Two 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.
The 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.
The 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}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H 3 on a single point \lambda\in{\bb R}^{3}. The H 3 symmetry is reduced to the symmetry of its two-dimensional subgroups H 2, A 1 × A 1 and A 2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) 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}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) 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}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N , C20+10N is presented.
A 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.
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