A generalization of the second Pappus–Guldin theorem
This paper deals with the question of how to calculate the volume of a body in R3 when it is cut into slices perpendicular to a given curve. The answer is provided by a formula that can be considered as a generalized version of the second Pappus–Guldin theorem. It turns out that the computation beco...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-02-01
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| Series: | Results in Applied Mathematics |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S2590037425000019 |
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| author | Harald Schmid |
| author_facet | Harald Schmid |
| author_sort | Harald Schmid |
| collection | DOAJ |
| description | This paper deals with the question of how to calculate the volume of a body in R3 when it is cut into slices perpendicular to a given curve. The answer is provided by a formula that can be considered as a generalized version of the second Pappus–Guldin theorem. It turns out that the computation becomes very simple if the curve passes directly through the centroids of the perpendicular cross-sections. In this context, the question arises whether a curve with this centroid property exists. We investigate this problem for a convex body K by using the volume distance and certain features of the so-called floating bodies of K. As an example, we further determine the non-trivial centroid curves of a triaxial ellipsoid, and finally we apply our results to derive a rather simple formula for determining the centroid of a bent rod. |
| format | Article |
| id | doaj-art-b6db7fad8f7b4ed78cca3d403008fb50 |
| institution | DOAJ |
| issn | 2590-0374 |
| language | English |
| publishDate | 2025-02-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Applied Mathematics |
| spelling | doaj-art-b6db7fad8f7b4ed78cca3d403008fb502025-08-20T02:47:33ZengElsevierResults in Applied Mathematics2590-03742025-02-012510053710.1016/j.rinam.2025.100537A generalization of the second Pappus–Guldin theoremHarald Schmid0University of Applied Sciences Amberg-Weiden, Amberg, GermanyThis paper deals with the question of how to calculate the volume of a body in R3 when it is cut into slices perpendicular to a given curve. The answer is provided by a formula that can be considered as a generalized version of the second Pappus–Guldin theorem. It turns out that the computation becomes very simple if the curve passes directly through the centroids of the perpendicular cross-sections. In this context, the question arises whether a curve with this centroid property exists. We investigate this problem for a convex body K by using the volume distance and certain features of the so-called floating bodies of K. As an example, we further determine the non-trivial centroid curves of a triaxial ellipsoid, and finally we apply our results to derive a rather simple formula for determining the centroid of a bent rod.http://www.sciencedirect.com/science/article/pii/S2590037425000019Pappus–Guldin theoremCentroid curveFloating bodyVolume distanceElastic rod |
| spellingShingle | Harald Schmid A generalization of the second Pappus–Guldin theorem Results in Applied Mathematics Pappus–Guldin theorem Centroid curve Floating body Volume distance Elastic rod |
| title | A generalization of the second Pappus–Guldin theorem |
| title_full | A generalization of the second Pappus–Guldin theorem |
| title_fullStr | A generalization of the second Pappus–Guldin theorem |
| title_full_unstemmed | A generalization of the second Pappus–Guldin theorem |
| title_short | A generalization of the second Pappus–Guldin theorem |
| title_sort | generalization of the second pappus guldin theorem |
| topic | Pappus–Guldin theorem Centroid curve Floating body Volume distance Elastic rod |
| url | http://www.sciencedirect.com/science/article/pii/S2590037425000019 |
| work_keys_str_mv | AT haraldschmid ageneralizationofthesecondpappusguldintheorem AT haraldschmid generalizationofthesecondpappusguldintheorem |