Re: [PyX-devel] Curves with infinite curvature

[Joerg L. <jo...@us...>]

Hi Michael,

[ sorry, I forgot to CC the list before, here it goes again ]

n 09.09.05, Michael Schindler wrote:
> On 09.09.05, Joerg Lehmann wrote:
> > On 09.09.05, Michael Schindler wrote:
> > > On 09.09.05, Joerg Lehmann wrote:
> > > > Hi Michael!
> > > > 
> > > > On 09.09.05, Michael Schindler wrote:
> > > > > I have hit a problem concerning the geometry things of the normpaths.
> > > > > Consider a Bezier curve that contains a point with infinite curvature.
> > > > > This is very easy to produce by setting e.g. the last two points to
> > > > > the same coordinates
> > > > > 
> > > > >   n = normcurve_pt(0,0, 1,1, 1,0, 1,0)
> > > > 
> > > > I think, we should return the correct rotation instead of None.
> > > 
> > > That is my problem. If you have infinite curvature, there is no
> > > geometrically defined rotation. Because the curvature is the change of
> > > the normal vector along the curve length, projected onto the tangent
> > > vector (or equally the other way round), you see that both vectors
> > > change infinitely fast. Thus, they do not exist in this point.
> > 
> > Maybe I'm missing the point, but for me the rotation doesn't have
> > anything to do with the curvature, but is solely defined in terms of
> > the tangent.
> 
> Correct. The tangent is not defined in such point.

Ok, this sounds reasonable.

> This is what I have learned some years ago in a lecture on
> differential geometry: Consider a parameterization of a curve. The
> parameterization can be C-infty. But: if the length "velocity" (dx/dt,
> dy/dt) is zero somewhere, you can easily create a cusp in that curve.
> So, the curve is not smooth anymore, and in that cusp the tangent
> vector is not continuous: From both sides the tangents approach
> different values.

Ah, ok, the limits from above and below do not agree. But in the output,
there is no cusp because we do not see the behaviour for param > 1.
Hmm...

> > So I would assume that we take the limiting case of the rotation defined
> > by the tangent as param->1. This you can easily calculate via l'Hopital
> > and it gives you (-1, 0) in the present case.
> 
> In the present case we have only one side to do this. It would be very
> nice if already this worked. Nevertheless, the problem remains if the
> parameterization velocity vanishes somewhere in the middle.

Can this happen in the case of Bézier curves?

> > Having said all that, I just want to remark that Ghostscript also
> > isn't quite a fan of a Bézier curve with two identical control points.
> > Just try to stroke the above normpath and view the resulting EPS
> > file...
> 
> Well, my ghostscript does not complain. But the tangent does not seem
> to be (-1, 0) at the end.

Ah, sorry, of course it works, there was another problem. And yes, the
tangent does not looks like being (-1, 0) at then end, although this is
the limit from below. I'm really getting confused now.

> > So, maybe we're discussing a point which is quite academical anyway.
> 
> I tapped into this while writing the parallel deformer where infinte
> curvatures are a big issue.

I suspected something like this...

    Jörg