Celestial Navigation Revisited

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Here are the results of all 23 navigation stars that I shot from my position at N50° W0° in the English Channel. Most of the differences from Almanac data were positive which made me suspect another Index error, just like a real sextant. Applying this new correction drove the errors down to an average of 6 minutes of arc or 12 km. The largest error was 12.4', or just over a tenth of a degree. My tired eyes and SH3's low resolution seem to be good for about 10' or so...

The picture shows Procyon in the sights with a particularly good measurement!

Now we have two accurate sextants.

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Here are some examples of the star charts that I made. They are screen shots from the flak station with the navigation stars labeled. I took them every 120° of bearing with one shot tilted down and the other tilted up. I didn't check my work here, so there could be an error. Comparison with bearing information in the almanac data will reveal any error.

There are 12 pictures in all showing all 23 navigation stars visible at the time and also Polaris.

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Next I am going to post some screen shots giving an example of fixing your position using the sextant and 3 star shots. This will take a little while. Then I will upload all the minimods and the other materials that will allow you to do the same. And that will conclude the second prewar voyage of the U-13 in its secret mission to develop navigation tools for the upcoming war...

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When you decide to shoot the stars, click on the stopwatch immediately after the minute on the clock changes. This allows you to know Greenwich Mean Time to about 5 second accuracy in game. Earlier in the thread is a discussion on how the game clocks work, and how to get GMT from your local time and the nautical time zone. Here I am using my nautical chart with 1° grid spacing. SH3 world is a cylinder with 120 km between latitudes and longitudes. We will eventually need to correct our result for this map projection.

Select navigation stars with bearings roughly 120° apart, so you will get a nice shaped error triangle when you are done. I chose the first star on the list, Alioth, and then used its bearing to help choose the other 2 stars. You will need a minimum of 2 stars, but everyone uses 3. When you shoot a star, you measure its altitude or height above the horizon. This is called Hs. You then make the reading more accurate by applying correction factors. We have identified two corrections for measurements done with the in-game sextant. The index correction adjusts for a the sextant not being zeroed. It gives us the Ha, or apparent altitude of the star. The second corrects for the distortion in the SH3 camera. I am calling that the Altitude correction. You look it up on the table I created based on the calibration curve I measured. Now you know the observed altitude or height above the horizon Ho. You don't need to record the bearing to the star. But you do need the time of measurement. My measurements were taken 2 minutes apart.

Our position is shown on the map, so we can judge the result.

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Since we don't know where we are, we need to assume a position. Since Bernard was the helmsman, we chose a nearby position on land. My nautical chart has 1° grid spacing with latitude/longitude labels on the edges, so I know the assumed position is 51° N 001° W. You need to know the latitude and longitude of the assumed position and its precise location on the chart for the method to work.

A nautical almanac will tell you the altitude of a navigation star from the assumed position at the time of your observation. Those heights above the horizon are called Hc. The almanac will also tell you the bearing from the assumed position to the star. If your boat was at the assumed position, your observed altitudes Ho would match the calculated altitudes Hc. Comparison of Ho and Hc determines which location is closer to the given star, your observing location O or the assumed position AP. Think of it this way...if the star stood still and you sailed toward it, it would appear to rise higher in the sky. So whatever position is closer to the star yields the higher altitude for the star.

For each star shot, subtract the Hc from the Ho and note the difference and its sign. Multiply the arc minutes by 2 km to turn them into distances. Use the sign to determine whether you are closer or further than AP from the star. Here is what we can say right now based on our measurements...

Our unknown position is nearly the same distance away from Alioth as the known assumed position.
Our unknown position is much closer to Gienah than the known assumed position.
Our unknown position is much further from Elnath than the known assumed position.

Let's put that information on the chart using the in-game mapping tools.





I have a typo in the picture above. The second blue line should read Hc > Ho...I wil fix it later...

[11Bravo]

We start with Alioth. Construct a bearing line from AP in Alioth's direction. Advance along the bearing toward Alioth 2 km because our Ho was 1 arcminute greater than the Hc. Construct a line at a right angle to the star's bearing. This is called the Line of Position for Alioth. All points on this line are 2 km closer to Alioth than is AP. If we have done everything right, our unknown position would be somewhere on the Alioth LOP...

If we were charting on a sphere. But we are charting on a cylinder. And that causes an error we will have to deal with later.

The Alioth LOP is the first side of our error triangle.

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Now we handle Elnath.

Plot a bearing to Elnath from the AP. Retreat in the opposite direction 108 km because our Ho was 54 arcminutes lower than the Hc. Construct a right angle line of position on the star bearing. This is the Elnath LOP. All positions on this line are 108 km further from Elnath than is AP. If we have done everything right, our unknown position is also on this line.

The Elnath LOP is the second leg of our error triangle.

The intersection of the Alioth LOP and the Elnath LOP is our emerging determined position...on a sphere, but not on this cylinder yet.

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Now we close the error triangle with Gienah. Construct a bearing to Gienah from AP. Advance along this bearing 108 km because our Ho was 54' greater than the Hc. Construct a right angle Line of Position for Gienah. All points on this line are 108 km closer to Gienah than is AP. If we have done everything right, we are also on the Gienah LOP.

The Gienah LOP closes the error triangle. The size of the triangle gives us an estimate of the error of our methods. The triangle is very small. 3 independent measurements of star altitudes in game have yielded the same position to within about 2 km. It might not be the correct position yet, but it is remarkably consistent.

This error triangle was produced using almanac values calculated with spherical trigonometry. Our best measurement of our position says we are in that error triangle. And we would be pretty close if we had charted the result on a sphere.

So we need to correct the position of this error triangle to determine where it would chart on a cylinder. Like our navigation chart.

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Let's compare our earth to planet SH3.

Earth is approximately a sphere. We plot our position using latitude lines and longitude lines. Latitudes are parallel to the equator. The distance between 1° of latitude on earth is the same everywhere, and is about 111 km.

Planet SH3 is a cylinder. We can plot our position using latitude and longitudes. The latitude lines are also parallel to the equator. The game editor reveals that 1° = 120 km everywhere.

So the distance between 50° N and 51° N is 111 km on earth.
And the distance between 50° N and 51° N in SH3 is 120 km.
There is no problem charting latitudes in the game.

Now we examine longitudes.

On earth the longitude lines converge at the poles. This causes the spacing between longitude lines to decrease as latitudes increase away from the equator.

So the distance between 0° W and 1°W is 111 km on earth at the equator.
And is 71.3 km at 50°N
And is 19.3 km at 80°N
And converges to zero at the poles.

On planet SH3 the distance between 0°W and 1°W is 120 km at the equator.
And is 120 km at 50°N
And is 120 km at 80°N
And stays constant at the poles.

We need to account for this difference in behavior between the two different shaped worlds.

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Celestial navigation depends on the mathematical principles of spherical trigonometry. All the inputs and outputs are angles. Latitude is an angle. Longitude is an angle. The altitudes of the heavenly bodies are angles. Only the final result is converted to a linear distance using the scale factor of that spherical world.

We have been working in km using the SH3 scale factor of 1°=120km, but we were always really working with angles...minutes of arc. And we plotted our error triangle using the arcminute difference between Ho and Hc.

Our error triangle is located a certain distance from a longitude line. That means it is located a certain angle from that line. And that angle was calculated with the assumption the world was a sphere. So we need to distort that angle onto the surface of a cylinder.

The mathematical function that causes earth's longitude lines to converge at the poles is the cosine of the latitude.

The cosine of 0° is 1
The cosine of 50° is 0.6428
And the cosine of 80° is 0.1736

Recall the 1° distances between longitudes on earth.

111 km at 0° = 111 cosine 0°
71.3 km at 50° = 111 cosine 50°
19.3 km at 80° = 111 cosine 80°

So when spherical trigonometry gives you an angle between longitudes, the distances get shrunk according to the cosine of the latitude.

So it stands to reason we need to stretch them back so we can chart them on a cylinder. Since spherical trigonometry has already multiplied the angle by the cosine of the latitude, we are going to undo that by dividing by the cosine of the latitude.

Our scale factor will depend on latitude and the function will be 1/cos(latitude).

The 1/cosine of 0° is 1
The 1/cosine of 50° is 1.556
And the 1/cosine of 80° is 5.759

[11Bravo]

Our error triangle was computed on the basis of spherical trigonometry because that is the how celestial navigation was developed on earth. Here is what we need to do to get the result to plot on a cylinder that SH3 plays on.

We need to measure the distance between the error triangle and the longitude of the assumed position. That distance is 73 km. At a scale of 1 arcminute = 2 km, that suggests our error triangle is 36.5 arcminutes East of the AP longitude.

We need to divide that number by the cosine of the latitude. The result is 56.8 arcminutes. East of the AP longitude. Then we convert back to km and get 114 km East of the AP longitude.

Or just look up the value in the chart. We need to multiply by 1.556. So our 73 km is really 114 km.

Just a little painful, but necessary to maintain accuracy, especially at higher latitudes.

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Here we see the our determined position after shifting the error triangle to compensate for the cylinder map projection. Also shown is our actual position, and the actual error of 18.6 km. My original goal was to develop the method of celestial navigation in-game with an accuracy of 30 km.

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BDU is cautiously optimistic about the results and not amused at the money back guarantee. This will require testing. I am uploading the files in a few minutes. That ends this voyage of the U-13.

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Third times the charm...

OK, someone go out there and blow this thing up!