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There is an arc a to b. The length is say 1 unit. There is an another arc c to d, drawn just below a to b, such that it sticks with it (ie. No vertical y space).

Let us say i have my world coordinate system, where measurements with precision of more than 2 decimal "CANNOT" be found out. For example i can have a measurement 1.55 (2 decimal places) but not 1.551 (3 decimal places). That is 2 decimal places is the limit of complexity i can dive into.

Now, in such a world, can i say by intuition that the arc that goes from c to d (which by intuition must be smaller than a to b) must be of length 0.99 unit? (0.99 is something immediate before "1", when i say i have restricted my self to 2 decimal places)

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It sounds like you want to limit yourself to curves that pass through lattice points of a grid with squares .01 by .01? Is that accurate? – orlandpm Jan 29 at 5:55
Ya. That seems correct. – Vishwas Gagrani Jan 29 at 6:09
Where are $c$ and $d$ located? the arc $cd$ could be significantly longer than $ab$. – nbubis Feb 6 at 15:09
cd is just inside ab. It's something like a circle inside another circle. AB is the arc of bigger circle. And CD is the arc of inner circle. They start togather and end togather. – Vishwas Gagrani Feb 7 at 5:10

1 Answer

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If a rectifyable arc (e.g. a polyline in a lattice) from $a$ to $b$ passes through $c\ne a$ and $d\ne b$ on its way (in this order) and the points can be distinguished in your world, then the arc from $c$ to $d$ is shorter by at least $0.02$ units ($0.01$ per end).

However, if the underlying world is not a lattice with grid size $0.01$, but rather there is some "true" distance between $a$ and $c$ that you only measure as $0.01$, say, because it is truely between $0.005$ and $0.015$, then the path $c$ to $d$ is shorter only by at least $0.01$ units ($0.05$ per end). Another big caveat: The result of rectifying an arbitrary curve with imperfect measuring tools may depend on the starting point in a nontrivial way, i.e. we cannot fully predict, how close your imperfect rectification is to the true arc length; it may even be the case that the measuring obtained for the truely shorter $cd$ exceeds the measuring obtained for the truely longer $ab$!

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Well.. i still have some doubts on this. I will try to rephrase question once i arrive home. But, thanks for providing attention to the question. :) – Vishwas Gagrani Feb 7 at 5:14

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