A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula>
The current work aims to develop an approximation of the slice of a Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semant...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/1/137 |
| _version_ | 1797406799429632000 |
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| author | Amirreza Fahim Golestaneh |
| author_facet | Amirreza Fahim Golestaneh |
| author_sort | Amirreza Fahim Golestaneh |
| collection | DOAJ |
| description | The current work aims to develop an approximation of the slice of a Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> that, to the best of the author’s knowledge, has not been addressed yet. This approximation of the actual slice is in a closed form of an explicit parametric equation in the case that the slice is not passing through the zones of the Minkowski surface with high curvatures, namely the “corners”. An alternative computational algorithm is introduced for the cases that the plane slices the corners, in which a family of ellipsoidal inner and outer bounds of the Minkowski sum is used to construct a “narrow strip” for the actual slice of Minkowski sum. This strip can narrow persistently for a few more number of constructing bounds to precisely coincide on the actual slice of Minkowski sum. This algorithm is also applicable to the cases with high aspect ratio of ellipsoids. In line with the main goal, some ellipsoidal inner and outer bounds and approximations are discussed, including the so-called “Kurzhanski’s” bounds, which can be used to formulate the approximation of the slice of Minkowski sum. |
| first_indexed | 2024-03-09T03:31:50Z |
| format | Article |
| id | doaj.art-c93c4996eb39404c96e2959bb2951a72 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T03:31:50Z |
| publishDate | 2022-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c93c4996eb39404c96e2959bb2951a722023-12-03T14:54:50ZengMDPI AGMathematics2227-73902022-12-0111113710.3390/math11010137A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula>Amirreza Fahim Golestaneh0Department of Mechanical Engineering, National University of Singapore, Singapore 117575, SingaporeThe current work aims to develop an approximation of the slice of a Minkowski sum of finite number of ellipsoids, sliced up by an arbitrarily oriented plane in Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> that, to the best of the author’s knowledge, has not been addressed yet. This approximation of the actual slice is in a closed form of an explicit parametric equation in the case that the slice is not passing through the zones of the Minkowski surface with high curvatures, namely the “corners”. An alternative computational algorithm is introduced for the cases that the plane slices the corners, in which a family of ellipsoidal inner and outer bounds of the Minkowski sum is used to construct a “narrow strip” for the actual slice of Minkowski sum. This strip can narrow persistently for a few more number of constructing bounds to precisely coincide on the actual slice of Minkowski sum. This algorithm is also applicable to the cases with high aspect ratio of ellipsoids. In line with the main goal, some ellipsoidal inner and outer bounds and approximations are discussed, including the so-called “Kurzhanski’s” bounds, which can be used to formulate the approximation of the slice of Minkowski sum.https://www.mdpi.com/2227-7390/11/1/137slice of Minkowski sumellipsoidsclosed-form parametrizationapproximationcomputational algorithmKurzhanski’s bounds |
| spellingShingle | Amirreza Fahim Golestaneh A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> Mathematics slice of Minkowski sum ellipsoids closed-form parametrization approximation computational algorithm Kurzhanski’s bounds |
| title | A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> |
| title_full | A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> |
| title_fullStr | A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> |
| title_full_unstemmed | A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> |
| title_short | A Closed-Form Parametrization and an Alternative Computational Algorithm for Approximating Slices of Minkowski Sums of Ellipsoids in <inline-formula><math display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup></semantics></math></inline-formula> |
| title_sort | closed form parametrization and an alternative computational algorithm for approximating slices of minkowski sums of ellipsoids in inline formula math display inline semantics msup mi mathvariant double struck r mi mn 3 mn msup semantics math inline formula |
| topic | slice of Minkowski sum ellipsoids closed-form parametrization approximation computational algorithm Kurzhanski’s bounds |
| url | https://www.mdpi.com/2227-7390/11/1/137 |
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