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I've been racking my brain about this problem, but can't seem to figure it out. It seems like it should be easy, but I keep getting confused. Let's say I have an arbitrary parametric curve [itex]r(t)=<x(t), y(t)>[/itex]. I want the velocity in the tangential direction to be constant. That clearly means that the x and y components of velocity cannot be constant as the curve changes direction. I realize that the speed tangent to the path is considered [itex]|r'(t)|[/itex] (i.e. the magnitude of the velocity). That would imply that if I want a zero acceleration speed along the path I would need [itex](x''(t))^2+(y''(t))^2=0[/itex] (I removed the square root that is present in magnitude since it's zero)... Does that mean to solve for x(t) and y(t) I need to solve this second order differential equation? This is where I get confused, should the position function I get from calculating this differential equation be different than the original function I am given? Or will the original parametric function run at a constant speed to begin with?

The idea is that given a certain curve I'd like to get out x(t) and y(t) functions that run a constant speed along that curve. That means that the curves may be gotten via regression (i.e. someone draws the curve and it is modeled via splines or polynomials)

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# Constant tangential speed along arbitrary parametric curve

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