Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in $$\textbf{Nil}$$ Geometry

Csima, Géza; Szirmai, Jenő

doi:10.1007/s00025-023-01961-z

Cited by 3 publications

(2 citation statements)

References 23 publications

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“…equioptic curves in [22] by Odehnal or secantopics in [21,25] by Skrzypiec. An algorithm for convex polyhedrons has been given by the authors in [8] in order to generalize the notion of isoptic curve into the space and it has been developed by Kunkli et al for non convex cases in [13]. The spatial case encompasses many applications in both physical and architectural aspects, see [9].…”

Section: Introductionmentioning

confidence: 99%

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Isoptic curves of cycloids

Csima

2024

AMI

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The history of the isoptic curves goes back to the 19th century, but nowadays the topic is experiencing a renaissance, providing numerous new results and new applications. First, we define the notion of isoptic curve and outline some of the well-known results for strictly convex, closed curves. Overviewing the types of centered trochoids, we will be able to give the parametric equation of the isoptic curves of hypocycloids and epicycloids. Furthermore, we will determine the corresponding class of curves. Simultaneously, we show that a generalized support function can be given to these types of curves in order to apply and extend the results for strictly convex, closed curves. The calculation methods used during the procedure provide an excellent example of the application of univariate calculus, parametric curves, and vector calculus in geometry and can therefore be processed by either advanced high school students or university students.

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Section: Introductionmentioning

confidence: 99%

“…There are some results in three dimensional Thurson geometries as well. The isoptic surface of segments has been determined in [10] in Nil geometry and in [3] for S 2 ×R and H 2 ×R geometries.…”

Section: Introductionmentioning

confidence: 99%

Isoptic curves of cycloids

Csima

2024

AMI

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Isoptic surfaces of segments in $$\textbf{S}^2\!\times \!\textbf{R}$$ and $$\textbf{H}^2\!\times \!\textbf{R}$$ geometries

Csima

2023

J. Geom.

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In this work, we examine the isoptic surfaces of line segments in the $$\textbf{S}^2\!\times \!\textbf{R}$$ S 2 × R and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R geometries, which are from the 8 Thurston geometries. Based on the procedure first described in Csima and Szirmai (Results Math 78:194-19, 2023), we are able to give the isoptic surface of any segment implicitly. We rely heavily on the calculations published in Szirmai (Bul Acad Ştiinţe Repub Mold Mat 2:44–61, 2020; Q J Math 73:477–494, 2022). As a special case, we examine the Thales sphere in both geometries, which are called Thaloid. In our work we will use the projective model of $$\textbf{S}^2\!\times \!\textbf{R}$$ S 2 × R and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R described by Molnár (Beitr Algebra Geom 38:261–288, 1997).

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Fibre-Like Cylinders, Their Packings and Coverings in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ Space

Szirmai

2024

Results Math

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In this paper we define the notion of infinite or bounded fibre-like geodesic cylinder in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ space, develop a method to determine its volume and total surface area. We prove that the common part of the above congruent fibre-like cylinders with the base plane are Euclidean circles and determine their radii. Using the former classified infinite or bounded congruent regular prism tilings with generating groups $$\mathbf {pq2_1}$$ pq 2 1 we introduce the notions of cylinder packings, coverings and their densities. Moreover, we determine the densest packing, the thinnest covering cylinder arrangements in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ space, their densities, their connections with the extremal hyperbolic circle arrangements and with the extremal fibre-like cylinder arrangements in $$\textbf{H}^3$$ H 3 and $$\textbf{H}^2\!\times \!\textbf{R}$$ H 2 × R spaces. We prove that in these three previous Thurston geometries, the densities of the optimal fiber-like cylinder packings are equal and the same is true for optimal coverings. In our work we use the projective model of $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ S L 2 R ~ introduced by Molnár (Beitr Algebra Geom 38(2):261–288, 1997).

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Translation-Like Isoptic Surfaces and Angle Sums of Translation Triangles in $$\textbf{Nil}$$ Geometry

Abstract: After having investigated the geodesic and translation triangles and their angle sums in $$\textbf{Sol}$$ Sol and $$\widetilde{{\textbf{S}}{\textbf{L}}_2{\textbf{R}}}$$ S L 2 R ~ geometries we consider the analogous problem i… Show more

Cited by 3 publications

References 23 publications

Isoptic curves of cycloids

Isoptic curves of cycloids

Isoptic surfaces of segments in $$\textbf{S}^2\!\times \!\textbf{R}$$ and $$\textbf{H}^2\!\times \!\textbf{R}$$ geometries

Fibre-Like Cylinders, Their Packings and Coverings in $$\widetilde{\textbf{S}\textbf{L}_2\textbf{R}}$$ Space

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