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Length of cubic polynomial (spline) ?
Posted:
Sep 26, 1996 10:13 PM


I have a computer problem.
I need to know the length of a cubic polynomial.
I'm actually using CatmullRom splines to describe the path of an object with a known velocity. I need to be able to travel along this spline a given distance for each frame.
/ P(i1) \ C(t) = [tÃÂ³ tÃÂ² t 1] B  P(i)   P(i+1)  \ P(i+2) /
where P is the interpolated points and B is the matrix
/ 0.5 1.5 1.5 0.5 \ B =  1.0 2.5 2.0 0.5   0.5 0.0 0.5 0.0  \ 0.0 1.0 0.0 0.0 /
Equation form
C(t) = (0.5*tÃÂ³ + 1.0*tÃÂ²  0.5*t) * P(i1) + ( 1.50*tÃÂ³  2.5tÃÂ² + 1.0) * P(i) + (1.5*tÃÂ³ + 2.0*tÃÂ² + 0.5*t) * P(i+1) + (0.5*tÃÂ³  0.5*tÃÂ²)* P(i+2)
Since P are constant numbers. Polynomials in x and in y can be formed.
I need to know the length of the curve for any t value (especially the entire length i.e. from t=0 to t=1).
Please note that this is required for realtime computer processing so speed is essential.
Any help will be much appreciated, thank you.
From Julian Bushell jbush@globalnet.co.uk



