Hi Andrew,

OK, with a little thought the problem reduces down to finding the
integral of the square-root of a fourth order polynomial, ie:

int(sqrt(A*t^4 + B*t^3 + C*t^2 + D*t + E), t);

where A to E are scalars derived from the coefficients of the x, y and z
cubic functions. So if you can get an analytic integration of this
please make it known ... I couldn't figure it out ...

Bye for now,
	Mike Andrews.


Andrew Clinton wrote:
> 
> Michael,
> 
> Ahh, I was thinking of this, but thought that maybe one could take the magnitude of
> the vector returned by that (terribly ugly) formula to get the distance traveled.  I
> understand how this is done with vectors now (there is a good page at
> http://iq.orst.edu/mathsg/vcalc/arc/arc.html)
> 
> Thanks for clarifying
> Andrew C
> 
> Michael Andrews wrote:
> 
> > Hi Andrew,
> >
> > This is the length of a simple parametric spline where a, b, c and d are
> > scalars, yes?
> >
> > The problem is that it would usually be used to produce a spline in 3d,
> > so a, b, c and d would be 3-component vectors. This would probably add a
> > whole new level of complexity, something like
> >
> > f_x := a_x * t^3 + b_x * t^2 + c_x * t + d_x ;
> > f_y := a_y * t^3 + ... ;
> > f_z := ... ;
> >
> > int(sqrt(diff(f_x,t)^2 + diff(f_y,t)^2 + diff(f_z,t)^2),t) ;
> >
> > Try feeding that to Maple and see if it chokes :-)
> >
> > Bye for now,
> >         Mike Andrews.
> >
> > Andrew Clinton wrote:
> > >
> > > Exact length of a cubic f(x) = a*x^3 + b*x^2 + c*x + d:
> > > http://www.eng.uwaterloo.ca/~ajclinto/test.html
> > >
> > > assuming you have x=0 and x=1 as the bounds of the segment (and maybe a=1) it
> > > would become SLIGHTly simpler, but I still doubt whether there would be any
> > > practical use for this mess.
> > >
> > > Andrew C
> > >
> > > "Tony[B]" wrote:
> > >
> > > > How can I get an exact/closely approximated measure of the distance traveled
> > > > along a spline and the total length of it? I am using a cubic_spline in
> > > > MegaPOV.

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