# Scale factor

As **μ scale factor** is used in Geodetic the aspect ratio of two versions of a survey network or an entire earth model referred.

The cartography knows correction factors for different maps and their projections, as well as noticeable paper arrears .

## In geodesy

Since surveying networks have only very little distortion and the earth's dimensions are well known, the possible scale factors are always close to the value *1* . It is therefore also written in the form

- µ = 1 + m

and calls the small size *m* the scale correction. In modern route networks, *m is* at most a few ppm (millimeters per km), so the scale factor is almost always within the limits

- 0.99999 <µ <1.00001.

Newer networks - even if they are based on terrestrial measurements - are often controlled and reinforced with data derived from satellite geodetic systems . A 7-parameter transformation with 3 shifts and 3 small angles of rotation is often used ; the 7th parameter is the scale factor. It means that the routes in network model A best match those in network B if they are lengthened or shortened by the factor µ.

The fact that µ is not *exactly* 1 even with modern network measurements can be due to several factors. The two most important are:

- different models of the earth's atmosphere , whose ground-level refraction index is around 1,00030
- different positioning of the two network variants on a reference ellipsoid or a mean earth ellipsoid (see also geodetic datum ).

Historically, the scale factor appeared in several ways from around 1800:

- for the first time with the transition from graphical to arithmetical measurement methods - for example when the measuring table was replaced by universal instruments or theodolites . Here the scale correction
*m*- depending on the quality of the distance measurement - was around 0.0001 to 0.001 (0.01 to 0.1 percent). From around 1800 the standard accuracy was almost always better than 10 ^ -4, today it is between 10 ^ -5 (for simple property measurements ) and 10 ^ -8 for satellite and earth measurements . - Next ellipsoid : here the question had to be discussed theoretically how the dimensions of an earth ellipsoid to be derived are related to the curvature of the geoid .

## In cartography

Cartography also speaks of the scale factor when z. B. an area is shown in different map projections or if the map scale changes noticeably within a map sheet or system for mathematical reasons. Usually the nominal scale only applies in the center of the map or (in the case of cone projections ) in the two reference parallels , whereas µ <1 in between.

Changes in scale can also occur through the entry of the paper - especially as a result of changing humidity . Here, empirical values for the scale correction *m* are sometimes given.