Re: [ft-devel] Re: preliminary analysis of the problem with cubic spline optimisation

[Graham Asher]

"It can happen, though". In fact I now don't think it can happen. If 
there are any good mathematicians out there (better than me at this, 
which sets quite a low bar), please confirm that no point on a cubic 
spline curve with both control points on the same side of the straight 
line from start to end can be further from the line than the curve's 
midpoint as defined by bisection of the curve.

Graham


Graham Asher wrote:
> No, in nearly all cases the midpoint is the point on the curve 
> furthest from the straight line when both control points are on the 
> same side. It can happen, though. I'm now working on a fix depending 
> on finding how far the furthest control point is from the straight 
> line. No point on the curve can be further from the straight line than 
> both control points.
>
> Graham
>
> James Cloos wrote:
> > > > > > > "GA" == Graham Asher <address@hidden> writes:
> > > > > > >             
> >
> > GA> Correction: it's not monotonicity that matters, but having the two
> > GA> control points on different sides of the straight line.
> >
> > Wouldn't the midpoint also fail to be the furthest point from the line
> > whenever the two off-curve control points are on the same side of the
> > midpoint?
> >
> > -JimC
> >   
>
>
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