Crofton formulas and convexity condition for secantoptics

Mozgawa, Witold; Skrzypiec, Magdalena

doi:10.36045/bbms/1251832370

Cited by 10 publications

(6 citation statements)

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“…There are some generalizations of the isoptics as well e.g. equioptic curves in [29] by Odehnal or secantopics in [28,34] by Skrzypiec. We can extend the very first question to the space: "What is the locus of points where a given segment subtends a given angle?" Or a question equivalent to the former: "For the given spatial points A and B, what is the locus of the points P for which the internal angle at P of the triangle ABP is a given angle?"…”

Section: Isoptic Surfacesmentioning

confidence: 99%

Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

Szirmai

2021

The Quarterly Journal of Mathematics

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In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ tetrahedron. Moreover, we generalize the famous Menelaus’s and Ceva’s theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries described by E. Molnár in [6].

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Section: Isoptic Surfacesmentioning

confidence: 99%

Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

Szirmai

2021

The Quarterly Journal of Mathematics

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“…There are some generalizations of the isoptics as well e.g. 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%

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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“…Thaloids are also analyzed as a special case. Similar investigations in this topic have only been carried out in spaces with constant curvature (see [8,9,27,28,5]).…”

Section: Introductionmentioning

confidence: 99%

“…There are some generalizations of the isoptics as well e.g. equioptic curves in [28] by Odehnal or secantopics in [27,31] by Skrzypiec. We can extend the very first question to the space: "What is the locus of points where a given segment subtends a given angle?" Or a question equivalent to the former: "For the given spatial points A and B, what is the locus of the points P for which the internal angle at P of the triangle ABP △ is a given angle?"…”

Section: Introduction To Isoptic Curvesmentioning

confidence: 99%