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Re: How to Calculatelength of an Spline curve between two points?


On Nov 10, 6:04 am, alfaeco <alfa... at gmail.com> wrote:
> I need some help.
>
> Given an Interpolating function with method-> Spline.
>
> sp = Interpolation[tbl, Method -> "Spline"];
>
> How can I calculate the distance between two arbitrary points of the
> splines?

I'm assuming you want to compute the distance along the spline. In
principle it should be easy, but because of a bug (that produces a
kernel crash) it's a bit harder.

Here's the setup:

data = Table[{i, RandomReal[{-10, 10}]}, {i, 5}];
ifun = Interpolation[data, Method -> "Spline"];

Now let's simply use the formula for arc length to compute the
distance from x = 1.5 to x = 3.1:

NIntegrate[Sqrt[1 + ifun'[x]^2], {x, 1.5, 3.1}]

But on my system (Windows XP, Mathematica Version 7.0.1) this crashes the
kernel. So does FunctionInterpolation[ifun'[x], {x, 1, 5}], so that's
out too.

OK, we can do this "by hand":

difun = Interpolation[Table[{x, ifun'[x]}, {x, 1, 5, .01}]]

Then use the arc length formula:

NIntegrate[Sqrt[1 + difun[x]^2], {x, 1.5, 3.1}]

On the other hand, if this is not what you want, then never mind.

--Mark


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