# Local Linear Scale Factors in Map Projections of an Ellipsoid

## Abstract

**:**

## 1. Introduction

- A map distorts distances (linear distortion) wherever the quotient between the lengths of an infinitesimally short line as projected onto the projection surface and as it originally is on the Earth model deviates from 1.
- A map distorts angles wherever the angles measured on the model of the Earth are not conserved in the projection. This is expressed by an ellipse of distortion, which is not a circle.
- A map distorts areas wherever areas measured in the model of the Earth are not conserved in the projection. This is expressed by ellipses of distortion whose areas vary across the map.

## 2. Ellipsoid, Map Projection, and Local Linear Scale Factor

## 3. Local Linear Scale Factors in the Directions of Coordinate Axes

## 4. Local Linear Scale Factor in a Given Direction

_{1}, a

_{2}, and a

_{3}are given by (27) and H by (16).

## 5. Examples

#### 5.1. Local Linear Scale Factors in the Mercator Projection

#### 5.2. Local Linear Scale Factors in the Transverse Mercator Projection

#### 5.3. Local Linear Scale Factors in the Web-Mercator Projection

#### 5.4. Local Linear Scale Factors in the Albers Equal-Area Conic Projection

## 6. Instead of Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**An ellipsoidal differential quadrangle (

**left**) and its image in the plane of projection (

**right**).

**Figure 3.**General case of Tissot’s indicatrix showing local linear scale factors along a meridian $h=c\left(d\lambda =0\right)$, along a parallel $k=c\left(d\phi =0\right)$, in the direction of the x-axis $c\left(dy=0\right)$, in the direction of the y-axis $c\left(dx=0\right)$, and extremal values ${c}_{min}$ and ${c}_{max}$. The angle β is the angle between the images of a meridian and a parallel at a point under consideration.

**Figure 4.**Mercator projection map of the earth’s sphere with Tissot’s indicatrices. Source [11].

**Figure 5.**The world on a transverse Mercator projection of the Earth’s sphere, with 10° graticule and Tissot’s indicatrices overlaid. Source [13].

**Figure 6.**The world on an Albers projection of the Earth’s sphere, with 10° graticule and Tissot’s indicatrices overlaid. Standard parallels are at 45° N and 15° N. Source [16].

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**MDPI and ACS Style**

Lapaine, M.
Local Linear Scale Factors in Map Projections of an Ellipsoid. *Geographies* **2021**, *1*, 238-250.
https://doi.org/10.3390/geographies1030014

**AMA Style**

Lapaine M.
Local Linear Scale Factors in Map Projections of an Ellipsoid. *Geographies*. 2021; 1(3):238-250.
https://doi.org/10.3390/geographies1030014

**Chicago/Turabian Style**

Lapaine, Miljenko.
2021. "Local Linear Scale Factors in Map Projections of an Ellipsoid" *Geographies* 1, no. 3: 238-250.
https://doi.org/10.3390/geographies1030014