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Image: ubuntu2004
We are exploring arc length. We want to understand what is happening. (We do NOT care whether we can integrate symbollically! Any program will integrate for us.)
YouTube Videos: Parameterization Arc Length
Related Sage Pages: Arc Length of Explicit Curves in 2D Arc Length of Curves in 3D
Related Wiki Pages:
Look at the curve above and estimate a minimum and maximum value for its length L.
Arc Length of a Curve given parametrically : for is
So the arc length of this weirdo curve using the formula (which we are calling our "exact" result even though it is being calculated numerically) is
Let us approximate this length by finding tangent line segments at regularly spaced values of t along the curve.
This algorithm is exactly the same as for parametric curves in 3d!
- We decide how many steps.
- The program calculates the stepsize of t.
- We draw points on the curve regularly spaced with repect to stepsize. They are the points: (s(j))
We draw pieces of tangent line segments starting at these points.
- Start point is s(j)
- Slope is value of the derivative vector ds(j).
- Length is stepsize = stepsize.
So parametrically these line segments are: s(j)+λ·ds(j) for λ=[0, stepsize].
We sum the length of these pieces. They are each of length: stepsize = stepsize, where the derivative is evaluated at the starting points of the pieces.
So the approximate arc length of this "simple" curve using 4 tangent pieces is: .
We calculate our error.
Our error is %.
Let us try more or less step sizes - change the value of steps2 and revaluate.
We sum the length of these pieces.
So the approximate arc length of this weirdo curve using 12 tangent pieces is: .
We calculate our new error.
Our error is now smaller than: .
We note that small errors are delicate things and in "real life" we must keep track of all of the possible types of errors that can occur here.