Timeline for Conditions for surface area of surface of revolution to be product of arclengths
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2022 at 16:27 | vote | accept | locally trivial | ||
| Dec 13, 2022 at 5:58 | answer | added | Michael Engelhardt | timeline score: 1 | |
| Dec 12, 2022 at 19:55 | history | edited | LSpice | CC BY-SA 4.0 |
`$…-$` -> `$…$-`
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| Dec 12, 2022 at 18:49 | comment | added | Michael Engelhardt | Think of the surface area of $S$ as the average $\langle 2\pi r \rangle_{C} $, where $\langle \cdot \rangle_{C} =\int_{C} \cdot \, d\gamma $ and $d\gamma $ is the arc length element on $C$. If almost all of your arc length is at a certain $r$, say, $r=b$, then $\langle r\rangle \approx b$ (I'm being cavalier about normalization by the length of $C$). I suppose curvature is necessary to concentrate most of the arc length around a certain $r$. | |
| Dec 12, 2022 at 18:16 | history | edited | locally trivial | CC BY-SA 4.0 |
Fixed definition of ``center" of C in the $xz-$plane
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| Dec 12, 2022 at 18:04 | history | edited | locally trivial | CC BY-SA 4.0 |
Fixed defintion of $b$
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| Dec 12, 2022 at 17:42 | comment | added | locally trivial | Is there a relationship here between curvature [either of C, or total curvature of S] with the surface area of S? | |
| Dec 12, 2022 at 17:36 | comment | added | locally trivial | I am confused: If $C$ is a circle then b is not the minimal distance of C to the z-axis; it's the distance from the z axis to the center of the circle. The surface area for the torus being (2pi a)(2pi b) only works for this specific value of b. What do you mean by ``having many wrinkles there?" Do you mean that if we concentrate curvature of C near this point for b (in the xz-plane) [i.e., nearest the z-axis, or furthest away from the z-axis, respectively] then we can have these given values for b? What are your assumptions on C? [just continuous, for instance?] | |
| Dec 12, 2022 at 6:52 | comment | added | Michael Engelhardt | You can make $b$ arbitrarily close to the minimal distance of $C$ from the z-axis (by having $C$ have many wrinkles there), or arbitrarily close to the maximal distance of $C$ from the z-axis (by having $C$ have many wrinkles there), or anywhere in between. | |
| Dec 12, 2022 at 2:50 | comment | added | locally trivial | I suppose the theorem I'm reaching for is along the lines of ``the n-volume of a product of n 1-manifolds is given by the product of their 1-volumes," in the smooth case. And then I'm wondering under what conditions this still holds in the non-smooth case, for a product of piecewise-smooth curves [or manifolds with corners]. More generally, given manifolds M_1, \dots, M_r of dimension d_1, \dots, d_r with D=∑di then relating the D volume of the product to the product of the d_i-volumes, and then how-far into the non-smooth case does this still hold? [Is there a theorem related to this?] | |
| Dec 12, 2022 at 2:24 | comment | added | locally trivial | I have in mind a definition of center here as something more like "barycenter," or the centroid of the convex hull of $C$ | |
| Dec 12, 2022 at 2:05 | comment | added | Michael Engelhardt | If you're going to allow $b$ to be as vague as "the distance from the z-axis to the center of $C$ in some sense", then of course there will be a suitable value of $b$. You're just defining it by asking it to reproduce the surface area of $S$. | |
| Dec 12, 2022 at 1:53 | history | asked | locally trivial | CC BY-SA 4.0 |