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parabolic trough edge's length

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Posted by Morris Dovey on July 8, 2010, 5:09 am

Someone came to my web site looking for a way to calculate the length of
a parabolic trough edge. On the off-chance that it was someone who reads
this group, I've written an MS-DOS program to do that job and am willing
to email a copy of the executable to anyone wanting it.

It provides as close an approximation as anyone is likely to want, and
it gives the length of y = Ax^2 + Bx + C between a pair of x limits to a
user-specified error threshold (on my machine an error threshold of zero
causes it to run for a ghastly 13 seconds).

It isn't beautiful, but the price is right. :)

Morris Dovey

Posted by dow on July 10, 2010, 12:10 am

13 seconds ghastly?!

Remember that thing I did a few months ago, calculating the dimensions
of a paraboloidal reflector that has its focus coincident with its
centre of mass? I made the program do the calculation to a precision
of ten significant digits (Why? Because I wanted to see if I could.)
On the ancient machine I used, the program took nearly ten *minutes*
to run.

Good programs give you time to go and make coffee. Really good
programs let you go see a movie. Excellent programs let you take a


Posted by Morris Dovey on July 10, 2010, 1:29 am
 On 7/9/2010 7:10 PM, dow wrote:

Eureka, and thank you (I think). After all these years of programming, I
finally understand why my coffee is _always_ cold - and why I've taken
so few vacations... :-/

Morris Dovey

Posted by Steve D on July 10, 2010, 1:50 am
Are you talking the length along a parabolic curve?  If you are, I know
there is an exact math solution to this one.  It involves a curve integral.
I'm sure of it since I calculated it when I was in college.  Of course I
can't remember how to do it any more, but if you find an appropriate egghead
I'm sure you can get the formula.  If I was more ambitious I'd try to
re-fire those old neurons (it's been more than 20 years).

I remember using the calculation to make something resembling a parabolic
dish.  Knowing the length along the parabolic curve I cut paper into wedges
that varied in width based on the length along the parabolic curve.  I taped
the wedges together and got a dish.


Posted by Morris Dovey on July 10, 2010, 2:51 am
 On 7/9/2010 8:50 PM, Steve D wrote:

Sigh - me too, except that I took the course 45 years ago, and my old
textbook was ruined in a basement flooding 30 years ago.

It should be a fairly simple definite integral, but I'm down to my last
two neurons - one is for breathing and the other... I forget what the
other one does. :)

Between the time I finished the program and the time I posted here I
visited Amazon and ordered a used copy of my old textbook (Abraham
Schwartz's "Calculus and Analytic Geometry"). Perhaps it'll stick better
the second time around.

My iterative approach is to approximate the curve with straight line
segments and sum their lengths. At each iteration I increase the number
of segments by a factor of ten, and when the difference between
successive approximations is less than or equal to the error threshold,
the program outputs the sum and quits. On this old IBM PC the last
non-zero difference (before running out of bits) was on the order of
10^-13 - which is probably close enough for most DIY work.

I suspect that project would have exceeded my attention span, but I did
write a Z-80 program to control three lead screws to spin dishes from
aluminum disks back in C- and K-band days.

Morris Dovey

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