[Tutor] [OT] Re: Floating Point Craziness

[Emile van Sebille]

On 6/12/2011 1:55 PM Andre' Walker-Loud said...
> Hi Alan,
>
>>> * Or you just get used to the fact that some numbers are not exact in
>>> floating point.
>>
>> This got me thinking. How many decimal places do you need to
>> accurately, say, aim a laser somewhere in a 180 degree arc accurately
>> enough to hit a dime on the surface of the moon?
>
> Here is a quick back of the envelope estimate for you.  (While I am still learning the Python, I can answer this one!)
>
> The angle subtended by a dime on the earth is (approximately) given by
>
> sin( theta ) = d / sqrt( R^2 + d^2 )
>
> where
>
> d = 1 cm (the diameter of a dime)
> R = 384,403 km (the average distance from the center of the earth to the center of the moon - the moon traverses an elliptical path about the earth)
>
> To make the approximation simple, take advantage of the series expansion for sin (theta) and 1 / sqrt(R^2 + d^2)
>
> first
>
> d / sqrt( R^2 + d^2 ) = d / R * 1 / sqrt(1 + d^2 / R^2 )
> 	~= d / R * (1 - 1/2 * d^2 / R^2 + ...)
>
> now
>
> d / R = 1 * e-2 / (384403 * e3)
> 	~= 3 * e-11
>
> so the d^2 / R^2 correction will be very small, and won't effect the determination.  So we now have
>
> sin (theta) ~= d / R
>
> This will be a very small angle.  The next approximation to make is for small angles
>
> sin (theta) ~= theta + ...
>
> leaving us with
>
> theta ~= d / R
>
>
> To be approximate, assume the precision you need is equal to the size of the dime.  This means you need an precision of
>
> d theta ~= d/R ~= 3 * e-11 ( = 3 * 10^{-11} if you aren't familiar with the "e" notation)
>
> this is the minimum precision you would need in both the "x" and "y" direction to accurately hit the dime on the moon with your laser (at its average distance).
>
> Corrections to this estimate will come from the fact that the moon's radius is ~1737 km and the earth's radius is ~6370 km, so you are actually this much closer (R is this much smaller).
>
> Of course both the earth is spinning and the moon is moving relative to us, so you would have to account for those extra corrections as well.
>
>
> Hope that wasn't too much info,
>


Of course not.  I enjoyed it.  However, don't you need to work 
divergence in, as per wikipedia, "...At the Moon's surface, the beam is 
only about 6.5 kilometers (four miles) wide[6] and scientists liken the 
task of aiming the beam to using a rifle to hit a moving dime 3 
kilometers (two miles) away."

(http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_experiment)

Emile