On Thu, 26 Jun 2003 11:43:50 -0700, David Hale <[log in to unmask]>
wrote:

>Feesh! Your calculations treat the sausage surface as
>if the entire 1,357 sq inch surface of the cylinder
>(sausage) can be seen at one time. I know that you
>know better than that.

That's for simplicity sake of presentation.  The flag can
be seen in ONE direction (of zero measure), and partially
seen in most other directions (slanted) so that the visible
part would be LESS than the surface of taking only the
rectangle of 6" times the height.

I knew that.  :-)    I was all factored into the equation.

That was actually the reason for my advanced math digression
of what's a set of measure zero.   :-))  That is, theoretically
the full surface of the flag is NEVER seen.

>
>I think the relevant figure is the two dimensional
>projection of the cylinder, which would be its
>diameter times height or 6 x 6 x 12 = 432. This would
>be the same no matter what angle it is viewed from, as
>long as you are in a plane that is perpendicular to
>the cylinder's axis.

That's correct.  That wuold be the CONSTANT view of the area
when a sausage is used.  To get the comparable figure for
the flag, you would have to "integrate" as in differential
calculus, the area seen, over the entire range of 360 degrees,
many of which is LESS than the rectangle area you mentioned,
and much would be only slivers when viewed in very obligue
angles.

>
>The viewable area of Andy's flag is 558, which is
>actually about 29% larger.

Seen o.000000000000000000001% of the time (just to make it
non-zero).   :-)

>But you have to be at
>exactly the right place to see that viewable area, so
>in nearly all cases the sausage will present a larger
>two dimensional projection than the flag.

Thank you for saying the same thing without mentioning
integral calculus.  :-))  What you said is exactly
right (though approximate) that you see a larger
two-dimentinoal view (for the sausage) MOST of the time.

A use of calculus would be able to let one say EXACTLY
what the "average" viewing area is, integrated over
the entire (uncountable infinity) of viewing directions.

>
>My geometry is rusty beyond belief, but I think the
>two dimensional projected sizes are what should be
>compared. I think I'm correct about this - please
>think it over. Of course if I'm wrong, I know this is
>really going to cost me!  :D
>
>David H.

This is actually "integral calculus", not really "geometry",
though it has the trivial "area" part in it.

In any event, the bottom line is that Andy's flag has much
less AVERAGE (calculus sense) viewable area than any 6' long
6" diameter sausage -- without having to do the detailed
calculus.

A "tangent" of this geometric discussion is the OPJECTIVE
FUNCTION of the gadget which has not been brought into play,
yet.  :-0

If I want something that can be seen by a rescuer, I want to
know the MINIMUM area that will be seen, no matter what
direction the rescuer is.

In that respect, the flag would FLUNK that criterion completely
becasue the minimum viewing area is ZERO.   :-)

Thus we wander into Optimization Theory in Operations Research
as well as in statistics and other fields, in which have well-
known terms to describe these CRITERIA, such as Minimax stragegy,
Masmin strategy, and strategies involving all kinds of LOSS
FUNCTIONS, integrated over the entire region of relevance.

Relativity Theory, Anyone?  :-))

ElPezNeuvo.