[metapost] The strangest issues?

[Giuseppe Bilotta]

Thursday, March 17, 2005 Daniel H. Luecking wrote:

> 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.

So my error was assuming that solutions of the
minimax problem should have null derivatives?

Because this was my initial line of thinking (which led me
to the same equations as considering directly the distance
function in R^4):

Fix the lambdas. For every t I can find the corresponding
tau: the point of \omega with minimal distance from
\gamma(t). This gives me the equation

(1) (\gamma(t)-\omega(tau)).\omega'(tau) = 0

from which I can define a tau(t) such that

\gamma(t) - \omega(tau(t))

give the "distance vector" between the two curves.

I need to find the minimum and maximum norm of this thing,
giving me the equations

(2) (\gamma(t) - \omega(tau)).\gamma'(t) = 0

(there would be an additional term, but it's null because of
(1), which must hold at the same time).

So far, so good. This meas that for every choice of \lambda0
and \lambda1 (in [0,1] or [1,+\infty] depending on which
side of the dilation I'm studying, and on the curvature at
the initial and final point) I can define
t(\lambda0,\lambda1) and tau(\lambda0,\lambda1) such that

crit(lambdas) = \gamma(t(lambdas)) - \omega(tau(lambdas))

are vectors of extremal distance.

Assuming L_\infty metric, the whole point of this work is to
find lambdas such that

Abs(crit(t).crit(t) - r^2)

isn't too big. Once again, I do this by finding the values
of the lambdas which nihil the derivative of this expression
and what do I get?

(\gamma(t) - \omega(\tau)).(P1 - P0) = 0
(\gamma(t) - \omega(\tau)).(P2 - P3) = 0

(again) to be used in combination with (1) and (2).

Which of these steps is wrong?

-- 
Giuseppe "Oblomov" Bilotta