Timeline for The area of a rectangle containing a curved line of a known length
Current License: CC BY-SA 4.0
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| when toggle format | what | by | license | comment | |
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| 5 hours ago | answer | added | Alex Jones | timeline score: 2 | |
| 7 hours ago | comment | added | David K | Note that if the outer curve always has length $L$, the area of the region covered by the "curve with thickness" depends on how curved it is. The greater the curvature, the less the area. | |
| 7 hours ago | comment | added | David K | Another thing that we have to guess at is what each curve really is. Each "curve" is actually a region of the plane with an "inner" (shorter) curved segment and an "outer" (longer) curved segment. We can also construct a "midline" curve exactly halfway between the inner and outer curve. If this were PostScript or similar graphics software I'd expect $L$ to be the length of the midline curve, but it seems you want the outer curve to have length $L$ because otherwise the semicircle wouldn't fit in the box you specified for it. | |
| 7 hours ago | comment | added | David K | You don't just transform a line to a semicircle. You have a whole family of curves that you parameterize by a parameter $x,$ and you want to say something about each value of $x.$ This should be made clear earlier in the question. I think you imagine that "transform" implies all these intermediate forms, but it does not: mathematically, to "transform" something describes a relationship between the starting state and the final state only and says nothing about anything "in between." | |
| 8 hours ago | answer | added | naturallyInconsistent | timeline score: 4 | |
| 9 hours ago | history | became hot network question | |||
| 14 hours ago | answer | added | Ethan Bolker | timeline score: 6 | |
| 15 hours ago | answer | added | Andrei | timeline score: 2 | |
| 15 hours ago | comment | added | Temani Afif | @EthanBolker not sure what else I can say. I explained that I need a formula/function and that I know how to calculate only two values of that formula/function. | |
| 15 hours ago | comment | added | Ethan Bolker | Please edit the question to tell us what "smallest rectangle" means. The one with the smallest area is the one on the left with height 0. That's the one with the smallest height. The smallest width rectangle is the one on the right when the curve is a semicircle. (Don't try to explain in a comment.) | |
| 15 hours ago | comment | added | Temani Afif | @ApexPandora which refer to the "line" I talked about in the previous sentence that goes from "straight" to "curved". | |
| 15 hours ago | comment | added | ApexPandora | I meant in the sentence, "Is there a way to find a formula for the height of the smallest rectangle containing that line? (the width of that rectangle will be the variable)" | |
| 15 hours ago | comment | added | Temani Afif | @ApexPandora I am already using the world "curved" and I added a figure showing the rectangle and I already added the result for both edge cases | |
| 15 hours ago | comment | added | Temani Afif | @Andrei I know, that's already in the question. I am looking for the function. I already know the easy two edge cases. I want the in-between. | |
| 15 hours ago | comment | added | ApexPandora | @TemaniAfif, I think curve might be a better suited word here since line implies it being straight, which only refers to the trivial case. | |
| 15 hours ago | comment | added | Andrei | If it's a semicircle with radius $R$, the width is $2R$ and the height is $R$. If you bend the line such as the outer part has length $L=\pi R$, you get the width $2L/\pi$ and the height $L/\pi$ | |
| 15 hours ago | comment | added | Temani Afif | @ApexPandora that's when the line is a straight one. I want to find the formula when we curve the line (from straight to curved) | |
| S 15 hours ago | history | suggested | ApexPandora | CC BY-SA 4.0 |
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| 16 hours ago | comment | added | ApexPandora | I feel like I am misunderstanding your question, the smallest rectangle containing that line would be a rectange with height $1$ (the width of the line) and base $L$. | |
| 16 hours ago | review | Suggested edits | |||
| S 15 hours ago | |||||
| 17 hours ago | history | asked | Temani Afif | CC BY-SA 4.0 |