Length of cubic polynomial (spline) ?

*To*: mathgroup at smc.vnet.net*Subject*: [mg4886] Length of cubic polynomial (spline) ?*From*: "Julian Bushell" <jbush at globalnet.co.uk>*Date*: Thu, 26 Sep 1996 22:42:34 -0400*Organization*: Inspira Limited*Sender*: owner-wri-mathgroup at wolfram.com

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 object with a known velocity. I need to be able to travel along this spline a given distance for each frame. / P(i-1) \ C(t) = [t³ t² 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*t³ + 1.0*t² - 0.5*t) * P(i-1) + ( 1.50*t³ - 2.5t² + 1.0) * P(i) + (-1.5*t³ + 2.0*t² + 0.5*t) * P(i+1) + (0.5*t³ - 0.5*t²)* 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 real-time computer processing so speed is essential. Any help will be much appreciated, thank you. jbush at globalnet.co.uk ==== [MESSAGE SEPARATOR] ====