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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 Parametric Curves in 2D Line Integral of Function - SOLVER
Related Wiki Pages:
Remember that a curve in 3d can ONLY be defined parametrically!
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 2d!
- 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.
So the approximate arc length of this weirdo 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 16 tangent pieces is: .
We calculate our new error.
Our error is now %.