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I've been looking at elementary cubic equations for curves and seem to understand them well enough. Going the other way and driving parametric equations has been mystifying. For example: given a simple cubic equation with 3 real roots, I'd like to derive a pair of parametric equations f(t), g(t) for the curve such that the velocity of a particle along the curve (in the direction of the curve) is a constant.

Perhaps a line integral will help? I'm not sure how to approach this.

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I recommend you look articles similar to the following if you do not know how to perform a reparametrization so as to get a constant speed:

https://en.wikipedia.org/wiki/Differential_geometry_of_curves#Length_and_natural_parametrization

https://www.geometrictools.com/Documentation/MovingAlongCurveSpecifiedSpeed.pdf

http://web.cs.iastate.edu/~cs577/handouts/curves.pdf

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