Wikipedia:Reference desk/Archives/Mathematics/2009 August 5

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August 5

A question about geographical coordinates and significant digits

There are a number of different ways in which a geographical position can be identified using coordinates. For example the location of Mount Whitney in California is listed in different ways by different sources.

The Geographic Names Information System (GNIS) uses two formats:

36.5782684, -118.2934244 (i.e. 36.5782684°N, 118.2934244°W) — Decimal Degrees (DD)

363442N, 1181736W (i.e. 36°34′02″N, 118°17′73″W) — Degrees, Minutes, Seconds (DMS)

The U.S. National Geodetic Survey (NGS) uses the following format:

36 34 42.89133(N) 118 17 31.18182(W)

Others use:

N 36° 05.885 W 115° 05.583 — Degrees, Minutes (DM)

For example when converting DMS to DD format how may decimal places are significant:

36°36′36″ then (36*3600 + 36*60 + 36)/3600 = 36.61°

How can I determine how many decimal places are significant when converting (e.g. from NGS data) to DD format. Is there some general rule I can follow. What about the inverse function (i.e. converting DD to DMS). Assume 36° implies >= 35.5° and < 36.5° etc. --droll [chat] 00:18, 5 August 2009 (UTC)

Find the precision of the initial value and convert it into the new units. For example 36°34′02″ implies uncertainty on the order of 1". Converting 1" to degrees gives you .000277°. You don't want the precision of your new value to be any higher than that of the initial value, so round up your uncertainty to the nearest power of 10 (.001° in this case) and get rid of any digits beyond that one. In this case the result would be 36.567°. Rckrone (talk) 02:59, 5 August 2009 (UTC)

Sorry if being a bit off-topic: When comparing geodesic data from two different sources, it is also convenient to verify that coordinates are expressed with the same reference datum. Pallida  Mors 05:38, 5 August 2009 (UTC)

True, I use decimal degrees with 4 decimal places in any maps and give a selection of maps a user could use. They're not too badly correlated now but in the past you could get to completely the wrong building with different maps. By the way I try to indicate the entrance rather than the body of a building. Dmcq (talk) 09:33, 5 August 2009 (UTC)

By the way I'd convert the precision up rather than down if comparing things, then subtract the two and see if the difference is smaller than what you're checking for. Who is going to actually read the numbers? Normally they'd just go to another computer. Dmcq (talk) 09:42, 5 August 2009 (UTC)

Mandelbrot set

If one iteration is done on the whole Mandelbrot set, is the result the whole of the set or just a subset? --Masatran (talk) 09:43, 5 August 2009 (UTC)

Clarification: Let the points that constitute the Mandelbrot set be the set S1. Take each point of this set and apply the iteration just once. Let the resultant set of points be S2. From the definition, S2 is a subset of S1. Is S2 a proper subset, or is it the whole of S1? --Masatran (talk) 05:45, 6 August 2009 (UTC)

An iteration of what? (Isn't that you mean the Julia set of a map, instead?) --pma (talk) 10:14, 5 August 2009 (UTC)

Did you mean to take test the iterated condition only once - eg using

zn+1 = zn2 + c

Only to go as far as z₁

That I think would be a subset, or more specifically a single quadratic equation.83.100.250.79 (talk) 13:10, 5 August 2009 (UTC)

And what do you mean? I can't understand neither of you. Can you formulate your question precisely?--pma (talk) 14:38, 5 August 2009 (UTC)

See Mandelbrot set first paragraph - using the formula - one iteration

Mathematically, the Mandelbrot set can be defined as the set of complex values of c for which the orbit of 0 under iteration of the complex quadratic polynomial z_n+1 = z_n² + c remains bounded.

The first iteration (for a general point [x,y]) ie x+iy yields a quadratic formula for the boundary condition, the second iteration a quartic, and so on...

Actually I think it's a subset of a subset - the first subset being a specific value of c, the second subset being the set of iterations forming the first subset. The questioner mentioned "iterations" - one obvious use of iterations in the mandelbrot set is that decribed above.HappyUR (talk) 15:07, 5 August 2009 (UTC)

Who knows what the questioner meant. At the moment I see no question...--pma (talk) 15:31, 5 August 2009 (UTC)

To construct the Mandelbrot set you start with z₀ = 0, then iterate z_n+1 = z_n² + c. If |z_m| is greater than 2 for some m then you can be sure that z_n diverges to infinity for this value of c, so c is not in the Mandelbrot set. So after one iteration you throw out the points with |c| > 2 and you are left with a circle radius 2. After two iterations you throw out points such that |c² + c| > 2 and you are left with a cardioid (I think) that intersects that real axis at c = −2 and c = 1. At each iteration you throw out more and more points - so you are left with a subset of the points you had one iteration before, but this is a superset of the Mandelbrot set itself. Gandalf61 (talk) 16:00, 5 August 2009 (UTC)

Yes that's what I was thinking, and you are right it is a superset (I think) (not subset) since some points will not 'escape' until n is big (for a given c).

Not a cardioid though - a distorted dumbell (peanut) shape with one reflection symmetry - I think the number of inflexions increase by 2 (or maybe doubles) for the boundary with each iteration. maybe/maybe not - a quartic in x and y (from c=x+iy) anyway83.100.250.79 (talk) 16:15, 5 August 2009 (UTC)

You are right - it's not a cardioid. In polar co-ordinates it is the region inside the curve $r^{4}+2r^{3}\cos(\theta )+r^{2}=4$ Gandalf61 (talk) 10:19, 6 August 2009 (UTC)

When the moment comes of interpreting meaningless questions, the RD\M becames like a school of Rabbinic studies in Yiddish jokes. --pma (talk) 22:21, 5 August 2009 (UTC)

Or the Three Stooges as people fall over each other and get into fights.83.100.250.79 (talk) 23:31, 5 August 2009 (UTC)

twice as wide

We have a question in the german wiki reference which can only be answered by a native speaker. How would you understand the sentence: "Now find at least three positions where you can put the light and the card to make a shadow twice as wide as the card." Do they mean a shadow width twice as wide as the edge width of the card or twice as wide as the whole card? --Ian Dury^{Hit me} 15:50, 5 August 2009 (UTC)

I'm not sure what you mean by "edge width". Do you mean "thickness"? In which case, it's not what is meant - width and thickness are completely different words in English. If they meant thickness, they would have said thickness. --Tango (talk) 18:10, 5 August 2009 (UTC)

Clever bloke... In fact I can see now what my unreflectet reproduction from someones expression at our reference desk did. What he meant with "edge width" is the length of the edge... Anyway, thank you very much. --Ian Dury^{Hit me} 18:35, 5 August 2009 (UTC)

I still don't understand... surely the length of the short edge is the same as the width of the card, and the length of the long edge is the same as the length of the card. --Tango (talk) 19:36, 5 August 2009 (UTC)

Looking at the original de: thread, the OP there asked whether the expression can be used to mean a card with doubled area ("Fläche") as well as a card with doubled width. Which, of course, it cannot. —JAO • T • C 19:54, 5 August 2009 (UTC)

Gravity and Mass

Question already posted on the RD/Science, where it fit better indeed. It should maybe be removed from here... --pma (talk) 23:10, 5 August 2009 (UTC)

Indeed. See Wikipedia:Reference desk/Science#Gravity and_Mass. -hydnjo (talk) 00:40, 6 August 2009 (UTC)