Calculate combined scale factor from grid and elevation factors or ellipsoid height, then convert ground and grid distances and view ppm distortion.
- All Math and Numbers Calculators
- Scale Factor Area Calculator
- Scale Factor Dilation Calculator
- Scale Speed Calculator
Calculator Functions
| Function | What it does | What it requires |
|---|---|---|
| Combined scale factor | Multiplies the grid scale factor by the elevation factor | Requires the grid / projection scale factor and an elevation factor source |
| Ground distance → grid distance | Converts a horizontal ground distance into a grid distance | Requires the grid scale factor, an elevation factor source and the distance |
| Grid distance → ground distance | Converts a grid distance into a horizontal ground distance | Requires the grid scale factor, an elevation factor source and the distance |
Elevation factor source has three choices. Ellipsoid height computes the factor from the height h and a reference radius. Orthometric height + geoid separation adds the two heights to get h first. Known elevation factor accepts the factor directly. Heights and the radius can be in metres or international feet, and distances in m, ft, km or mi. Every result also lists the elevation factor, the linear distortion in parts per million and the reciprocal of the combined factor.
Combined Scale Factor Formulas
| CSF | Combined scale factor (no unit) |
| GSF | Grid scale factor of the map projection at the point or for the line (no unit) |
| EF | Elevation factor, which reduces a horizontal ground distance to the ellipsoid (no unit) |
Combined scale factor
The formula above applies directly once the elevation factor is known. With a height, the calculator finds it on a sphere:
R is the reference radius and h is the height above the ellipsoid. For Orthometric height + geoid separation, h = H + N, where H is the height above the geoid and N is the geoid height, which is negative across the conterminous United States [1]. With GSF = 0.9996 and h = 1,000 m, EF = 6,371,000 ÷ 6,372,000 = 0.999843063 and CSF = 0.9996 × 0.999843063 = 0.999443126. The linear distortion is (CSF − 1) × 1,000,000 = −556.874 ppm.
The radius field starts at 6,371,000 m, a mean Earth radius. The National Geodetic Survey manual uses 6,372,000 m (20,906,000 ft) for the conterminous United States [1]. The two radii change the elevation factor by about 0.02 ppm at a height of 1,000 m.
Ground distance → grid distance
A horizontal ground distance of 1,000 m with CSF = 0.999443126 is 999.443126 m on the grid, which is 0.556874 m shorter.
Grid distance → ground distance
A grid distance of 1,000 m with the same factor is 1,000 ÷ 0.999443126 = 1,000.557184 m on the ground. For areas, a grid area divided by CSF² gives the area at ground elevation [1].
Limits of This Estimate
- The elevation factor from a height uses a sphere with one mean radius. The National Geodetic Survey calls this adequate for most surveying and engineering work and points to a more rigorous reduction for the most precise surveys [1].
- One factor is applied to the whole distance. The grid scale factor changes from point to point, so a long line needs a line factor, such as (k₁ + 4kₘ + k₂) ÷ 6 from the factors at both ends and the midpoint, and a mean height [1].
- Distances must already be horizontal. Slope distances are not reduced, and no projection, zone or geoid model is looked up.
- Feet are international feet of 0.3048 m. Values in US survey feet need converting first.
Sources
- National Oceanic and Atmospheric Administration, National Geodetic Survey, NOAA Manual NOS NGS 5, State Plane Coordinate System of 1983, Sections 4.1 and 4.2: elevation factor, mean radius, combined factor, area rule and line scale factor
