Re: continued fraction program

Jun 20, 2009 68 Replies

I remember hearing that story decades ago. However, looking at the current maps showing the difference between the geoid and the reference ellipsoid, it is hard to think about pear.

To be pear-shaped, there would have to be a deep depression at Antarctica, which does not exist, according to current measurements.

Looking only at geometry, a bulge before or around the target would cause a premature impact point only if the missile approached the target in a _very_ shallow angle. A 100 m bulge and 3 degree approach angle (e.g. airport ILS approach) would cause an impact 2 km prematurely.

In my understanding the ballistic missiles of those days approached the target nearly vertically, so the bulge would affect the impact point only a few meters, although the impact would happen perhaps 100 ms earlier than calculated.

The reason for the "bulge" is that the mass below the Earth's surface is not evenly distributed, i.e. there are low density materials (and hence less mass and less gravitation) below the bulges. The varying gravitational force along the whole flight path will cause deviations in the flight path of the missile.

This is exactly what the French Geodesic Mission did in the 18th century :-). However, you can only determine the diameter of the Earth measured on the equator and on the poles, but you can not map local variations in the gravitational field by that method.

Fortunately the two triangulation networks used by the FGM (one in Ecuador and the other in Scandinavia) happened to be on two separate "bulges", compensating for some of the errors. However, if one network had been on a bulge and the other in a depression, the error would have been larger.

Paul

This is easier said than done.

If you only perform these measurements over small distances, or only have limited accuracy, you only end up with the parameters for a spheroid which fits the region being surveyed.

This is why you have multiple geodetic datums (for whatever reason, no-one uses the latin plural "data" in this context), each of which provides a spheroidal approximation to the geoid which is a close fit for a specific area.

The following diagram shows the location of a specific coordinate in a variety of datums:

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[The Tokyo datum is so far out as Japan lies on a relatively steep slope in the geoid, so the best-fit ellipsoid to that area is rather skewed.]

Tying these together requires surveying large areas with a high degree of accuracy. The first major attempts at tying regional datums together were the traverses conducted using the SHORAN and HIRAN distance measuring systems starting in the 1950s.

Obtaining this level of accuracy over long distances simply wasn't feasible until quite recently.

More information:

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You mean, 2 parts in 10^6? Otherwise the US mile would be about 200 feet longer than the British one.

Bartc

Google "usenet complaint form". Enjoy.

John

Learning to cooperate with the Usenet community is somewhat more useful than this silly childish attitude.

Some useful links on quoting:

[mail]: Chuck F (cbfalconer at maineline dot net) [page]: Try the download section.

I do. You should read up about them, before commenting to the net. CF is Continued Fraction obviously.

It is highly advisable to understand CF's before attempting approximation programs. I doubt the c-program takes advantage.

Striking fact about CF's. If you need a value to N digits, there is an approximation A/B where both A and B are limited to N/2 digits.

This has an application in 1 bit D/A converters. With 16 bit timers you can have a 32 bit precision average DC output for any value. (Provided your clock is jitter free enough to accomodate this.) (Oops giving away a patent ...)

Groetjes Albert

--

Albert van der Horst, UTRECHT,THE NETHERLANDS Economic growth -- being exponential -- ultimately falters. albert@spe&ar&c.xs4all.nl &=n http://home.hccnet.nl/a.w.m.van.der.horst

You can never "give away" a patent which requires disclosure such that anybody "skilled in the art" has the necessary knowledge.

... snip ...

You obviously didn't read either the ratapprx program or my message. The uncertainty was about "what you mean". Note the 'you'. ratapprx doesn't do approximations, it does exact calculations (within the limits of the floating point system). Each answer is the next best approximation with a larger numerator. Thus no need of convergents.

I saw no sign of continued fractions. I saw no need of approximations. Thus I asked questions and avoided snotty comments.

[mail]: Chuck F (cbfalconer at maineline dot net) [page]: Try the download section.

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