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
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