### Scale Factor

As explained at the “Point Scale Factor” section, The measurements of the lines must always be done on the topographic surface, first step convert the ground measured distances to the ellipsoid distances by multiply the ground distances to the elevation scale factor (ESF), the second step deriving the grid distances by uses the grid scale factor (GSF) multiply to the ellipsoid distances. The elevation scale factor can be combined with the grid scale factor by multiply them together to get the combined scale factor (CSF). The combined factor will move the line from the topographic surface of the earth to the map plane (aka grid). It must be remembered that the process must be reversed if it is to stake out for construction, on the ground surface.

Ellipsoid Distance = ESF x Ground Distance

Grid Distance = GSF x Ellipsoid Distance

or Grid Distance = CSF x Ground Distance

which CSF = ESF x GSF

To apply the** grid scale factor** (GSF) on the second step:

- For short lines (< 2 km), the grid scale factor will be reasonably constant along the whole length of the line, so a single grid scale factor can be applied.
- For lines 2 to 4 km, calculation a average scale factor of the two ends.
- For the lines over 4 km long, the grid scale factors should be found at both endpoints, and at the midpoint of the line; a mean grid scale factor can then be estimated using Simpson’s rule, i.e.,

K_{mean} = (K_{1} + K_{2} + 4 x K_{mid}) / 6

K_{1} is grid scale factor at the start of line, K_{2} is grid scale factor at the end of line and K_{mid} is the grid scale factor on the middle line.

### Line Scale Factor

The Line Scale Factor tool uses the Simpson’s rule to calculate the grid scale factor of the lines if you define the coordinate and the elevation/height of the two ends. When you start the Surveyor Pocket Tools you will see the Line Scale Factor tool on the main window as the highlighted.

Start the tool and the window will be populated as the screenshot below.

The short description of the tool as the screenshot attached below.

The another four small buttons are function to:

- The button use to store/save point to database.
- The button use to restore point from database.
- The button use to pin location to Google maps.
- The button use to save KML file and display location on Google Earth.

#### Working with Coordinate Reference Systems (CRS)

To specify the given coordinate reference system definition you can do easily by the steps that provided by **Surveyor Pocket Tools**. Please follow the tutorial:

How to select a Coordinate System?

#### Example 1 – Test on Myanmar Datum 2000 coordinate system

The first step you have to select the coordinate system. Please note the system coordinate must be projected coordinate system, then input the coordinate values of the two stations on the appropriate text boxes. The next step to select the vertical reference is it “*Orthometric height (H)*” or “*Ellipsoidal height (h)*” then enter the height on the text box.

Given:

- Myanmar Datum 2000 based on UTM on zone 47N & 48N
- use “Everest 1830” ellipsoid.
- transformation parameters to WGS84 datum: DX = 246.632; DY = 784.833; DZ = 276.923

- Station “0619”, Grid Northing = 1562929.360m, Grid Easting = 402828.340m Orthometric height = 5.056m
- Station “0612”, Grid Northing = 1532598.440m, Grid Easting = 420788.830, Orthometric height = 6.252m

Find the average combined scale factor.

The calculation will use **EGM2008** model to extract geoid height and then calculate ellipsoidal height, finally uses this ellipsoidal height to calculate elevation scale factor. Select coordinate system and enter the coordinate and height values.

Ready to compute click button and the results will be displayed.

Please notice the average grid scale factor as calculated by Simpson’s rule.

K_{1} = 0.9997168179, K_{2} = 0.9996776271, K_{mid} = 0.9996962250

K_{mean} = (K_{1} + K_{2} + 4 x K_{mid}) / 6

K_{mean }= (0.9997168179 + 0.9996776271 + 4 x 0.9996962250) / 6 = 0.9996965575

Average grid scale factor = 0.9996965575

Average elevation scale factor = 1.0000050916

Average combined scale factor = 1.0000050916 x 0.9996965575 = **0.9997016476**

The ground distance measured on the topography of the earth is 1000 m will be reduced to be 999.692 m on the map plane.

Pin on Google Maps

Click button to locate the point on Google Maps.

**Example 2 – Test on State Plane Coordinate System (NAD83)**

Given:

- Vermont / NAD83(NSRS2007) coordinate system
- Station 1, Grid Northing = 195854.310, Grid Easting = 490744.197, Orthometric height (NAVD88) = 161.0m
- Station 2, Grid Northing = 196070.049m, Grid Easting = 490343.673, Orthometric height (NAVD88) = 166.3m

Find the average combined scale factor.

Select coordinate system and enter the coordinate and height values. Please note this calculation will use EGM2008 model to extract the geoid height.

Click the button to compute.

Since this tool doesn’t use the USA geoid models, for example, 1999, 2003, 2009 and 2012, therefore the elevation scale factor will be a small difference (e.g. 7 in the eighth place) to the NGS published document. This effect will make the distances less than about 10 KM rounds to a millimeter. That is very small and no need to worry about it.

#### Pin to Google Maps

Click the button to see the location on Google Maps.

#### Conclusion

The Line Scale Factor is handy tool to calculate the average combined scale factor in one step.

#### Limitation of the tool

The tool is not yet supported another geoid models at this time. But the plans will be implemented USA geoid grids from 1999, 2003, 2009 and 2012.

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