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B-Spline derivatives

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A derivative of B-Spline at some parameter t is a vector with its magnitude and direction. This magnitude is changing is size over time. What I am interested is what this magnitude is a function of? Why is sometimes magnitude bigger and why is sometimes smaller?

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It's also function of 't'. That doesn't change when you take the derivative. The magnitude changes because the instantaneous rate-of-change at every point on the spline changes as a function of 't'. If you have dx/dt, dy/dt, and dz/dt for each direction, then combined you get [dx,dy,dz]/dt, or dP/dt, which you can think of as velocity. Then magnitude would be speed.

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Yes, the rate of change of the rate of change would be the acceleration.


X(t) = spline function
d/dt[X(t)] = V(t)
d/dt[V(t)] = d/dt[d/dt[X(t)]] = A(T)

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