RE: [ft-devel] Re: preliminary analysis of the problem with cubic spline optimisation
-Original Message- From: freetype-devel-bounces+david.bevan=pb@nongnu.org [mailto:freetype-devel-bounces+david.bevan=pb@nongnu.org] On Behalf Of Graham Asher Sent: 29 August 2010 20:01 To: James Cloos Cc: freetype-devel Subject: Re: [ft-devel] Re: preliminary analysis of the problem with cubic spline optimisation 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. The point at maximal distance is the midpoint ONLY if the control points are equidistant from the chord. David %^ ___ Freetype-devel mailing list Freetype-devel@nongnu.org http://lists.nongnu.org/mailman/listinfo/freetype-devel
Re: [ft-devel] Re: preliminary analysis of the problem with cubic spline optimisation
GA == Graham Asher graham.as...@btinternet.com 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 -- James Cloos cl...@jhcloos.com OpenPGP: 1024D/ED7DAEA6 ___ Freetype-devel mailing list Freetype-devel@nongnu.org http://lists.nongnu.org/mailman/listinfo/freetype-devel
Re: [ft-devel] Re: preliminary analysis of the problem with cubic spline optimisation
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 graham.as...@btinternet.com 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 ___ Freetype-devel mailing list Freetype-devel@nongnu.org http://lists.nongnu.org/mailman/listinfo/freetype-devel ___ Freetype-devel mailing list Freetype-devel@nongnu.org http://lists.nongnu.org/mailman/listinfo/freetype-devel
