keywords = {Coxeter groups, dual polytopes, orbit decompositions, fullerenes, nanotubes, stellated polyhedra}, }
2020
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Central Splitting of A₂ Discrete Fourier–Weyl TransformsJiří Hrivnák, Mariia Myronova, and Jiří PateraSymmetry, Jul 2020Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A₂ constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A₂ is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.
@article{sym12111828, author = {Hrivnák, Jiří and Myronova, Mariia and Patera, Jiří}, title = {Central Splitting of A₂ Discrete Fourier–Weyl Transforms}, journal = {Symmetry}, volume = {12}, year = {2020}, number = {11}, article-number = {1828}, url = {https://www.mdpi.com/2073-8994/12/11/1828}, issn = {2073-8994}, doi = {10.3390/sym12111828}, } -
Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic TypeMariia Myronova, Jiří Patera, and Marzena SzajewskaSymmetry, Jul 2020The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H₂, H₃ and H₄. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H₂ and H₃. The geometrical structures of nested polytopes are exemplified.
@article{sym12101737, author = {Myronova, Mariia and Patera, Jiří and Szajewska, Marzena}, title = {Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type}, journal = {Symmetry}, volume = {12}, year = {2020}, number = {10}, article-number = {1737}, url = {https://www.mdpi.com/2073-8994/12/10/1737}, issn = {2073-8994}, doi = {10.3390/sym12101737}, }
2019
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Dynamical Generation of GrapheneMariia Myronova, and Emmanuel BourretIn Geometric Methods in Physics XXXVI, Jul 2019In recent years, the astonishing physical properties of carbon nanostructures have been discovered and are nowadays being intensively studied. We introduce how to obtain a graphene sheet using group theoretical methods and how to construct a graphene layer using the method of dynamical generation of quasicrystals. Both approaches can be formulated in such a way that the points of an infinite graphene sheet are generated. The main objective is to describe how to generate graphene step by step from a single point of ℝ².
@inproceedings{10.1007/978-3-030-01156-7_35, author = {Myronova, Mariia and Bourret, Emmanuel}, editor = {Kielanowski, Piotr and Odzijewicz, Anatol and Previato, Emma}, title = {Dynamical Generation of Graphene}, booktitle = {Geometric Methods in Physics XXXVI}, year = {2019}, publisher = {Springer International Publishing}, address = {Cham}, pages = {341--346}, isbn = {978-3-030-01156-7}, } -
The orthogonal systems of functions on lattices of SU(n + 1), n < ∞Mariia Myronova, and Marzena SzajewskaIn Geometric Methods in Physics XXXVII, Jul 2019The definitions of orbit functions, their orthogonality relations, congruence classes and decomposition matrices are recalled. The orthogonality of the symmetric C- and antisymmetric S-orbit functions, which are given on the fundamental region FM of the weight lattice, for simple Lie group SU(n + 1) of any rank n is defined. The splitting of the weight lattice of An into congruence classes is shown.
@inproceedings{10.1007/978-3-030-34072-8_21, author = {Myronova, Mariia and Szajewska, Marzena}, editor = {Kielanowski, Piotr and Odzijewicz, Anatol and Previato, Emma}, title = {The orthogonal systems of functions on lattices of SU(n + 1), n < ∞}, booktitle = {Geometric Methods in Physics XXXVII}, year = {2019}, publisher = {Springer International Publishing}, address = {Cham}, pages = {195--203}, isbn = {978-3-030-34072-8}, }