Timeline for Does the centroid depend continuously on the curve?
Current License: CC BY-SA 3.0
15 events
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| Mar 5, 2014 at 17:04 | history | edited | Wlodek Kuperberg | CC BY-SA 3.0 |
minor clarifying corrections
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| Mar 5, 2014 at 4:58 | comment | added | user44143 | It's a pretty construction, it deserves to end in a pretty expression for the centroid: $\left(\frac{1}{\sqrt{2}}-\frac{1}{4}, \frac{1}{\sqrt{2}}-\frac{1}{4}\right)$ | |
| Mar 5, 2014 at 4:42 | comment | added | Nate Eldredge | The picture is very helpful, thanks. I had been visualizing something different and now your construction is clear (and clearly correct). | |
| Mar 5, 2014 at 4:36 | comment | added | Wlodek Kuperberg | @NateEldredge: Done. Hope it helps. | |
| Mar 5, 2014 at 4:35 | history | edited | Wlodek Kuperberg | CC BY-SA 3.0 |
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| Mar 4, 2014 at 23:12 | comment | added | Mathieu Baillif | Ah, I see, ok, thanks Wlodek and Paul. | |
| Mar 4, 2014 at 23:07 | comment | added | Wlodek Kuperberg | @MathieuBaillif: Your staircase is slightly different, but in the limit you get the same point anyway. | |
| Mar 4, 2014 at 23:04 | comment | added | Paul Siegel | To make Wlodek Kuperberg's example work with the centroid as he claimed, you want to use a stair case that goes above and below the diagonal at each step; as long as it is symmetric, you can arrange for the centroid to be $(1/4, 1/4)$ each time. | |
| Mar 4, 2014 at 22:59 | comment | added | Mathieu Baillif | I must have misunderstood the definition of the centroid, but there is something that I don't get. If you take a curve made of just one stair ``$\ulcorner$'', say joining $(0,0)$ to $(1,1)$, then $\int_\gamma x d\gamma = 1/2$, so $\bar x$ is $1/4$. With $2$ stairs the length is the same and one gets $\bar x = 3/8$, or I am completely lost ? | |
| Mar 4, 2014 at 22:37 | history | edited | Wlodek Kuperberg | CC BY-SA 3.0 |
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| Mar 4, 2014 at 22:37 | vote | accept | Paul Siegel | ||
| Mar 4, 2014 at 22:26 | history | edited | Wlodek Kuperberg | CC BY-SA 3.0 |
small corrections
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| Mar 4, 2014 at 22:19 | history | edited | Wlodek Kuperberg | CC BY-SA 3.0 |
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| Mar 4, 2014 at 22:03 | comment | added | Paul Siegel | This looks like a very nice idea. I'm about to try to calculate the centroid by hand just to make sure; if you happen to know a slick proof that the centroid is not the midpoint, could you share it? | |
| Mar 4, 2014 at 21:52 | history | answered | Wlodek Kuperberg | CC BY-SA 3.0 |