# The Lambert Conic Conformal projection¶

In this exercise we will introduce the Lambert Conic Conformal (LCC) projection as well as general map projection concepts such as projection center, scale factor, false easting/northing and ellipsoid models.

We will do so by defining various versions of the LCC optimised for use in and around the Gulf of Finland.

In case you are not familiar with the LCC projection, Wikipedia describes the Lambert Conic Conformal projection as:

"A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten (Notes and Comments on the Composition of Terrestrial and Celestial Maps).

Conceptually, the projection seats a cone over the sphere of the Earth and projects the surface conformally onto the cone. The cone is unrolled, and the parallel that was touching the sphere is assigned unit scale. That parallel is called the reference parallel or standard parallel.

By scaling the resulting map, two parallels can be assigned unit scale, with scale decreasing between the two parallels and increasing outside them. This gives the map two standard parallels. In this way, deviation from unit scale can be minimized within a region of interest that lies largely between the two standard parallels. Unlike other conic projections, no true secant form of the projection exists because using a secant cone does not yield the same scale along both standard parallels"

Download the gie file for the exercise: `projections2.gie`.

## Exercise 1: Standard parallel¶

Set up a LCC projection with one standard parallel placed halfway between Helsinki and Tallinn.

Hint

• Approximate location of Helsinki: 60.171N 24.938E

• Approximate location of Tallinn: 59.437N 24.745E

• Approximate mid point between Helsinki and Tallinn: 59.8N 24.8E

Hint

See Lambert Conformal Conic for more on the Lambert Conformal Conic projection

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

accept      24.938              60.171  # Helsinki
expect      1351962.01          298663.54

accept      24.745              59.437  # Tallinn
expect      1371811.19          218598.22
```

## Exercise 2: Distance distortion¶

Improve the distance distortion of the projection.

The projection defined above is conformal, i.e. has no angular distortions for infinitesimal objects - but there is a minor distortion of distances. This cannot be avoided in general, but locally distortions can be minimized by adjusting the scale factor.

With the default factor of 1, the distance error is 0 for (infinitesimal) line segments at the latitude of the standard parallel.

By reducing the scale factor, the line of zero distance distortion degenerates into two lines, north and south of the standard parallel.

For the abovementioned point in Tallinn, the distance error is around 0.002%, but this can be reduced to zero by varying the scaling factor.

Even by trial and error it is not hard to reduce it to a few millionths of a percent. So try to find a more optimal scaling factor by trial and error using the projection analysis functionality of the proj application.

Hint

Modify and use the command below to find a better scaling factor:

```echo 24.745 59.437 | proj -V <your projection setup here>
```

Hint

Look for the lines displaying the meridian and parallel scale.

Hint

Consult the LCC documentation to find out how to set the scaling factor.

Hint

Scaling factors usually deviates from unity by the order of 1e-3 to 1e-5

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

accept      24.745              59.437
expect      1371783.759883408   218593.850591891
```

## Exercise 3: Using two standard parallels¶

Set up a similar LCC projection, this time using two standard parallels. The first passing through Helsinki, the second through Tallinn.

Hint

• Approximate location of Helsinki: 60.171N 24.938E

• Approximate location of Talinn: 59.437N 24.745E

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

accept      24.8                59.8
expect      1359907.80          8024082.260
```

## Exercise 4: Projection center¶

Change the projection center to the midway point between Helsinki and Tallinn.

By changing the projection center we move the origin of the output coordinates to be coinciding with the position where the projection is most accurate.

Hint

Hint

The mid point between Helsinki and Tallinn is roughly 59.8N 24.8E

Hint

Lambert Conformal Conic explains how to change the projection center

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

accept      24.8            59.8
expect       0.0             0.0
```

## Exercie 5: False easting and northing¶

Introduce false eastings and northings.

Oftentimes it is preferable to keep coordinate values positive even when moving west and south of the projection center. This can be achieved by introducing false easting and northing. Add false easting and northing of 1000000

Hint

Lambert Conformal Conic explains how to set false easting/northing

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

accept      24.8            59.8
expect      1000000.00      1000000.00
```