[metapost] The strangest issues?

[Daniel H. Luecking]

On Wed, 16 Mar 2005, Giuseppe Bilotta wrote:

> (*)
> (\gamma(t) - \omega(\tau)).(P1 - P0) = 0
> (\gamma(t) - \omega(\tau)).(P2 - P3) = 0
>
> which are to be combined with
>
> (\gamma(t) - \omega(\tau)).\omega'(\tau) = 0
> (\gamma(t) - \omega(\tau)).\gamma'(t) = 0
>
> (where . represents the dot product). Now have a
> look at (*): since we're on a plane, from it we can deduce
>
> (P1 - P0) x (P2 - P3) = 0
>
> (where x represents the cross product). That is, the two
> sides of the original curve must be parallel! Isn't this
> ridiculous? Where did I go wrong?

Assuming the generic case, that this determinant is non-zero, you deduce
that the system (*) can be solved to get \gamma(t) = \omega(\tau).

It is wrong to allow t, \tau, \lambda0 and \lambda1 to vary
independently. What you get is the minimum distance over all
values of these variables (there can be no maximum since large
values of the \lambda's make the distance arbitrarily large).
The above seems to imply that this minimum can only be 0 (but the
minimum distance need not be at t, \tau between 0 and 1).

What you really need is to find the maximum M and and minimum m over
all t and \tau (the second set of equations). These values depend on
\lambda0 and \lambda1. Now you need to minimize (M - m). This is really
a mini-max problem (minimizing the maximum deviation), and those are
notoriously hard in general.

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