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"Flying" along several Catmull-Rom-spline segments "at constant speed"

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Hello, sorry for the long topic, but I don't know how to describe my problem more formal. Here's the setup: I have a path, along which I want to move an object at constant speed. The path is made up by several controlpoints. I interpolate between them using the Catmull-Rom formula. The problem is, that obviously, the distance covered by an object that moves along this spline will depend on the distance of the controlpoints. So I tried adjusting the interpolation weight by the distance between 2 controlpoints. This doesn't account for the curve of the spline though. Also, 4 controlpoints are involved when interpolating. I probably have to calculate 3 distances and weigh the acoording to the Catmull-Rom formula and adjust my interpolation time by that, but I'm really at a loss here. Does anyone understand what I mean/have advice for me? Thanks in advance.

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