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Anonymous
Equation of the longest distance of a one to one function 2016-10-19 21:00:57 Post No. 8423585
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Equation of the longest distance of a one to one function 2016-10-19 21:00:57 Post No. 8423585
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Equation of the longest distance of a one to one function
Anonymous
2016-10-19 21:00:57
Post No. 8423585
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Hey /sci/, so while I was driving yesterday I noticed that the left turn of an intersection had an unusual curve on the dotted guiding line; that got me thinking about what the equation of the absolute maximum distance between two points may be, while the function still remains one to one? I'm not very smart but this has kind of been on my mind for the last few hours - the line acts like a limit but it obviously does reach that limit. I believe it also is essentially 1/4th of a circle as well. Pic related of course.
I'd really appreciate any help.
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>>8423585
For some reason it uploaded the photo sideways. I apologize.
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>>8423585
>the function still remains one to one?
What do you mean by that ?
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>>8423585
There is no maximum length curve between two points. You can always add "squiggles" to make it longer
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>>8423595
A one to one function means it passes both the vertical and horizontal line test. You could have an infinitely oscillating line between these two points, but it wouldn't pass the horizontal line test.
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>>8423602
Then it would not pass the horizontal line test and therefore wouldn't be one to one.
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>>8423607
You are looking for bijections?
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>>8423614
Somewhat, but more specifically I'm looking for the equation that would model said line(s) since there technically would be two maximum distance lines of equal length (lines 1 and 2) that share the same slope but are on different sides of the direct line.
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>>8423585
Between (0,0) and (1,1) such a function is:
[eqn]\lim_{n \to \infty} x^{n}[/eqn]
As you increase n it becomes more and more like a square, but it's still one to one for all finite n.
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>>8423585
You have set no constraints on the function being continuous so the answer is it can be as long as you like it to be
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>>8423656
Thanks a bunch anon!
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>>8423656
How did you arrive at this?
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>>8423820
It's pretty obvious desu
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>>8423820
It's quite trivial desu
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