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Re: Curvature calculations
Since I'm not on the design team of Mathematica, I'm hardly in the position to comment on the presence or absence of any particular algorithm in the software. On the other hand, I think you may want to know that there is a huge difference in geometric significance between the ArcLengthFactor and the curvature of a curve. This is because, THE line element (not A line integral) defines the metric tensor of the underlying space in terms of the chosen coordinate system. Thus, all geometric properties of the space can be derived from the ArcLengthFactors. In contrast, the curvature of a line is just that: a local property of a specific object. It goes without saying that anyone interested in such things may expect to do some leg-work himself.