What bike computers measure elevation?

Booker C. Bense said:

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In article <[email hidden]>,

bill 'just use comcast' said:

Quoted message said:

The information is there for the taking since a GPS uses the
triangulation of information from 3 satellites meaning you can be at one
and only one point in space (hopefully on the Earth).

The GPS's that display altitude generally
use a internal barometer to do it. It is only very recently
with the addition of WAAS signal that GPS can have any
reasonable accuracy wrt altitude.

No, all the handheld GPS receivers I've ever used or seen have had the
ability to display altitudes, but very few of them have had barometric
sensors. And while WAAS has improved accuracy, it is by no means
necessary to get good enough altitude data to be useful for bicycling
or similar applications.

Quoted message said:

Even then most GPS's can
not approach the accuracy of a recently calibrated barometer.

Actually, most barometers only report atmospheric pressure and not
altitude at all, but I presume you mean pressure-based altimeters.
Then it depends on the specifications of the particular instrument and
how recently it was calibrated compared to weather-related changes in
the pressure.

The GPS model I've been using as my bicycle cyclometer for the last 7
years is a Garmin eMap, which is a receiver that does *not* support
WAAS. Studies I've done with it over the years have shown that the
altitude indications are within 35' over 95% of the time when the unit
has reasonable reception (at least 5 satellites in view). This is
consistent with what was found by David Wilson as reported at:
http://users.erols.com/dlwilson/gpsvert.htm
He gives an RMS accuracy of the eMap of 6 meters which corresponds to a
95% accuracy of just under 12 meters, or 39'. WAAS-equipped receivers
could improve on that accuracy by up to a factor of two when the
geosynch. satellites can be seen (much more frequently the case now
since the new ones have come on line).

Here's an example of the data the eMap gives on actual bike rides:
http://home.comcast.net/~prathman/Image33.gif
This shows two rides of the same course on two different days. One
ride is shown in red and the other in blue with the horizontal plots of
position superimposed on each other and the elevation plots below.
Small differences can be seen in the two plots due to measurement
variations, but they are in good agreement in general and are also in
good agreement with 1:24000 USGS topo maps with which I've compared
them.

Quoted message said:

If you want altitude data, the Garmin Edge 205/305 is a very nice if
pricy cyclometer.

But unfortunately does not provide any mapping capability to show you
where you are and what roads will take you to your destination or
useful information on a tour like where nearby restaurants or grocery
stores are located.
So I find other handheld GPS models to be far more useful - and they
also provide altitude data plus typical cyclometer functions (odometer,
total average and moving average speeds, etc.)

In article <[email hidden]>,

peter said:

Booker C. Bense said:

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In article <[email hidden]>, bill

just use comcast' said:

>
The information is there for the taking since a GPS uses the
triangulation of information from 3 satellites meaning you can be
at one and only one point in space (hopefully on the Earth).

The GPS's that display altitude generally use a internal barometer
to do it. It is only very recently with the addition of WAAS signal
that GPS can have any reasonable accuracy wrt altitude.

No, all the handheld GPS receivers I've ever used or seen have had
the ability to display altitudes, but very few of them have had
barometric sensors. And while WAAS has improved accuracy, it is by no
means necessary to get good enough altitude data to be useful for
bicycling or similar applications.

Precisely. In our brevet series in 2005, two riders used GPS units for
guidance as well as to record information about the rides. One unit
measured 10,500 feet of climbing and the other 9,817 over the 191 miles
on the day. That's quite a margin of error- a difference of 683 feet or
7%. By comparison, the mapping software predicted total climbing of
11,144 feet. The mapping software estimated at least 10% more climbing
than any of the GPS units measured on each of the four brevets.

Personally I have a much more calibrated gauge of climbing: flat,
rolling, hilly, and wowser. I don't really need to know the numbers.
It's just a bike ride. I'm generally more concerned with the fun than
the stats.

Quoted message said:

I guess you don't understand what "within their resolution" means,
which doesn't get any better if that resolution is poor. It just
collects greater errors.

Are we speaking of "resolution" or "accuracy?" A typical GPS unit has a
resolution of 1ft/1 meter. As to accuracy, I think if you had actually
used a GPS receiver you might have a substantially different
understanding. You have not used one, have you?

Quoted message said:

It's still not useful for bicycling altimeters.

Perhaps you should preface your statement with "I was under the
erroneous impression that..."
I'm not sure what your definition of "useful" is, but please don't be
making broad erroneous generalizations about the usefulness or accuracy
of something you appear to have no hands-on familiarity with.

See this example of a GPS "track" recently recorded on a simple
handlebar mounted consumer GPS receiver with the profile displayed
using Topofusion software.
http://members.aol.com/dirtroadie/bike/Cliffs_Profile.jpg
While there is some variation to be seen, there are no dramatic errors.

And here's the top portion of the the same track projected (via the
same software) onto the corresponding USGS topo.
http://members.aol.com/dirtroadie/bike/CliffsTopDetail.jpg

Note the elevation on the topo as compared to the GPs profile. I'm not
sure what you consider to be sufficient resolution or accuracy but the
GPS measurement and the map indication are virtually identical. (I am
actually less impressed by the GPS accuracy than by the accuracy of
ancient USGS topographic maps, which generally predate any form of
electronic technology.)
And the software (using only GPS data) accurately calculates the "total
climbing" to be essentially the same as that which is found by merely
subtracting the base elevation from that of the top, which accurately
reflects the steady nature of this particular climb.

Quoted message said:

Which instrument do you suggest as a bicycle altimeter and speedometer/
odometer?

I have primarily used various Garmin units. My favorite for cycling
use is the Foretrex 201, primarily for its convenient and compact size
and shape. It ofers all the basic functions of a typical cyclometer
and works well with the aforementioned TopoFusion software.

DR

Tim McNamara said:

In article <[email hidden]>,

peter said:

Booker C. Bense said:

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In article <[email hidden]>, bill
<"just use comcast", "you know the drill for no spam bots."> wrote:
> The information is there for the taking since a GPS uses the
> triangulation of information from 3 satellites meaning you can be
> at one and only one point in space (hopefully on the Earth).
The GPS's that display altitude generally use a internal barometer
to do it. It is only very recently with the addition of WAAS signal
that GPS can have any reasonable accuracy wrt altitude.

No, all the handheld GPS receivers I've ever used or seen have had
the ability to display altitudes, but very few of them have had
barometric sensors. And while WAAS has improved accuracy, it is by no
means necessary to get good enough altitude data to be useful for
bicycling or similar applications.

This is why some of the posters ragging on my 'lack of knowledge' have
gotten me testy. I had a friend in 2000 who had a cheap Radio Shack GPS
that showed altitude and it was within 5 feet of the posted altitude on
the welcome sign for the town we were in. I really doubt that WAAS was
on that Radio Shack unit back then.
My comment on barometric based units was actually at least semi-serious.
If your barometer pressure drops and it shows you going rapidly uphill
as a result you can be fairly sure a sudden storm is going to nail you.
Sailors live 'literally' by paying attention to this and a barometer
based altimeter could conceivably save your life if a thunderstorm was
coming fast. I have seen the effects of a big lightning strike and you
'DON'T' want to get hit.

Quoted message said:

Precisely. In our brevet series in 2005, two riders used GPS units for
guidance as well as to record information about the rides. One unit
measured 10,500 feet of climbing and the other 9,817 over the 191 miles
on the day. That's quite a margin of error- a difference of 683 feet or
7%. By comparison, the mapping software predicted total climbing of
11,144 feet. The mapping software estimated at least 10% more climbing
than any of the GPS units measured on each of the four brevets.

Personally I have a much more calibrated gauge of climbing: flat,
rolling, hilly, and wowser. I don't really need to know the numbers.
It's just a bike ride. I'm generally more concerned with the fun than
the stats.

Just have fun riding, or get serious and rig a Blackberry or something
for a data recorder. The possibilities are endless but depend on how
deep your pockets are lined with money.

--
Bill (Sleepless biker) Baka

In article
<[email hidden]>,
"Buck" <[email hidden]> wrote:
[...]

Quoted message said:

6. Many consumer-grade GPS units do not report much of the information
available in or calculable from the primary satellite signal. Measures
of error such as HDOP or PDOP are often left out. Even something as
simple as whether or not the device is receiving a differential
correction signal is often left out. It is not surprising that many
cell phones don't report an estimated elevation.

Buck, would you direct me to a document or two or three
on the web that discusses the different things a GPS
receiver can do? Also a document on the theory and
methods. Highly technical documents are good for me,
and I know how to distinguish precision from accuracy.
I am shopping for a general purpose receiver, but
prefer to get all the options that make sense for me.

Thank you.

--
Michael Press

Michael Press said:

would you direct me to a document or two or three
on the web that discusses the different things a GPS
receiver can do? Also a document on the theory and
methods. Highly technical documents are good for me,
and I know how to distinguish precision from accuracy.
I am shopping for a general purpose receiver, but
prefer to get all the options that make sense for me.

Here are some to get you started.

A good overview of the theory of operation of GPS is at the Trimble
website:
http://www.trimble.com/gps/index.shtml

A website with quite a few reviews of different GPS receivers and also
with extensive lists of additional GPS resources is:
http://gpsinformation.net

Technical details on the functions, signal formats, etc. are available
in this document:
http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf

peter said:

Michael Press said:

would you direct me to a document or two or three
on the web that discusses the different things a GPS
receiver can do? Also a document on the theory and
methods. Highly technical documents are good for me,
and I know how to distinguish precision from accuracy.
I am shopping for a general purpose receiver, but
prefer to get all the options that make sense for me.

Here are some to get you started.

A good overview of the theory of operation of GPS is at the Trimble
website:
http://www.trimble.com/gps/index.shtml

This would have been good except for the plug in requirement, which
never did work with my Firefox.

Quoted message said:

A website with quite a few reviews of different GPS receivers and also
with extensive lists of additional GPS resources is:
http://gpsinformation.net

Good site for buyers.

Quoted message said:

Technical details on the functions, signal formats, etc. are available
in this document:
http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf

Pretty good but looked like an old document. It did confirm my
suspicions about 3d wanting a fourth satellite in the loop for elevation
but that only resolves 2 data points into one. There should be no need
for a civilian GPS unit to need the fourth satellite since I know of no
biker riding at an altitude of 400 miles or more. It should be blatantly
obvious that a civilian, and most military personnel except for
astronauts will be on the data point within ten miles of sea level.
On the flip side of that comment, astronauts don't have a latitude or
longitude problem. As to the civilian accuracy of 100 meters, that is
wrong for today, or we would not have the 'ugh' Tom Tom units that can
tell what street you are on.

--
Bill (Sleepless biker) Baka

bill said:

peter wrote:

Quoted message said:

Quoted message said:

A good overview of the theory of operation of GPS is at the Trimble
website:
http://www.trimble.com/gps/index.shtml

This would have been good except for the plug in requirement, which
never did work with my Firefox.

Quoted message said:

Quoted message said:

Technical details on the functions, signal formats, etc. are available
in this document:
http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf

Pretty good but looked like an old document.

Yes, but still the best source I've found for the basic system details.
Most newer documents don't go into detail on the formats and signal
characteristics. Of course one does have to realize that some changes
have occurred since '96 - especially turning off the SA dithering and
the implementation of WAAS and other SBAS systems. Further changes
such as additional frequency channels are also being incorporated into
the latest GPS satellites, but not enough yet for them to be used by
our receivers.

Quoted message said:

It did confirm my
suspicions about 3d wanting a fourth satellite in the loop for elevation
but that only resolves 2 data points into one. There should be no need
for a civilian GPS unit to need the fourth satellite since I know of no
biker riding at an altitude of 400 miles or more.

You keep misinterpreting the reason why there's a need for the 4th
satellite signal. Too bad you can't get the Trimble tutorial above
since it goes into a more detailed explanation. It has nothing to do
with distinguishing between only two possible points, but is essential
to get a true 3D fix for any receiver that lacks a precision time
source (i.e. an atomic clock or something directly synched to one).
Receivers that lack atomic clocks need to solve for 4 things: latitude,
longitude, altitude, and time. To solve for 4 variables you need data
from 4 satellites. Now if only 3 satellites can be received, then the
receiver will make do as well as possible by *assuming* that the
altitude hasn't changed since the last fix and will determine the
remaining 3 variables under that assumption.

I've sometimes checked my GPS when I first turn it on upon arrival
after a flight. If it initially just gets a 3 satellite ("2D"😉 fix it
can be off by a few miles since it is still using the last determined
altitude which was likely at 35000' or so while up in the air.
*Assuming* an altitude that's off by miles results in the calculated
position also being off by miles. As soon as the receiver gets
additional satellite signals it corrects the altitude and position -
but until it gets at least 4 satellites it continues to give the wrong
information.

On bike rides the elevation changes are less drastic, but the same
effect occurs under conditions of marginal reception. If only 3
satellites are being received on a climb up a mountain, then the
reported altitude will stay fixed until it suddenly jumps up when the
receiver gets signals from additional satellites. Careful examination
of the tracklog shows that during the time that the reported altitude
was wrong, the position also drifts off from the correct location.

In article <[email hidden]>,

Quoted message said:

Bill Baka said:

Quoted message said:

I'm looking for a bike computer with a barometer that will give me
the total climb on a ride. I had an old Cateye that did this but
Cateye doesn't make such a beast any more. Any suggestions?

Quoted message said:

Anything with a GPS is capable of giving elevation since it will
resolve not only you latitude and longitude, but also you elevation.
This comes in handy for GPS enabled cell phones so the 911 operators
can tell not only where the address is you are calling from but also
what floor you are on. That might be more than you want to know,
but the stuff is out there if you want to pay for it.

GPS does not give reasonable vertical resolution for bicycling.

That's for sure. I have a Garmin Etrex Legend that I use for kayaking.
I stay within a couple of feet of sea level but the elevation profile it
spits out shows variations of 40 feet or more.

peter said:

bill said:

peter wrote:

Quoted message said:

Quoted message said:

A good overview of the theory of operation of GPS is at the Trimble
website:
http://www.trimble.com/gps/index.shtml

This would have been good except for the plug in requirement, which
never did work with my Firefox.

Quoted message said:

Quoted message said:

Technical details on the functions, signal formats, etc. are available
in this document:
http://www.navcen.uscg.gov/pubs/gps/gpsuser/gpsuser.pdf

Pretty good but looked like an old document.

Yes, but still the best source I've found for the basic system details.
Most newer documents don't go into detail on the formats and signal
characteristics. Of course one does have to realize that some changes
have occurred since '96 - especially turning off the SA dithering and
the implementation of WAAS and other SBAS systems. Further changes
such as additional frequency channels are also being incorporated into
the latest GPS satellites, but not enough yet for them to be used by
our receivers.

Quoted message said:

It did confirm my
suspicions about 3d wanting a fourth satellite in the loop for elevation
but that only resolves 2 data points into one. There should be no need
for a civilian GPS unit to need the fourth satellite since I know of no
biker riding at an altitude of 400 miles or more.

You keep misinterpreting the reason why there's a need for the 4th
satellite signal. Too bad you can't get the Trimble tutorial above
since it goes into a more detailed explanation. It has nothing to do
with distinguishing between only two possible points, but is essential
to get a true 3D fix for any receiver that lacks a precision time
source (i.e. an atomic clock or something directly synched to one).

I won't argue that since a Cesium clock and a hand held device aren't
going together any time soon.

Quoted message said:

Receivers that lack atomic clocks need to solve for 4 things: latitude,
longitude, altitude, and time. To solve for 4 variables you need data
from 4 satellites. Now if only 3 satellites can be received, then the
receiver will make do as well as possible by *assuming* that the
altitude hasn't changed since the last fix and will determine the
remaining 3 variables under that assumption.

That means that back in 2000 the Radio Shack unit actually hit 4
satellites to get an accurate elevation reading. Surprising.

Quoted message said:

I've sometimes checked my GPS when I first turn it on upon arrival
after a flight. If it initially just gets a 3 satellite ("2D"😉 fix it
can be off by a few miles since it is still using the last determined
altitude which was likely at 35000' or so while up in the air.

Doesn't that hit you as being a kid with a new toy, using it to
determine your altitude? Of course if you are cruising at 35,500 that
something is not good, but do you really want to know?

Quoted message said:

*Assuming* an altitude that's off by miles results in the calculated
position also being off by miles. As soon as the receiver gets
additional satellite signals it corrects the altitude and position -
but until it gets at least 4 satellites it continues to give the wrong
information.

Simple. Quit playing with it on the plane. Kind of like me using the on
board phone on a commuter flight to LA with my phone card and forgetting
that it was about $3 a minute. I got the bill and quit that in a hurry.
No more chatting with the wife at 30,000 feet.

Quoted message said:

On bike rides the elevation changes are less drastic, but the same
effect occurs under conditions of marginal reception. If only 3
satellites are being received on a climb up a mountain, then the
reported altitude will stay fixed until it suddenly jumps up when the
receiver gets signals from additional satellites. Careful examination
of the tracklog shows that during the time that the reported altitude
was wrong, the position also drifts off from the correct location.

The unfortunate conclusion is that we need about twice as many
satellites to ensure coverage in adverse conditions, which may not be
practical unless 6 satellites at once can be launched and then the
orbits juggled by on board booster rockets once they are up there.
Launching another constellation with single satellite rocket launches
would cost more than I feel like paying tax money for.

Quoted message said:

--
Bill (Sleepless biker) Baka

Harold Burton said:

In article <[email hidden]>,

Quoted message said:

Bill Baka said:

> I'm looking for a bike computer with a barometer that will give me
> the total climb on a ride. I had an old Cateye that did this but
> Cateye doesn't make such a beast any more. Any suggestions?
Anything with a GPS is capable of giving elevation since it will
resolve not only you latitude and longitude, but also you elevation.
This comes in handy for GPS enabled cell phones so the 911 operators
can tell not only where the address is you are calling from but also
what floor you are on. That might be more than you want to know,
but the stuff is out there if you want to pay for it.

GPS does not give reasonable vertical resolution for bicycling.

That's for sure. I have a Garmin Etrex Legend that I use for kayaking.
I stay within a couple of feet of sea level but the elevation profile it
spits out shows variations of 40 feet or more.

Are you going between mountains that can block a satellite or cause
multipath problems? It can guess with 3 birds in sight but needs 4 for a
really accurate fix. I suspect the military is happy with it the way it
is just to send the smart bombs down at x-y and not worry about z (alt).

--
Bill (Sleepless biker) Baka

Michael Press said:

Buck, would you direct me to a document or two or three
on the web that discusses the different things a GPS
receiver can do? Also a document on the theory and
methods. Highly technical documents are good for me,
and I know how to distinguish precision from accuracy.
I am shopping for a general purpose receiver, but
prefer to get all the options that make sense for me.

Michael,

Here are some links for you:
http://www.cmtinc.com/gpsbook/index.htm
http://users.erols.com/dlwilson/gps.html
http://edu-observatory.org/gps/gps_accuracy.html

As to what is available on consumer-grade GPS receivers, you might take
a look at this site:
http://www.gps4fun.com/main_compare.php

If you are interested in some advanced mapping and don't like the
software they offer, you can use a program called DNR Garmin to
download the GPS data into a format compatible with a mapping program
like ArcReader or FGIS. Both of these will easily add your GPS data
onto a topo map or aerial photography.

-Buck

Harold Burton said:

In article <[email hidden]>,
[email hidden] wrote:

Quoted message said:

Quoted message said:

GPS does not give reasonable vertical resolution for bicycling.

Quoted message said:

That's for sure. I have a Garmin Etrex Legend that I use for kayaking.
I stay within a couple of feet of sea level but the elevation profile it
spits out shows variations of 40 feet or more.

Guess it depends on what you're expecting. You view it as almost
worthless since it sometimes gives variations of 40' while I view it as
great that I finally have a tool that'll almost always give me the
altitude of the mountains I climb to within 40'. It's certainly much
better than the barometric altimeters that I used before (incl. the
Avocet 50) which had much larger errors when there were weather changes
since the last accurate recalibration - or even with constant weather
but conditions that differed from the "standard atmosphere'"
assumptions used to convert pressure readings into altitude values.

Buck said:

Dan said:

With USGS topographical maps
available on line these days, it is easy to check elevations too.

It depends on how accurate you want. The maps are classified according
to the scale at which the data was captured and the extent which the
map covers.

The vertical standard for USGS 1:24,000 topographic maps is 90% of all
points tested must be within 1/2 of one contour line interval.

The horizontal standard for USGS 1:24,000 topographic maps is 90% of
all points tested must be within 40 feet of their true location.

In many areas, 1:24,000 maps are available. The contour intervals vary
by location. In some places, they are as frequent as every ten feet, in
others it can be much more coarse.

The next scale available is 1:100,000 and after that, 1:250,000. These
are, of course, much less accurate than the 1:24,000 maps.

-Buck

Geez, you sound like a USGS guy. My Dad was a field man for the survey.
I spent many summer days helping him lug a theodolite up peaks and
recording angles in little books while he peered through the scope.

Anyway, The accuracy provided by topos is good enough to verify my bike
altimeter readings.

Roger Zoul said:

Buck wrote:
::
:: Bill, you are showing your ignorance. Here are a few points:
::

Thanks for the info, Buck!

Only on Planet Baka. The funny part is Billy's follow up "excuse
post."

Harold Burton said:

In article <[email hidden]>,

Quoted message said:

Bill Baka said:

>I'm looking for a bike computer with a barometer that will give me
>the total climb on a ride. I had an old Cateye that did this but
>Cateye doesn't make such a beast any more. Any suggestions?

Quoted message said:

Anything with a GPS is capable of giving elevation since it will
resolve not only you latitude and longitude, but also you elevation.
This comes in handy for GPS enabled cell phones so the 911 operators
can tell not only where the address is you are calling from but also
what floor you are on. That might be more than you want to know,
but the stuff is out there if you want to pay for it.

GPS does not give reasonable vertical resolution for bicycling.

That's for sure. I have a Garmin Etrex Legend that I use for kayaking.
I stay within a couple of feet of sea level but the elevation profile it
spits out shows variations of 40 feet or more.

And you think 40 feet is bad? I always have to smile when people say: 'I
climbed the Galibier today and my altitude (GPS or barometric) meter
said 2610 meters instead of the promised 2642 meters'.....

Lou
--
Posted by news://news.nb.nu

the Garmin Fortrex 101 is nice

all sorts of options, including altitude
check the Amazon listing for more info
under $100 [ebay]

waterboy

In article <[email hidden]>,

Lou Holtman said:

Harold Burton said:

In article <[email hidden]>,

Quoted message said:

Bill Baka writes:

>>I'm looking for a bike computer with a barometer that will give me
>>the total climb on a ride. I had an old Cateye that did this but
>>Cateye doesn't make such a beast any more. Any suggestions?

>Anything with a GPS is capable of giving elevation since it will
>resolve not only you latitude and longitude, but also you elevation.
>This comes in handy for GPS enabled cell phones so the 911 operators
>can tell not only where the address is you are calling from but also
>what floor you are on. That might be more than you want to know,
>but the stuff is out there if you want to pay for it.

GPS does not give reasonable vertical resolution for bicycling.

That's for sure. I have a Garmin Etrex Legend that I use for kayaking.
I stay within a couple of feet of sea level but the elevation profile it
spits out shows variations of 40 feet or more.

And you think 40 feet is bad? I always have to smile when people say: 'I
climbed the Galibier today and my altitude (GPS or barometric) meter
said 2610 meters instead of the promised 2642 meters'.....

OCD mumber-collecting data-freaks amuse me :-)

How many of 'em can go for a computerless ride
without suffering withdrawal pangs?

cheers,
Tom

--
-- Nothing is safe from me.
Above address is just a spam midden.
I'm really at: tkeats [curlicue] vcn [point] bc [point] ca

On Fri, 27 Oct 2006 09:49:06 -0700, [email hidden] (Tom Keats)
wrote:

[snip]

Quoted message said:

OCD mumber-collecting data-freaks amuse me :-)

How many of 'em can go for a computerless ride
without suffering withdrawal pangs?

cheers,
Tom

Dear Tom,

Estimates range from 15.2% to 27.3%, but there is considerable debate
about whether the stop-watch function on the wrist-watch should
eliminate many test subjects.

Further data is needed.

Cheers,

Carl Fogel

R Brickston said:

Roger Zoul said:

Buck wrote:
::
:: Bill, you are showing your ignorance. Here are a few points:
::

Thanks for the info, Buck!

Only on Planet Baka. The funny part is Billy's follow up "excuse
post."

Not really. 3 satellite CAN compute elevation but it needs a bit more
computing power than your average hand held device. Mathematically the
intersection of three spheres is two points and it takes just a little
more smarts to know you are on the earth. A 4th satellite makes it
easier for a 'dumb' chip and does add another reference point for a
little more accuracy.
No excuse from me. I just got tired of arguing what should have been
mathematically obvious.

--
Bill (Sleepless biker) Baka