## Azimuth and Distance with the Dot Grid

### Instrument: Wndsn NATO-MIL Telemeter

The dot grid on the NATO Telemeter can be used together with the Universal Transverse Mercator (UTM) grid on a topographic map.

The dot grid is used together with the string and the degree arc to calculate the distance and direction to a target as the crow flies. This can be useful when flying, sailing or crossing flat and open terrain. The angles and distances are read directly on the Quadrant. The trigonometry is precise and can complement or confirm other navigation methods.

We are looking for the distance and azimuth from P1 to P2. The UTM coordinates are given. The map in the figures is for visualization purposes.

- P1: 33U 0364496E 5767404N
- P2: 33U 0401555E 5744867N

The difference in the UTM coordinates in meters is 37,059 east and -22,537 north (negative; i.e. south) (Figure 1).

### Visualization

The Quadrant can be understood as a graphical map representation to visualize course and direction by rotating it to the corresponding quarter of the full circle. Note that the scales are unitless, and a 60 grid (as on the model in Figure 2) can also be used directly for decimal values.

The Quadrant can display both bearing and distance.

### The azimuth

The difference in meters calculated from the coordinates is divided by 1000^{[1]} and transferred to the dot grid. The values are placed on the outer scales and the string is pulled through the intersection of the north and east values at 37×22.5. This gives a bearing angle of 31° (Figure 3).

Calculation:

```
tan(bearing) = easting/northing
```

To determine the azimuth, we add 90° for a result of 121°.

Why 121°? Comparison with the map or a sketch shows that we are in the southeast quarter of the full circle and our bearing is towards the southeast. The compass direction is measured from north as 0° and from east as 90°. For the resulting *azimuth* of 121°, the determined *bearing* of 31° must be added to 90° (see the full circle sketch in Figure 2).

### The distance

For the distance we set the string to the bearing of 31° and the cursor at the crossing point at 37×22.5. Then we rotate the string and read the value on the side scale: 43. (See Figure 3. This value is **not the cosine**, but the hypotenuse length, which is the distance we are looking for.)

The distance triangle is -22.5 units to the south (approx. 22.5 km, so ×1000) and 37 units to the east (approx. 37 km, again ×1000), the hypotenuse is the distance and is 43 units (approx. 43 km, ×1000, calculated using the Pythagorean theorem.^{[2]}

Calculation:

`Distance = √(Easting`^{2} + Northing^{2})

### Result

To get from P1 to P2, we follow an azimuth of 121° for a distance of about 43 km as the crow flies.

**See also:**

**Footnotes:**