Pappus-Guldin theorems for weighted motions (Q2455388)

The Pappus-Guldin rule gives the volume of a domain \(D\) generated by rotating a plane domain \(D_0\) as the product of the area of \(D_0\) and the length of the circle generated by the center of mass of \(D_0\). A similar rule holds for the surface area of \(\partial D\) involving the center of mass of \(\partial D_0\). This rule has been extended to motions along a space curve. In the present paper the notion of motion along a curve is combined with the theory of tubes with non-constant radius by the notion of weighted motion along a curve in 3-space. Such motions are treated as curves in a Lie group or its corresponding Lie algebra. In the volume formula, a special role is played by motions associated with Frénet frames, whereas in the area formula motions associated with parallel frames appear.