# Parallels in Cartography: Standard, Equidistantly Mapped and True Length Parallels

## Abstract

**:**

## 1. Introduction

- standard line; standard parallel
- A line on a map along which the principal scale is retained.
- r линия нулевых искажений. Линия на карте, вo всех тoчках кoтoрoй
- сoхраняется главный масштаб.
- f isométre m
- s línea f de sección; paralelo m base
- g Berührungslinie f

## 2. Map Projections

## 3. Standard Parallels

## 4. Equidistantly Mapped Parallels

#### 4.1. Orthographic Projection—Normal Aspect

- From (22), we can compute$$\frac{\partial x}{\partial \phi}=-R\mathrm{sin}\phi \mathrm{sin}\lambda ,\frac{\partial x}{\partial \lambda}=R\mathrm{cos}\phi \mathrm{cos}\lambda ,\frac{\partial y}{\partial \phi}=R\mathrm{sin}\phi \mathrm{cos}\lambda ,\frac{\partial y}{\partial \lambda}=R\mathrm{cos}\phi \mathrm{sin}\lambda $$$$E={R}^{2}{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}\phi ,F=0,G={R}^{2}{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\phi .$$

#### 4.2. Orthographic Projection—Transverse Aspect

- From (31), we can compute$$\frac{\partial x}{\partial \phi}=-R\mathrm{sin}\phi \mathrm{sin}\lambda ,\frac{\partial x}{\partial \lambda}=R\mathrm{cos}\phi \mathrm{cos}\lambda ,\frac{\partial y}{\partial \phi}=R\mathrm{cos}\phi ,\frac{\partial y}{\partial \lambda}=0$$$$E={R}^{2}\left(1-{\mathrm{s}\mathrm{i}\mathrm{n}}^{2}\phi {\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\lambda \right),F=-{R}^{2}\mathrm{sin}\phi \mathrm{sin}\lambda \mathrm{cos}\phi \mathrm{cos}\lambda ,G={R}^{2}{\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\phi {\mathrm{c}\mathrm{o}\mathrm{s}}^{2}\lambda .$$

## 5. True Length Parallels

#### 5.1. Example 1

#### 5.2. Example 2

## 6. Conclusions

- A standard point is a point where linear distortions are equal to zero in all directions.
- A standard parallel is a parallel whose all points are standard;
- An equidistantly mapped point in the direction of the parallel passing through that point has the property $k=1$, where $k$ is the linear scale factor in the direction of the parallel at that point;
- An equidistantly mapped parallel in the direction of that parallel is a parallel whose all points are equidistantly mapped in the direction of that parallel;
- An equidistantly mapped point in the direction of the meridian passing through that point has the property $h=1$, where $h$ is the linear scale factor in the direction of the meridian at that point;
- An equidistantly mapped parallel in the direction of the meridian is a parallel whose points are all equidistantly mapped in the direction of that meridian;
- An equidistantly mapped meridian in the direction of the meridian, as well as an equidistantly mapped meridian in the direction of the parallels, can be defined analogously;
- A parallel of true length is a parallel whose length in the projection plane is equal to the length of that parallel on the sphere (ellipsoid).

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**World map in an equidistant cylindrical projection for which $n=R\mathrm{cos}30\xb0$. The length of all parallels in the projection is $2\pi R\mathrm{cos}30\xb0=\pi \sqrt{3}R$. This length is equal to the length of the parallel on the sphere to which latitude 30° corresponds.

**Figure 7.**Every standard parallel is an equidistantly mapped parallel, and every equidistantly mapped parallel in the direction of the parallel is a true length parallel. The opposite need not be true.

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**MDPI and ACS Style**

Lapaine, M.
Parallels in Cartography: Standard, Equidistantly Mapped and True Length Parallels. *Geographies* **2024**, *4*, 52-65.
https://doi.org/10.3390/geographies4010004

**AMA Style**

Lapaine M.
Parallels in Cartography: Standard, Equidistantly Mapped and True Length Parallels. *Geographies*. 2024; 4(1):52-65.
https://doi.org/10.3390/geographies4010004

**Chicago/Turabian Style**

Lapaine, Miljenko.
2024. "Parallels in Cartography: Standard, Equidistantly Mapped and True Length Parallels" *Geographies* 4, no. 1: 52-65.
https://doi.org/10.3390/geographies4010004