Geometric Properties of Planar and Spherical Interception Curves
In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemn...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2023-07-01
|
| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/12/7/704 |
| _version_ | 1797590342563790848 |
|---|---|
| author | Yagub N. Aliyev |
| author_facet | Yagub N. Aliyev |
| author_sort | Yagub N. Aliyev |
| collection | DOAJ |
| description | In this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></semantics></math></inline-formula> and Gauss’s constant <i>G</i>, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results. |
| first_indexed | 2024-03-11T01:19:05Z |
| format | Article |
| id | doaj.art-f1833e98a25240c9b2bf8ca8e1e4212a |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T01:19:05Z |
| publishDate | 2023-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-f1833e98a25240c9b2bf8ca8e1e4212a2023-11-18T18:18:18ZengMDPI AGAxioms2075-16802023-07-0112770410.3390/axioms12070704Geometric Properties of Planar and Spherical Interception CurvesYagub N. Aliyev0School of IT and Engineering, ADA University, Ahmadbey Aghaoglu Str. 61, Baku AZ1008, AzerbaijanIn this paper, some geometric properties of the plane interception curve defined by a nonlinear ordinary differential equation are discussed. Its parametric representation is used to find the limits of some triangle elements associated with the curve. These limits have some connections with the lemniscate constants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></semantics></math></inline-formula> and Gauss’s constant <i>G</i>, which are used to compare with the classical pursuit curve. The analogous spherical geometry problem is solved using a spherical curve defined by the Gudermannian function. It is shown that the results agree with the angle-preserving property of Mercator and Stereographic projections. The Mercator and Stereographic projections also reveal the symmetry of this curve with respect to Spherical and Logarithmic Spirals. The geometric properties of the spherical curve are proved in two ways, analytically and using a lemma about spherical angles. A similar lemma for the planar case is also mentioned. The paper shows symmetry/asymmetry between the spherical and planar cases and the derivation of properties of these curves as limiting cases of some plane and spherical geometry results.https://www.mdpi.com/2075-1680/12/7/704plane curvespherical curvenon-linear ODEpursuit curveMercator projectionGudermannian function |
| spellingShingle | Yagub N. Aliyev Geometric Properties of Planar and Spherical Interception Curves Axioms plane curve spherical curve non-linear ODE pursuit curve Mercator projection Gudermannian function |
| title | Geometric Properties of Planar and Spherical Interception Curves |
| title_full | Geometric Properties of Planar and Spherical Interception Curves |
| title_fullStr | Geometric Properties of Planar and Spherical Interception Curves |
| title_full_unstemmed | Geometric Properties of Planar and Spherical Interception Curves |
| title_short | Geometric Properties of Planar and Spherical Interception Curves |
| title_sort | geometric properties of planar and spherical interception curves |
| topic | plane curve spherical curve non-linear ODE pursuit curve Mercator projection Gudermannian function |
| url | https://www.mdpi.com/2075-1680/12/7/704 |
| work_keys_str_mv | AT yagubnaliyev geometricpropertiesofplanarandsphericalinterceptioncurves |