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A generalization of the second Pappus-Guldin theorem

Harald Schmid

TL;DR

This work provides a generalized framework for computing the volume of a 3D body by slicing along a curved path, yielding the key formula $\operatorname{vol}(K)=\int_\gamma A(s)\,ds$ when cross-sections are centroidal along $\gamma$. It derives this via a ribbon frame and a diffeomorphism, allowing for non-congruent cross-sections and identifying a centroid-curve condition that preserves volume. By linking centroid curves to Dupin's floating bodies and the volume distance, the authors prove local existence of centroid curves near the boundary for convex bodies and illustrate nontrivial examples in ellipsoids. They further apply the theory to elastic rods, deriving explicit centroid formulas and showing practical implications for geometric quantites like the barycenter under deformations, while outlining open questions for global extension and non-convex shapes.

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.

A generalization of the second Pappus-Guldin theorem

TL;DR

This work provides a generalized framework for computing the volume of a 3D body by slicing along a curved path, yielding the key formula when cross-sections are centroidal along . It derives this via a ribbon frame and a diffeomorphism, allowing for non-congruent cross-sections and identifying a centroid-curve condition that preserves volume. By linking centroid curves to Dupin's floating bodies and the volume distance, the authors prove local existence of centroid curves near the boundary for convex bodies and illustrate nontrivial examples in ellipsoids. They further apply the theory to elastic rods, deriving explicit centroid formulas and showing practical implications for geometric quantites like the barycenter under deformations, while outlining open questions for global extension and non-convex shapes.

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 by using the volume distance and certain features of the so-called floating bodies of . 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.

Paper Structure

This paper contains 6 sections, 7 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Volume calculation using perpendicular cross-sections
  3. On the existence of centroid curve s in a convex body
  4. Example: The centroid curve s of an ellipsoid
  5. Application: The centroid of an elastic rod
  6. Conclusion

Key Result

Theorem 1

Let $\gamma:I\longrightarrow\mathbb{R}^3$ be a regular curve on some open interval $I\subset\mathbb{R}$ with unit tangent vector $T(s)=\gamma'(s)$, $s\in I$, and let $(T(s),N(s),B(s))$ be an orthogonal moving frame along $\gamma$ with normal and geodesic curvature $\kappa_n(s)$ and $\kappa_g(s)$, re

Figures (4)

  • Figure 1: Two toroidal bodies with the same volume. A circle of the same size forms a centroid curve in both solids, where the vertical cross-sections on the left-hand side are circular disks with varying radii, while on the right-hand side they are rotating ellipses having the same area as the circular disks.
  • Figure 2: In this sectional view, $p_0$ and $s$ are centroids of plane cross-sections tangential to the floating body $K_{[\delta_0]}$. The hatched segments cut off from $K$ by these cross-sections have the same volume $\delta_0$. Moreover, $p$ is the centroid of the plane cross-section $H(n,p)$ tangential to $\partial K_{[\delta]}$ and perpendicular to $n$, where $H(n,p)$ and $\partial K$ enclose the gray shaded segment with volume $\delta$.
  • Figure 3: The centroid curve $\gamma$ passing through $p_0$ in an ellipsoid with semi-axes $a=1$, $b=0.625$, $c=0.5$. The centroid curve is given by $y(x)=-0.3\,(x/0.8)^{2.56}$, $z(x)=0.18\,(x/0.8)^{4}$ for $x\in[0,0.8]$, and it has been continued symmetrically to the origin. The gray-shaded ellipse is the cross-section $\Gamma(s)$ perpendicular to the centroid curve at $p_0$.
  • Figure 4: An elastic rod, above in the initial state and below in bent form with centroid curve $\gamma$ and unit normal field $N$.

Theorems & Definitions (13)

  • Theorem 1
  • Definition 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • proof
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • ...and 3 more