Comments on: π and ∞ http://www.cryptohedonology.com/2007/11/12/%e2%88%8f-and-%e2%88%9e/ Has a polemic ever run over your memeplex? Sun, 07 Dec 2008 21:17:47 +0000 hourly 1 http://wordpress.org/?v=3.0.1 By: Cryptohedonologist http://www.cryptohedonology.com/2007/11/12/%e2%88%8f-and-%e2%88%9e/comment-page-1/#comment-56 Cryptohedonologist Sun, 02 Dec 2007 02:02:04 +0000 http://www.cryptohedonology.com/2007/11/12/%e2%88%8f-and-%e2%88%9e/#comment-56 Hey Josh! Good question. A modest number of sig figs of pi will usually do the job for most worldly applications (including high precision surveying). Computer programs and calculators usually use between 9 and 16 sig figs for pi (or more). It is unlikely you are limited by rounding pi in a computer program or calculator, but a quick use of pi=3.14 in the field could be a problem. To see the rough effects of rounding pi in a survey, we’ll consider a 100 meter diameter circular sample property. If we included radial error too, we would have to fold that into the analysis below. But here are some rounding errors when calculating the circumference for different values of pi assuming we know the radius exactly: pi=3 (one sig fig), error=about 14.16 meters in the circumference (ouch!) or about 9735 arcseconds for any radius. If you use pi=3 and are seeing lots of errors, well there's your problem! As a bit of trivia, the <a href="http://www.straightdope.com/classics/a3_341.html" rel="nofollow">state of Indiana actually tried (unsuccessfully) to pass a law in 1897</a> which “declared” pi to be valid to only a couple sig figs. It is an <a href="http://mysite.du.edu/~jcalvert/humor/pi.htm" rel="nofollow">urban myth they were successful in decreeing pi=3</a>, but the thought is still amusing. It is based on the bible passage <a href="http://www.skepticsannotatedbible.com/1kg/7.html" rel="nofollow">I Kings 7:23</a>: "And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits; and a line of thirty cubits did encompass it round about." Pretty amusing stuff. Apparently god thinks pi=3; so much for intelligent design. pi=3.142 (4 sig figs), error=about 0.041 meters in the circumference or about 26.74 arcseconds for any radius. Still not good enough for your work. pi=3.1416 (5 sig figs), error=about 0.7 millimeter error in the circumference or about 0.4823 arcseconds for any radius. This is hitting your 1 arcsecond limit. You should use at least 5 sig figs to get your required angular resolution if you know your radius perfectly. pi=3.1415926536 (11 sig figs), error = roughly a nanometer (atomic distances -- way overkill, but calculators can do it easily). As a fun bit of trivia, about 37 sig figs of pi (3.14159265358979323846264338327950288420) can calculate the circumference of any circle in the UNIVERSE to at least one atom's width! Anyway, hope this gives you some ideas and addresses your questions. Tom Hey Josh!
Good question. A modest number of sig figs of pi will usually do the job for most worldly applications (including high precision surveying). Computer programs and calculators usually use between 9 and 16 sig figs for pi (or more). It is unlikely you are limited by rounding pi in a computer program or calculator, but a quick use of pi=3.14 in the field could be a problem. To see the rough effects of rounding pi in a survey, we’ll consider a 100 meter diameter circular sample property. If we included radial error too, we would have to fold that into the analysis below. But here are some rounding errors when calculating the circumference for different values of pi assuming we know the radius exactly:

pi=3 (one sig fig), error=about 14.16 meters in the circumference (ouch!) or about 9735 arcseconds for any radius. If you use pi=3 and are seeing lots of errors, well there’s your problem! As a bit of trivia, the state of Indiana actually tried (unsuccessfully) to pass a law in 1897 which “declared” pi to be valid to only a couple sig figs. It is an urban myth they were successful in decreeing pi=3, but the thought is still amusing. It is based on the bible passage I Kings 7:23: “And he made a molten sea, ten cubits from the one brim to the other; it was round all about, and his height was five cubits; and a line of thirty cubits did encompass it round about.” Pretty amusing stuff. Apparently god thinks pi=3; so much for intelligent design.

pi=3.142 (4 sig figs), error=about 0.041 meters in the circumference or about 26.74 arcseconds for any radius. Still not good enough for your work.

pi=3.1416 (5 sig figs), error=about 0.7 millimeter error in the circumference or about 0.4823 arcseconds for any radius. This is hitting your 1 arcsecond limit. You should use at least 5 sig figs to get your required angular resolution if you know your radius perfectly.

pi=3.1415926536 (11 sig figs), error = roughly a nanometer (atomic distances — way overkill, but calculators can do it easily).

As a fun bit of trivia, about 37 sig figs of pi (3.14159265358979323846264338327950288420) can calculate the circumference of any circle in the UNIVERSE to at least one atom’s width!

Anyway, hope this gives you some ideas and addresses your questions.
Tom

]]> By: jmartin http://www.cryptohedonology.com/2007/11/12/%e2%88%8f-and-%e2%88%9e/comment-page-1/#comment-55 jmartin Sat, 01 Dec 2007 23:49:54 +0000 http://www.cryptohedonology.com/2007/11/12/%e2%88%8f-and-%e2%88%9e/#comment-55 NOT SO PERFECT PI? Firstly I’ll have to apologize because I feel I’m about to dumb down this discussion of pi. As a surveyor, we utilize pi in calculating curves (is there another way). Now, we (surveyors) try to keep our “math” in the realm of trig and geometry, and try to avoid any “higher” math – “Keep it simple”-- Although it seems that every once in a while we have to delve into analyzing what the numbers are actually doing. For the most part surveyors would rather go hiking than analyze numbers. A common task of the Surveyor is in creating “closed” geometric figures, using Cartesian Coordinate Systems, such as a property boundary or a closed traverse. These figures need to be portrayed on a map which is to become a public document and potentially used by the public and/or retraced by other surveyors. The units of these maps have a legal standard of tens and hundredths of a survey foot(00.00’) and seconds of a degree(00°00’00”). We dimension curves using 3 pieces of info. Radius, Delta and Length. A problem occurs when the geometric figure does not close (to within legal standards-hundredths or seconds). Sometimes this is human error which means you’re just a bad surveyor. But this mis-closure often happens when there is a curve involved in the figure. I am not a mathematician but I get the feeling this has something to do with the significant figures and/or the precision of pi (if that would be applicable). What in the world is pi doing to my map? So, I know what the problem is but not really the answer. Are the seconds of a degree more precise than the hundredths of an arc length? We chalk this up to a rounding problem. Are the characteristics of a curve not “precise” due to pi? Or can I blame this on the various programs that are trying to build curves using pi incorrectly. (It’s probably just me). NOT SO PERFECT PI?

Firstly I’ll have to apologize because I feel I’m about to dumb down this discussion of pi. As a surveyor, we utilize pi in calculating curves (is there another way). Now, we (surveyors) try to keep our “math” in the realm of trig and geometry, and try to avoid any “higher” math – “Keep it simple”– Although it seems that every once in a while we have to delve into analyzing what the numbers are actually doing. For the most part surveyors would rather go hiking than analyze numbers.

A common task of the Surveyor is in creating “closed” geometric figures, using Cartesian Coordinate Systems, such as a property boundary or a closed traverse. These figures need to be portrayed on a map which is to become a public document and potentially used by the public and/or retraced by other surveyors. The units of these maps have a legal standard of tens and hundredths of a survey foot(00.00’) and seconds of a degree(00°00’00”). We dimension curves using 3 pieces of info. Radius, Delta and Length.

A problem occurs when the geometric figure does not close (to within legal standards-hundredths or seconds). Sometimes this is human error which means you’re just a bad surveyor. But this mis-closure often happens when there is a curve involved in the figure. I am not a mathematician but I get the feeling this has something to do with the significant figures and/or the precision of pi (if that would be applicable).

What in the world is pi doing to my map?

So, I know what the problem is but not really the answer. Are the seconds of a degree more precise than the hundredths of an arc length? We chalk this up to a rounding problem. Are the characteristics of a curve not “precise” due to pi? Or can I blame this on the various programs that are trying to build curves using pi incorrectly.
(It’s probably just me).

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