Trigonometry question...

[conachair]

Originally Posted by OldFrog75

I guess the moral to the story, if there is one, is to.........

Sail downwind

[Ericson38]

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?

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)

[OldFrog75]

Originally Posted by Ericson38

Staying on either side of the equator, where a degree=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 (on a flat plate)

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

great circle nm distance = 4.242398336379212 (on a sphere)


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

Maybe picking a nit here but given the Earth is not a sphere, is one degree anywhere between the Tropic of Cancer and the Tropic of Capricorn still equal to 1 nm?

[Ericson38]

Originally Posted by OldFrog75

Maybe picking a nit here but given the Earth is not a sphere, is one degree anywhere between the Tropic of Cancer and the Tropic of Capricorn still equal to 1 nm?

I used the ellipsoid for WGS84-

referenceEllipsoid with defining properties:

Code: 7030
Name: 'World Geodetic System 1984'
LengthUnit: 'meter'
SemimajorAxis: 6378137
SemiminorAxis: 6356752.31424518
InverseFlattening: 298.257223563
Eccentricity: 0.0818191908426215

[OldFrog75]

Originally Posted by Ericson38

I used the ellipsoid for WGS84-

referenceEllipsoid with defining properties:

Code: 7030
Name: 'World Geodetic System 1984'
LengthUnit: 'meter'
SemimajorAxis: 6378137
SemiminorAxis: 6356752.31424518
InverseFlattening: 298.257223563
Eccentricity: 0.0818191908426215

Thanks but I don't understand if that answers my question.

[Ericson38]

Originally Posted by OldFrog75

Thanks but I don't understand if that answers my question.

Those are the coefs for the ellipsoid. I used an equator illustration to keep the cosine falloff with increasing latitude from shortening the nominal and approximate 1 nm=1 minute of arc relationship.

[dougdaniel]

At one time a nautical mile was defined at one minute of longitude. But it has since been defined as 1,852 meters. So a nautical mile is never exactly a minute of longitude or a minute of latitude at the equator.

[OldFrog75]

Originally Posted by dougdaniel

At one time a nautical mile was defined at one minute of longitude. But it has since been defined as 1,852 meters. So a nautical mile is never exactly a minute of longitude or a minute of latitude at the equator.

Really? So the factor converting statute miles to nautical miles is 1.14 rather than 1.15?

I guess it's safe to say one nm is exactly one degree in very few places on the entire planet.

Does that mean all paper charts have a scale of 1852 meters per one minute of latitude to measure distances on a chart with a pair of dividers?

Do GPS systems also define a nm as 1852 meters?

Are boat speeds calculated by transducers or GPS coordinates based upon 1852 meters/nm?

Sorry for all the questions but one just sort of lead to another...

[dougdaniel]

A lot of good questions.

I am not sure the exact conversion factor from statute to nautical miles. Let me give you a little background.

Even though we use the English system of weights and measures here in the US, we are in fact on the metric system. The government organization responsible for these standards is NIST (National Institute of Standards and Technology). They have decided that an inch is exactly 2.54 cm. They have agreed via treaty with the international governing body for maps (sorry, I can't think of its name) that a nautical mile is 1852 meters. So for me to tell you the conversion factor, I would have to get out my trusty calculator and do some multiplying and dividing. ... I'll let you do that.

My point about 'defining' inches and nautical miles relative to cm and meters is that the only way two things in the macro world that we live in are exactly equal is when they are defined to be that way. (No doubt someone will enter this discussion with quantum mechanics and Heisenberg. That's not what I'm talking about. I'm talking about things that we can see, feel and measure with a ruler.)

These definitions have been in effect for many decades. So GPS systems, maps made even before 1984, etc. agree on these definitions.

And, just from number theory, there are no places on earth where a minute of arc can be exactly one nautical mile.

If I sound like a self-styled expert, I don't mean to. My qualifications are that I worked professionally as a mathematician for one of our national laboratories for ten years and hold a RYA yacht master theory certificate.

I would point out that, from a practical point of view, that considering one minute of longitude to be one nautical mile is accurate to within the width of a tick mark on the map. That's how I measure distances on a chart.

[OldFrog75]

Originally Posted by dougdaniel

A lot of good questions.

I am not sure the exact conversion factor from statute to nautical miles. Let me give you a little background.

Even though we use the English system of weights and measures here in the US, we are in fact on the metric system. The government organization responsible for these standards is NIST (National Institute of Standards and Technology). They have decided that an inch is exactly 2.54 cm. They have agreed via treaty with the international governing body for maps (sorry, I can't think of its name) that a nautical mile is 1852 meters. So for me to tell you the conversion factor, I would have to get out my trusty calculator and do some multiplying and dividing. ... I'll let you do that.

My point about 'defining' inches and nautical miles relative to cm and meters is that the only way two things in the macro world that we live in are exactly equal is when they are defined to be that way. (No doubt someone will enter this discussion with quantum mechanics and Heisenberg. That's not what I'm talking about. I'm talking about things that we can see, feel and measure with a ruler.)

These definitions have been in effect for many decades. So GPS systems, maps made even before 1984, etc. agree on these definitions.

And, just from number theory, there are no places on earth where a minute of arc can be exactly one nautical mile.

If I sound like a self-styled expert, I don't mean to. My qualifications are that I worked professionally as a mathematician for one of our national laboratories for ten years and hold a RYA yacht master theory certificate.

I would point out that, from a practical point of view, that considering one minute of longitude to be one nautical mile is accurate to within the width of a tick mark on the map. That's how I measure distances on a chart.

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

[El Pinguino]

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 Traverse tables ( based on the plane sailing formula) are designed for use up to 600 miles... after that dmp etc etc raises its head.... that is if you choose to work with rhumb lines rather than GC.... which is what most of us used to do except on E/W courses in high latitudes.

[El Pinguino]

You could of course just use the versine...

This is what S.M.Burton had to say about the versine...

'Nautically, the versine more commonly expresses distance lost by deviation. Thus, if a ship steers 25* off the courseline along which she wishes to advance for (say) 6 miles, the amount of advance lost would be 6 x Vers(ine) 25* = 6 x .09 = .54 miles . On the other hand she would be displaced off the courseline by the sine i.e. 6 x sin 25* = 6 x .42 = 2.52 miles.
It is interesting that a ship has to steer 60* off her courseline before she halves her rate of advance'

[hellosailor]

OldFrog-
I've since passed it on, but IIRC you want an old book called "Mixter's Primer of Navigation" written by some guy named Mixter. He starts you out, in very simple basic terms that you won't have to look up, with concepts like "this is distance, in a straight line" and works it up to bigger issues in navigation, like why we use "radians" for spherical trigonometry for sextant navigation.
First time I read that I went "OH!" and all the stuff that never made sense in high school trig, all the stuff that seemed totally useless about that, suddenly became totally clear and incredibly welcome.
The guy just presents a wealth of information and DOES IT WITH EXCEPTIONAL CLARITY. Find a used copy, or ask your library about an interlibrary loan.


All this stuff is most easily resolved with trig and simple multiplication, and thank the gods we have tiny cheap powerful computers now, because when I went to school we had big, heavy, expensive, tedious BOOKS that listed values and calculations page after page for 500-1000 pages. And if your slide rule wasn't close enough, yes, you had to look every number up.


Much easier to punch it on a calculator, or a smartphone app, or a PDA, if you remember PDAs. If you have an old Palm, there were some great nav apps for it. Cheap and reliable, too.

[StuM]

Originally Posted by El Pinguino

You could of course just use the versine...

This is what S.M.Burton had to say about the versine...

'Nautically, the versine more commonly expresses distance lost by deviation. Thus, if a ship steers 25* off the courseline along which she wishes to advance for (say) 6 miles, the amount of advance lost would be 6 x Vers(ine) 25* = 6 x .09 = .54 miles . On the other hand she would be displaced off the courseline by the sine i.e. 6 x sin 25* = 6 x .42 = 2.52 miles.
It is interesting that a ship has to steer 60* off her courseline before she halves her rate of advance'

That's if you have a versine table or calculator with that function (I've never seen one). In normal practice, the equivalent "1 - cos()" would be a lot easier to derive.

[secrabtree]

Originally Posted by StuM

, So If I sail 90° to my course, that would be 8,100 feet sailed to make a mile?
How about if I sail more than 90° to my course?

You've an Error here. The answer to your question as you posed it is NEVER. If you sail a straight line at 90° to your desired course you will not get to your destination. You will continually get further from the rhumb line and, as you get further from the rhumb line you will get further from your destination at an ever increasing rate.
On the other hand, if you continually sail (motor) at 90° to the course to the destination, continually adjusting your course so that it is always 90° off the course to the desired destination you will sail (motor) in a circle around your desired destination, never getting closer nor further. (And to do this your desired destination must have open water all around it.)
Increase to angle to ANYTHING greater than 90° and you will ALWAYS get further from your destination.
Many of the "rules of thumb" given here as so much per degree ar rough approximations that are only valid for certain, usually small, angles.
People call me a nit picker because as a civil engineer (and navigation instructor) I often insist on things to be correct about numbers.