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 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) = [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(i-1) +

( 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 real-time computer processing so

speed is essential.

Any help will be much appreciated, thank you.

From Julian Bushell

jbush@globalnet.co.uk