Length of a Curve

Compute the length of curves.

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Question

How can the length of a curved path be estimated and then expressed as an integral?

The length estimate improves as a polygonal path follows the curve more closely.
The length estimate improves as a polygonal path follows the curve more closely.

Goals

  • Approximate a curve by short line segments and observe convergence as the sampling becomes finer.
  • Connect the numerical approximation with the arc-length formula.

Method

Replace the curve by a polygonal path and progressively shorten its segments.

  • Sample points from a graph or parametric curve and add the distances between consecutive points.
  • Repeat with finer samples and compare the estimates.
  • Evaluate the corresponding arc-length integral numerically and discuss the error.

Try an example

Curve Length!

Source code ↗ Open full screen ↗

Approximate a smooth curve with straight segments and watch the total length settle as the segments become shorter.

Controls: Use + and − to change the number of line segments. Compare the gold approximation with the cyan curve and the reference length.