Wndsn Quadrant Telemeter Tutorials
Making the most out of our graphical telemetry computers.
Like with many complex instruments, there are multiple ways to solve certain problems and to measure the required inputs. Combining the various functions leads to a multitude of advanced uses. See also the printed manual.
Course and Distance with the Dot Grid
The dot grid on the Astronomy Quadrant (or on the Sine Quadrant, as well as on the NATO-Telemeter) can be used together with the grid on a terrain map. These grids are called Universal Transverse Mercator (UTM) and avoid the problem of longitude degrees getting squished at high latitudes. In Sweden the grid is called SWEREF and is mapped by the Lantmäteriet authority.
This map grid can be used with the Quadrant dot grid to calculate the distance and direction to a destination as the crow flies. This can be useful for flying, sailing, or crossing flat and open terrain. The angles and distances are calculated with trigonometry. The calculations are precise, and can supplement or confirm other methods of navigation.
Figure 1. Coordinates of harbor on the island of Ven from SWEREF.
Figure 2. Coordinates of Landskrona harbor from SWEREF.
Let’s say I want to visit the Tycho Brahe museum on the island of Ven. The nearest mainland harbor is Landskrona. I look up the coordinates of the origin and destination harbors and find the difference is 3939 meters north and 5890 meters west (Figures 1 and 2).
Figure 3. The sides of my triangle are 20, 30, and 36.
Method 1: If the grids match
My Quadrant can now tell me the distance and course. The difference in meters, taken from the map grid, can be divided by 200[1] to nicely fit on my dot grid Quadrant, touching the 20×30 cross. This gives an angle of 34°. Since I am travelling mostly west, I add 270° to get the course 304°.
For distance, I set the green string to the 34° I found, and the red cursor on the 20×30 cross. Now I sweep the green string to the side and read the value on the cosine scale: 36. (See Figure 3. This value is not the cosine, it is the hypotenuse length.) From this we can deduce that to go 20 units north (actually 3939 meters, so ×200) and 30 units west (actually 5890 meters, again ×200), the shortest distance is 36 units (actually 7086 meters, ×200, calculated with Pythagoras’[2] theorem).
So to get from Landskrona to Ven I set the course to 304° and travel about 7200 meters. It’s that simple!
Sidebar. Using the Sine Quadrant as a map tool: Why 304°? The compass direction is measured from north as 0° and west as 270°. The determined bearing of 34° has to be added to 270° for the resulting azimuth of 304°. Also note how the Sine Quadrant can be understood as a fully graphical map overlay to visualize the course and directions by rotating it to the relevant quadrant of the full circle. Further note that the scales are unitless and that the 60-division grid of the Sine Quadrant can be used directly for the decimal values.
Method 2: Using the decimal scale
If the west and north differences do not nicely fit on the Quadrant’s dot grid, the decimal-sexagesimal converter offers an alternate fit. The sine and cosine scales along the edge of the instrument are sexagesimal, that is, they go up to sixty, like minutes in an hour. The sine scale has extra marks on the inside for every sixth value, making a decimal-sexagesimal converter. It is useful for general time and navigation calculations. For example, the seventh decimal mark indicates that 0.7 hours is 42 minutes.
Figure 4. My triangle has the sides 39, 59, and 71 as measured along the decimal scale.
I start by rounding the north and west differences to 39 and 59 and finding these on the decimal scale. Guided by the dot grid, I use string, or a pencil and a straight edge, to draw perpendicular lines (Figure 4). Then I pull the string through the origin and the intersection of my lines. This time, the intersection I pencilled in is a little closer to the degree scale than before, and so gives a little more precision: 33.8°. I slide the cursor knot to the intersection, and then sweep the string to the decimal scale and read the hypotenuse length: 71.
So now I get to Ven from Landskrona by setting the course to 303.8° and travelling about 7100 meters. Again, this is fairly simple and can be done with barely any written or electronic calculations. This distance estimate, 7100 meters, matches the previous estimate of 7200 meters as well as the exact length of 7086 meters. The errors are small, 1.6% for method one and less than 1% for method two.
Actual shipping lanes
Figure 5. Sea chart with leading lights indicating 125° (measured from south).
In real life, shoals and other obstructions may prevent shipping lanes from taking a direct route, especially near land. In our case, the sea chart says there is a pair of leading lights and a channel leading most of the way back to the mainland harbor, course 125° (Figure 5). This implies a 305° course toward Ven, which is very close to the 304° course I calculated on my Quadrant. While ships follow shipping lanes and can not just go as the crow flies, small craft such as kayaks or sailing dinghies may instead avoid shipping lanes and take a more direct route away from marine traffic.
Once the direction to the destination is known, a compass can guide you there. If you know the declination of a few stars, and if your Quadrant or astrolabe has azimuth curves for your latitude, you can even use it as a star compass to get where you are going.
Thank you to Andrej and Ken for helping me simplify the math and clarify the explanation.
(Tutorial written by Cute Puppy, first published on his blog.)
See also:
Footnotes: