```Date: Sep 26, 1996 10:13 PM
Author: Julian Bushell
Subject: Length of cubic polynomial (spline) ?

I have a computer problem.I need to know the length of a cubic polynomial.I'm actually using Catmull-Rom splines to describe the path of an objectwith a known velocity.I need to be able to travel along this spline a given distance for eachframe.                                      /  P(i-1) \                C(t) = [tÃÂ³ tÃÂ² 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 formC(t) = (-0.5*tÃÂ³ + 1.0*tÃÂ² - 0.5*t) * P(i-1) +       ( 1.50*tÃÂ³ - 2.5tÃÂ² + 1.0) * P(i)   +       (-1.5*tÃÂ³ + 2.0*tÃÂ² + 0.5*t) * P(i+1) +       (0.5*tÃÂ³ - 0.5*tÃÂ²)* 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 theentire length i.e. from t=0 to t=1).Please note that this is required for real-time computer processing sospeed is essential.Any help will be much appreciated, thank you.From Julian Bushelljbush@globalnet.co.uk
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