A generalization of the second Pappus-Guldin theorem

TL;DR

This paper generalizes the second Pappus-Guldin theorem to calculate volumes of bodies sliced by a curve, simplifying computations when the curve passes through cross-section centroids, and explores the existence of such curves.

Contribution

It introduces a generalized volume formula for bodies sliced by a curve and investigates the existence of centroid curves, applying the theory to ellipsoids and bent rods.

Findings

Derived a generalized volume formula for bodies with slicing curves.

Identified conditions for the existence of centroid curves in convex bodies.

Provided explicit centroid curves for a triaxial ellipsoid.

Abstract

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.