Timeline for Conditions for surface area of surface of revolution to be product of arclengths
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2022 at 16:27 | comment | added | Michael Engelhardt | @Locallytrivial - I fully agree. | |
| Dec 13, 2022 at 16:27 | vote | accept | locally trivial | ||
| Dec 13, 2022 at 16:21 | comment | added | locally trivial | So if we define ``center" as centroid of $C$, and $b$ to be the distance from the $z-$axis to the center of $C$, then you are saying that surface area will be invariant under rotating $C$ in the $xz$-plane before revolving $C$ around the $z$-axis, provided that such rotation fix the centroid of $C$. There are no smoothness assumptions on $C$ [for instance, continuity suffices]. I suspect that if we rotate as we revolve, then surface area can change, but that is a separate question. | |
| Dec 13, 2022 at 6:17 | comment | added | Michael Engelhardt | The correct alternative definition of $b$ which will make $\Sigma =2\pi b L$ (not just for this example, but in general) is to take $b$ to be the distance of the centroid of $C$ itself from the axis, not the one of its convex hull. | |
| Dec 13, 2022 at 5:58 | history | answered | Michael Engelhardt | CC BY-SA 4.0 |