[metapost] The strangest issues?

[Daniel H. Luecking]

On Fri, 18 Mar 2005, Giuseppe Bilotta wrote:

> Thursday, March 17, 2005 Daniel H. Luecking wrote:
>
> > (\gamma(t) -\omega(\tau)).\gamma'(t) = 0
> > (\gamma(t) -\omega(\tau)).\omega'(\tau) = 0
>
> > will be a minimum (or a saddle point). And minimums happen when the

Actually, I fogrgot to include local maxima.

> > curves cross. Minimizing over the lambdas will now likely give you the
> > crossing points. Of course, if \gamma is a straight line, the
> > alternative is that \omega is a parallel line and (P1-P0) x (P3-P2) = 0.
>
> Well, t and \tau are restricted to the [0,1] range and since
> I'm for local, not global, extremal points I really don't
> care if they are minima or maxima until I get down to
> analyzing them. Which I cannot do if I cannot find them, of
> course :)
>
> Plus, the curves \omega and \gamma are not entirely
> arbitrary. While I'm trying not to put any constraints on
> \gamma, the endpoints of \omega and its initial and final
> tangents are in very well fixed relations to \gamma.
> Constraints on the lambdas can be also easily created so
> that the two curves don't cross in the [0,1]^2 box, unless
> the original curve is knotted itself or the radius is higher
> than a critical value.
>
> Does this mean I only have to look at the values for which
> t and \tau are not differentiable?

Since the two equations to be solved for t and \tau are both cubic
in one variable and quintic in the other, if they could be solved
for t and \tau in terms of the lambdas they would be multiple-valued.
There would therefore be the possibility (even likelyhood) of "branch
points" where these different values come together and differential
analysis is impossible.

And here we have rapidly converged to the limits of my knowledge of
these things.

-- 
Dan Luecking
Dept. of Mathematical Sciences
University of Arkansas
Fayetteville, AR 72101