This article was automatically translated from the original Turkish version.

Map Projections

Map projection is the process of transforming a curved reference surface such as a sphere or ellipsoid defined for the Earth’s surface onto a plane through a geometric transformation. Mathematically, it is defined by a pair of functions that establish the relationship between the reference surface coordinate system and the plane coordinate system. This transformation is known as the “forward transformation”; the transition from plane coordinates back to reference surface coordinates is called the “inverse transformation”.

Projecting the Earth’s surface onto a plane offers advantages in terms of map usage, perception, and storage. However, since the reference surfaces used for the Earth are closed surfaces, their transfer to a plane is not a simple scaling problem.

Core Issue: Distortion

Since it is geometrically impossible to unfold a curved surface such as a sphere or ellipsoid onto a plane, changes in area, angle, and length are unavoidable during the transformation process. These changes are termed “distortion” or “deformation”.

It is impossible to preserve all three fundamental cartographic properties—area, angle, and length—simultaneously. Therefore, map projections have been developed to preserve one of these properties according to the intended use of the map.

Distortions are visualized and analyzed using a method known as Tissot’s Indicatrix (or distortion ellipse). It is based on the projection of an infinitesimally small circle on the original surface as an ellipse on the projection plane. The major axis (a) and minor axis (b) of this ellipse represent the maximum and minimum length distortions at that point.

The fundamental classifications of projections according to the properties they preserve are as follows:

Conformal (Angle-Preserving) Projections: Projections that preserve angles in a differential (infinitesimal) sense. This implies that shapes remain similar, but this does not hold for finite-sized angles.

Equal-Area Projections: Projections that preserve the areas of finite-sized regions.

Equidistant Projections: Projections that preserve lengths along specific directions. In normal-position projections, meridian lengths are typically preserved.

Classification of Projections

Map projections are classified according to the projection surface used, the orientation of that surface, its contact condition with the reference surface, and the property preserved.

By Projection Surface (Auxiliary Surfaces)

A projection can be made directly from the reference surface onto a plane, or it can be performed via auxiliary surfaces that can be unfolded into a plane.

Azimuthal (Planar) Projections: The projection is made directly onto a plane.

Planar Projection (Generated by Artificial Intelligence)

Cylindrical Projections: The projection is made onto a cylindrical surface tangent to or intersecting the reference surface, which is then unfolded.

Cylindrical Projection (Generated by Artificial Intelligence)

Conic Projections: A conical surface is used for the projection.

Conic Projection (Generated by Artificial Intelligence)

By Surface Orientation

This refers to the position of the projection surface (plane, cylinder, or cone axis) relative to the reference surface.

Normal Position: The surface axis coincides with the Earth’s rotation axis.

Transverse Position: The surface axis lies in the equatorial plane (perpendicular to the rotation axis).

Oblique Position: The surface axis lies between the normal and transverse positions.

By Contact Condition (Points of Contact)

Tangent: The projection surface touches the reference surface at a single point (azimuthal) or along a circle (conic or cylindrical). No distortion occurs at the point of tangency or along the circle of tangency.

Secant: The projection surface intersects the reference surface. In this case, two circles of zero distortion can be obtained. Secant projections are often preferred to reduce overall distortion.

By Mathematical Structure

True (Perspective) Projections: A geometric projection center (projection point) is defined. Examples include gnomonic (center at the sphere’s center), stereographic (center at the opposite pole), and orthographic (center at infinity).

False (Pseudo) Projections: No true projection surface is defined. The projection is defined solely by a mathematical function pair relating curved surface coordinates to plane coordinates. They are typically used to represent the entire Earth’s surface.

Projection Selection

The choice of an appropriate projection depends on factors such as the map’s intended use, target audience, location, shape, size of the area being mapped, and the map scale. The primary goal is to select the method that produces the smallest possible distortion for the region of interest.

Some widely accepted selection rules are as follows:

By Region Location: Azimuthal (planar) projections are preferred for polar regions, conic projections for mid-latitudes, and cylindrical projections for equatorial regions.

By Region Shape (Young’s Rule): If the area to be projected is approximately circular, a planar (azimuthal) projection is recommended. If the area is elongated (rectangular strip), a cylindrical or conic projection aligned with the longer dimension is preferred.

By Map Purpose: Conformal (angle-preserving) projections are generally preferred for large-scale topographic maps and medium-scale physical maps. For small-scale thematic maps covering large areas (e.g., population distribution), equal-area projections are recommended.

By Region Size:

World Maps (Single Piece): Projections such as Mollweide, Eckert (equal-area), Mercator, Lagrange (conformal), or Robinson (distortion-compromise) are commonly used to represent the entire Earth.

Hemisphere Maps: Commonly used in atlases. Stereographic (conformal), Lambert Azimuthal (equal-area), or azimuthal equidistant projections are used.

Continental and National Maps: For east-west oriented areas in mid-latitudes, Lambert Conformal Conic or Albers Conic projections are used; for north-south oriented areas, Transverse Mercator is used.

Standardization and Coordinate Reference Systems (CRS)

Theoretically, an infinite number of map projections can be defined, and advances in information technology and Geographic Information Systems (GIS) have created a need for a standardized system of definition and coding.

The most widely used coding system in this field is EPSG. EPSG (European Petroleum Survey Group) has published a dataset containing geodetic parameters, coordinate reference systems (CRS), units of measurement, and ellipsoid definitions. These codes are known as SRID (Spatial Reference System Identifier).

Some commonly used EPSG codes are:

EPSG:4326: Defines two-dimensional latitude/longitude coordinates on the WGS84 ellipsoid. This is not a projection but a geographic coordinate system, primarily intended for data storage.

EPSG:3857: Web Mercator Projection. Used by web-based mapping services such as Google Maps, Bing Maps, and OpenStreetMap.

EPSG:23036: Defines UTM 6° zones on the ED50 ellipsoid; previously used for large and medium-scale topographic maps.

Theoretical Approaches and Numerical Analysis

Map projections are grounded in mathematical principles. The complexity of projections necessitates the use of numerical analysis methods in modern cartography.

Numerical Differentiation

To calculate projection distortions (such as Tissot’s Indicatrix elements), the partial derivatives of the projection equations (∂x/∂φ, ∂y/∂λ, etc.) must be known. Analytically computing these derivatives can be complex. Numerical differentiation methods allow approximate values of these partial derivatives to be calculated directly from the function itself, without analytical derivation—for example, using f′(x)≈2tf(x+t)−f(x−t). A small parameter value such as t=10⁻⁶ has been found to yield satisfactory results for map projection equations.

Iterative Solutions and Auxiliary Variables

Some pseudo-cylindrical projections (e.g., Mollweide projection) do not define a direct relationship between latitude (φ) and the plane coordinate (y), but instead use an auxiliary variable (θ) defined by an implicit function. For example, for Mollweide, the equation 2θ+sin(2θ)=πsin(φ) holds. Finding the value of θ for a given φ requires iterative numerical methods such as the Newton-Raphson method.

Numerical Inverse Projection

For some projections (e.g., Winkel Tripel), the inverse transformation equations that convert plane coordinates (x,y) back to geographic coordinates (φ,λ) form nonlinear equation systems that cannot be solved analytically. In such cases, the inverse transformation can only be performed using numerical methods such as Newton-Raphson. This approach, combined with numerical differentiation, enables the calculation of inverse projections using only the forward projection equations.

Cartograms (Density-Preserving Projections)

As a theoretical approach, cartograms differ from traditional map projections. Their purpose is not to preserve geographic area (in square meters) but to “adjust” the representation of a phenomenon (population, income, etc.) within that area. The goal of these transformations is to scale the area of a region on the map proportionally to a value assigned to it (e.g., population or retail sales volume). Mathematically, this can be expressed as a transformation where Tissot’s area distortion measure (S) is set equal to the given density distribution (D): D=S.

Common Projections and Application Errors

Mercator Projection

A normal-position, conformal cylindrical projection introduced by Gerardus Mercator in 1569. Its key feature is the accurate representation of loxodromes—curves that cross meridians at constant angles—as straight lines, preserving angles. This property made it useful for compass navigation. However, it causes extreme area distortion near the poles and is unsuitable for representing the entire Earth.

The Web Mercator variant (commonly EPSG:3857) is widely used in online mapping services such as Google Maps. Its conformal nature ensures that square or rectangular shapes like buildings remain undistorted at high zoom levels and that geographic grids intersect at right angles, making it favorable for computer graphics. Nevertheless, this adaptation has faced criticism for severe area distortion at global scales.

Gauss-Krüger (Transverse Mercator) Projection

This is the application of the Mercator projection in transverse position using an ellipsoidal reference surface. It is a conformal projection used for large-scale mapping. The projection is constructed on a cylinder tangent to the ellipsoid along a chosen central meridian. To avoid increasing distortion away from the central meridian, the projection is applied in narrow “zones”.

Direct Use of Geographic Coordinates (EPSG:4326)

A common problematic practice is the direct use of geographic coordinates (latitude, longitude) as if they were planar (x,y) coordinates in a map projection. This system, commonly known by the EPSG:4326 (WGS84) code, is primarily intended for data storage.

When these data are displayed on a map, the result is equivalent to an equidistant cylindrical projection (also called Plate Carrée). This projection is only suitable for regions near the equator. When used in mid-latitude regions such as Türkiye, it causes severe compression in the north-south direction, resulting in an inaccurate representation. Additionally, when horizontal units are in degrees and vertical units (if elevation data is present) are in meters, software such as QGIS produces incorrect results in three-dimensional operations like hillshading or hydrological analysis. For such analyses, data must be transformed into a metric CRS.

Applications in Türkiye

In Türkiye, standard projection systems defined by legal regulations and guidelines are used for large and medium-scale map production. Both standards are based on the conformal transverse cylindrical Gauss-Krüger projection.

TM (Transverse Mercator) - 3° Zones:

Used for mapping projects at scales of 1:5000 and larger (as per the Regulation on Production of Large-Scale Maps and Map Information).
The reference surface is the GRS80 ellipsoid.
Zone width is 3°.
No scale factor is applied at the central meridian (m0=1).
In this system, at zone boundaries (1.5° away), length increases of 190 ppm to 231 ppm (19 cm to 23 cm per 1000 meters) occur at Türkiye’s latitudes.

UTM (Universal Transverse Mercator) - 6° Zones:

Used for medium-scale topographic mapping between 1:25,000 and 1:250,000 scales (by the General Command of Mapping).
The reference surface is the WGS84 ellipsoid.
Zone width is 6°.
A scale factor of m0=0.9996 is applied at the central meridian. This factor compensates for increasing distortion away from the central meridian; therefore, length is not preserved along the central meridian.

Author Information

AuthorYunus Emre YüceDecember 1, 2025 at 12:03 AM

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Core Issue: Distortion
Classification of Projections
- By Projection Surface (Auxiliary Surfaces)
- By Surface Orientation
- By Contact Condition (Points of Contact)
- By Mathematical Structure
Projection Selection
Standardization and Coordinate Reference Systems (CRS)
Theoretical Approaches and Numerical Analysis
- Numerical Differentiation
- Iterative Solutions and Auxiliary Variables
- Numerical Inverse Projection
- Cartograms (Density-Preserving Projections)
Common Projections and Application Errors
- Mercator Projection
- Gauss-Krüger (Transverse Mercator) Projection
- Direct Use of Geographic Coordinates (EPSG:4326)
Applications in Türkiye