# The Transverse Mercator projection¶

In this exercise we will introduce the Transverse Mercator projection, the UTM projection and the relationship between the two.

In addition we will investigate the differences between the two Transverse Mercator implementations available in PROJ and when one should be used in favour of the other.

Download the gie file for the exercise: `projections3.gie`.

## Exercise 1. Transverse Mercator with default parameters¶

Set up a Transverse Mercator projection using the default parameters.

Hint

Consult Transverse Mercator

```operation   <your answer here>
tolerance   1 mm

accept      24.745          56.437       # Talinn, Estonia
expect      1506742.2481    6535299.3398
```

## Exercise 2: Use the Transverse Mercator to model the UTM projection¶

The backbone of the UTM projection is a Transverse Mercator projection. In this exercise we will model the behavior of the UTM projection using the Transverse Mercator.

Hint

Remember that the UTM projection on the Northern Hemisphere uses a scale factor of 0.9996, a false easting of 500000 and a false northing of 0. UTM on the Southern Hemisphere is similar but with a false northing of 10000000.

Hint

The projection center is determined from the UTM zone and can be determined by zone*6 - 183

Hint

Tranverse Mercator parameters are documented at Transverse Mercator

```operation  <your answer here>
tolerance   1 mm

accept      24.745          56.437       # Talinn, Estonia
expect      360965.5942     6256998.5609
```

## Exercise 3: The less accurate, but faster, version of the Tranverse Mercator¶

As mentioned in the introduction to this set of exercises, two versions of the Transverse Mercator is implemented in PROJ. The default uses the Engsager/Poder algorithm which is accurate far away from the central meridian of the projection. The downside to this accuracy is that the algorithm is slower. The alternative algorithm, which is toggled by the +approx parameter, is faster but usage is recommended only within a few degrees away from the central meridian.

In this and the following exercises we will explore the accuracy of the two algorithms by checking the roundtrip stability of a number of transformations. A coordinate in Greenland will be used, as it is common practice to store geospatial data covering the whole country in the same UTM zone. This is only possible when using the correct algorithm. For the sake of simplicity, all operations in the following exercises are expressed as UTM projections. The UTM projection also has the +approx parameter which toggles the use of the faster, less accurate transverse mercator algorithm.

We will try to determine the approximate roundtrip accuracy of the +approx algorithm several UTM zones away from the actual zone for the given coordinate. For all the exercises below the aim is to find the lowest tolerance for each roundtrip test. You can of course make all tests pass by setting a tolerance of 1000 km - that's not the point: How low can you go?

After you have answered all exercise 3 questions below, based on your findings consider in which situation use of each of the algorithms is appropriate.

Hint

gie accepts most common SI unit prefixes to the meter when specifying the tolerance, e.g. km, m, dm, cm, mm, um, nm.

Hint

Look at the output gie produces - the difference between the actual result and the expected result is reported when tests fail.

### Exercise 3a¶

As a baseline, determine the roundtrip accuracy of the default algorirthm using UTM zone 22.

```operation   +proj=utm +zone=22
tolerance   <your answer here>  # Hint: You can go *very* low here

accept      -20.0   74      # Daneborg, Greenland (~30 degrees from central meridian)
roundtrip   1000
```

### Exercise 3b¶

Determine the roundtrip accuracy of the `+approx` algorithm using UTM zone 22:

```operation   +proj=utm +zone=22 +approx

accept      -20.0   74      # Daneborg, Greenland (~30 degrees from central meridian)
roundtrip   1000
```

### Exercise 3c¶

Determine the roundtrip accuracy of the `+approx` algorithm using UTM zone 23.

```operation   +proj=utm +zone=23 +approx

accept      -20.0   74      # Daneborg, Greenland
roundtrip   1000
```

### Exercise 3d¶

Determine the roundtrip accuracy of the `+approx` algorithm using UTM zone 24:

```operation   +proj=utm +zone=24 +approx

accept      -20.0   74      # Daneborg, Greenland
roundtrip   1000
```

### Exercise 3e¶

Determine the roundtrip accuracy of the `+approx` algorithm using UTM zone 25:

```operation   +proj=utm +zone=25 +approx

accept      -20.0   74      # Daneborg, Greenland
roundtrip   1000
```

### Exercise 3f¶

Determine the roundtrip accuracy of the `+approx` algorithm using UTM zone 26:

```operation   +proj=utm +zone=26 +approx

accept      -20.0   74      # Daneborg, Greenland
roundtrip   1000
```

### Exercise 3g¶

Determine the roundtrip accuracy of the `+approx` algorithm using UTM zone 27:

```operation   +proj=utm +zone=27 +approx