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Length of cubic polynomial (spline) ?
Posted:
Sep 26, 1996 10:13 PM
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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) = [tó tò 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*tó + 1.0*tò - 0.5*t) * P(i-1) +
( 1.50*tó - 2.5tò + 1.0) * P(i) +
(-1.5*tó + 2.0*tò + 0.5*t) * P(i+1) +
(0.5*tó - 0.5*tò)* 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
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