A generalization of the second Pappus-Guldin theorem
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866917889371537408 |
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| author | Schmid, Harald |
| author_facet | Schmid, Harald |
| contents | This paper deals with the question of how to calculate the volume of a body in the three-dimensional Euclidean space 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 | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A generalization of the second Pappus-Guldin theorem Schmid, Harald Metric Geometry Classical Analysis and ODEs Differential Geometry 51M25, 52A15, 53A15, 74K10 This paper deals with the question of how to calculate the volume of a body in the three-dimensional Euclidean space 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. |
| title | A generalization of the second Pappus-Guldin theorem |
| topic | Metric Geometry Classical Analysis and ODEs Differential Geometry 51M25, 52A15, 53A15, 74K10 |
| url | https://arxiv.org/abs/2411.04488 |