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Motion Control: Computing the Arc Length of Parametric Curves
May/June 1990 (vol. 10 no. 3)
pp. 72-78

Specifying constraints on motion is simpler if the curve is parameterized by arc length, but many parametric curves of practical interest cannot be parameterized by arc length. An approximate numerical reparameterization technique that improves on a previous algorithm by using a different numerical integration procedure that recursively subdivides the curve and creates a table of the subdivision points is presented. The use of the table greatly reduces the computation required for subsequent arc length calculations. After table construction, the algorithm takes nearly constant time for each arc length calculation. A linear increase in the number of control points can result in a more than linear increase in computation. Examples of this type of behavior are shown.

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Brian Guenter, Richard Parent, "Motion Control: Computing the Arc Length of Parametric Curves," IEEE Computer Graphics and Applications, vol. 10, no. 3, pp. 72-78, May-June 1990, doi:10.1109/38.55155
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