Isoptic surfaces of polyhedra

Abstract

The theory of the isoptic curves is widely studied in the Euclidean plane E 2 (see [2] and [20] and the references given there). The analogous question was investigated by the authors in the hyperbolic H 2 and elliptic, but in the higher dimensional spaces there are only a few result in this topic.In [7] we gave a natural extension of the notion of the isoptic curves to the n-dimensional Euclidean space E n (n ≥ 3) which are called isoptic hypersurfaces. Now we develope an algorithm to determine the isoptic surface HP of a 3-dimensional polytop P.We will determine the isoptic surfaces for Platonic solids and for some semi-regular Archimedean polytopes and visualize them with Wolfram Mathematica.

Efficiently parallelised algorithm to find isoptic surface of polyhedral meshes

“…In this section, we recall the notion of the isoptic surface, defined in [1] and briefly describe the earlier sequential algorithm, presented in [9] that obtains the isoptic surface of a closed polyhedral mesh.…”

“…In [1], the isoptic in E 3 is defined as a surface by substituting the two-dimensional viewing angle for the appropriate three-dimensional measure of visibility (solid angle). The authors are also provided a formula and algorithm for convex shapes, but it is possible to solve and plot the isoptic surface only using computer algebra systems.…”

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The isoptic surface of a three-dimensional shape is defined in [1] as the generalization of isoptics of curves. The authors of the paper also presented an algorithm to determine isoptic surfaces of convex meshes.

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Embedding QR Code onto Triangulated Meshes using Horizon Based Ambient Occlusion

“…On the other hand, if the measure of the surface area from a given viewpoint is larger than the QR code, there is a good chance that we can place the code onto that part of the mesh. Recent studies provide a way of finding all viewpoints in the space around the given surface from where the actual visible area of the surface is of constant measure (here we suppose that the surface is given as a polyhedral mesh) [NKH18, CS16]. The set of these points are called an isoptic surface (of a given measure) of the mesh since this is somewhat similar to the well‐known elementary planar case of isoptic circular arcs from where a given line segment can be seen under a certain angle.…”

On curves with circles as their isoptics

“…We would like to emphasize that all the papers in the bibliography, that is [1][2][3][4][5][6][7][8][9][10]47,[49][50][51][52][53][54][55][56], with the exception of Santaló's and Su's books, [46,48], and the paper by Cyr, [11], present a wide spectrum of results in isoptics theory and are included here for the interested reader to have a complete overview of isoptics theory.…”