Trigonometry question...

[dougdaniel]

Originally Posted by OldFrog75

Thanks for the information. So if I understand this correctly (always a risky assumption), one minute of latitude on a chart is technically longer than one nm even though common convention is 1:1

To be honest, I have no idea. It is certainly close enough for us humans. As someone pointed out earlier, the earth is not a perfect sphere. So my gut feeling is that some places a minute of longitude is greater than 1852 meters and other places it is less. I just don't know for sure.

I suppose if we knew the exact circumference of the earth in meters and compared that number to 1852 X 360 X 60 we would have some idea. Of course the circumference is usually stated at the equator and not a great circle passing through the poles.

I just looked it up on space.com and at the equator the earth is 40,075km vs 40,008km through the poles. If you multiply it out you'll get 40,003.2 km. So it looks like a nautical mile is a bit short of a minute of longitude on the chart.

The error is only 4.8 km out of 40,008. So I wouldn't worry too much.

[StuM]

Originally Posted by Ericson38

Staying on either side of the equator, where a minute=1 nautical mile:

start coordinates = -1.5 degrees Lat, -1.5 degrees Lon

end coordinates = 1.5 degrees Lat, 1.5 degrees Lon

planar case nm distance =4.2426406 deg (on a flat plate) - 254.7300811534701e+002 nm

rhumb line case nm distance = 4.242398358524743 deg (on a sphere) - 254.7155368645547 nm

great circle nm distance = 4.242398336379212 deg (on a sphere) - 254.7155355349268 nm


difference between rumb and GC = 1.329627789644135e-006 nm (76.12 feet)

Assuming your previous figures are correct, that is not 76.12 feet.
1.329627789644135e-006 nm is about 1/10 in or 1/4 cm.


Also, your results are very precise! But are they accurate?

How many significant figures are there in those answers given the start value only has two, you have eight in an intermediate calculation, and the number of calculations involved?

I doubt that you answer for the rhumb v GC can be expressed more accurately than
254.71554 v 254.71554 i.e. no discernable difference.

[StuM]

Originally Posted by secrabtree

You've an Error here. The answer to your question as you posed it is NEVER.

Of course it is. You clearly missed that fact that I was pointing out the absurdity of applying a specific distance off course per degree. Maybe we need a /sarc smilie

[StuM]

Originally Posted by dougdaniel

To be honest, I have no idea. It is certainly close enough for us humans. As someone pointed out earlier, the earth is not a perfect sphere. So my gut feeling is that some places a minute of longitude is greater than 1852 meters and other places it is less. I just don't know for sure.

I suppose if we knew the exact circumference of the earth in meters and compared that number to 1852 X 360 X 60 we would have some idea. Of course the circumference is usually stated at the equator and not a great circle passing through the poles.

I just looked it up on space.com and at the equator the earth is 40,075km vs 40,008km through the poles. If you multiply it out you'll get 40,003.2 km. So it looks like a nautical mile is a bit short of a minute of longitude on the chart.

The error is only 4.8 km out of 40,008. So I wouldn't worry too much.

Note that "minute of longitude = 1NM" is only reasonably accurate at the equator. By the time you get to either pole, a minute of longitude = 0NM.
(At 45N or 45S, a minute of longitude is approx 40NM.)

It is only for latitude that the relationship remains reasonable constant.

[Robin3]

bugger the maths, what affordable boat sails at 30 degs to the true wind please and at what speed??

[secrabtree]

I think that here is the place to tell a little story that I heard many decades ago about the difference between a mathmetician and an engineer.

Assume that you are 20 feet from a wall. If you move half the distance to the wall with each jump, how many jumps do you have to take to reach the wall?

The mathmetician says "an infinate number" since you only get half way to the wall with each jump.

The engineer sats "let's see. The first jump gets you ten feet from the wall. The second jump gets you five feet from the wall. The third jump gets you 30 inches from the wall. The fourth jump gets you 15 inches from the wall. The next jump gets you 7-1/3 inches from the wall. Your foot is 8 inches long, so you are effectively at the wall."

[secrabtree]

Originally Posted by Robin3

bugger the maths, what affordable boat sails at 30 degs to the true wind please and at what speed??

The same boat that continually sails on a line that is always 90° from the line to the desired location.

[dougdaniel]

Originally Posted by StuM

Note that "minute of longitude = 1NM" is only reasonably accurate at the equator. By the time you get to either pole, a minute of longitude = 0NM.
(At 45N or 45S, a minute of longitude is approx 40NM.)

It is only for latitude that the relationship remains reasonable constant.

I think you may have confused latitude and longitude. The way I remember the difference is that longitude is always long. It is a great circle passing through the poles. So they are the lines that run north - south.

[monte]

That's what he said : p

[David M]

Originally Posted by StuM

So what's would the difference be on a say a 200NM run, given that the radius of the earth is nearly 3500NM?

The answer is not enough to worry about. At 200 NM planar trig works fine.

[OldFrog75]

Originally Posted by dougdaniel

I think you may have confused latitude and longitude. The way I remember the difference is that longitude is always long. It is a great circle passing through the poles. So they are the lines that run north - south.

Pretty sure he's not confused. Most of us measure distances on our charts using the latitude scale because latitude distances are a constant whereas longitude distances are an ever decreasing variable in either direction from the equator toward the poles.

Latitude measures north and south, up and down the longitude lines while longitude measures east and west along the latitude lines...

[StuM]

Originally Posted by David M

The answer is not enough to worry about. At 200 NM planar trig works fine.

My point exactly and I think we have now established that they are the same to within 6 significant figures or so.

[StuM]

Originally Posted by dougdaniel

I think you may have confused latitude and longitude. The way I remember the difference is that longitude is always long. It is a great circle passing through the poles. So they are the lines that run north - south.

Nope. You are confused

You are correct in your description of lines in longitude. I was correct in stating that the distance between lines of longitude one minute apart decreases as you go polewards, so you need to use line of latitude for measurement.

[dougdaniel]

Originally Posted by StuM

Nope. You are confused

You are correct in your description of lines in longitude. I was correct in stating that the distance between lines of longitude one minute apart decreases as you go polewards, so you need to use line of latitude for measurement.

you're quite right. I thought you were talking about a minute of longitude. I should have read your statement more carefully. Since the discussion was about minutes of longitude vs. nautical miles, I assumed that was your subject also.

[StuM]

Originally Posted by dougdaniel

you're quite right. I thought you were talking about a minute of longitude. I should have read your statement more carefully. Since the discussion was about minutes of longitude vs. nautical miles, I assumed that was your subject also.

????? I was talking about a minute of longitude v NM! As I said it gets smaller as you move poleward.