Map Projections: Scientific Overview and Classification

Description and Overview

Map projections are systematic mathematical transformations that convert the three-dimensional coordinates of locations on the Earth's surface (where the third dimension refers to the Z positions on the reference sphere) into two-dimensional representations suitable for flat maps. Because it is impossible to represent a curved surface on a plane without distortion, every map projection introduces some form of deformation in area, shape, distance, or direction.

Understanding these distortions is fundamental to accurate cartographic representation and spatial analysis. On a small-scale map of the whole world this matters far more than on a large-scale map of a single region. As Snyder notes, "On maps of countries or even of continents, distortion may not be evident to the eye, but it becomes apparent upon careful measurement and analysis" (Snyder, 1987, p. 3). The choice of projection represents a compromise; the selection requires consideration of the geographic extent being mapped, the purpose of the map, and the spatial relationships that need to be preserved (Snyder, 1987).

Classification of Map Projections (Snyder)

This software primarily utilizes the classification system established by John P. Snyder (USGS), which groups projections based on their geometric construction and the specific properties they preserve.

1. Geometric Construction (Developable Surfaces)

Projections are often categorized by the type of "developable surface" onto which the reference sphere is projected. A developable surface is one that can be flattened onto a plane without stretching or tearing (Snyder, 1987).

• Cylindrical: The globe is projected onto a cylinder. When unrolled, meridians and parallels usually form a rectangular grid.
• Conic: The globe is projected onto a cone. When unrolled, meridians appear as straight lines radiating from a point, and parallels appear as concentric arcs.
• Azimuthal (Planar): The globe is projected onto a flat plane. Meridians radiate from a central point (often the pole), and parallels are concentric circles.

2. Preserved Properties

In addition to construction, Snyder classifies projections by the metric properties they maintain (Snyder, 1987):

• Conformal (Orthomorphic): Preserves local angles and shapes. At any point, the scale is the same in every direction, though it varies across the map.
• Equal-Area (Equivalent): Preserves the area of displayed features. A coin placed anywhere on the map covers the same surface area of the Earth.
• Equidistant: Preserves true scale along specific lines (e.g., the equator or meridians) or from specific points.
• Azimuthal: Preserves true direction (azimuth) from a central point to all other points.

3. Aspect

The aspect describes the orientation of the projection surface relative to the Earth's axis (Snyder, 1987):

• Normal: The standard orientation (e.g., cylinder axis parallel to Earth's axis).
• Transverse: The projection surface is rotated 90 degrees (e.g., cylinder axis perpendicular to Earth's axis).
• Oblique: The projection surface is at any other angle.

Scientific Classification Frameworks (Lee and Tobler)

While Snyder provides a practical framework for software use, scientific literature offers deeper mathematical classifications. This manual incorporates the works of L.P. Lee and Waldo Tobler to provide a comprehensive theoretical background.

Lee's Dual Classification

Lee proposed a rigorous system that separates projections into two distinct sets of groups: Representation (based on property) and Graticule (based on the pattern of the grid) (Lee, 1944).

Representation Groups:

• Conformal: Preserves angles (same as Snyder).
• Authalic: Preserves area (synonymous with Equal-Area).
• Aphylactic: Projections that are neither conformal nor authalic (compromise projections).

Graticule Groups:

• Cylindric: Meridians and parallels are straight perpendicular lines.
• Pseudocylindric: Parallels are straight; meridians are curved (except the central one).
• Conic: Meridians are radiating straight lines; parallels are circular arcs.
• Pseudoconic: Parallels are circular arcs; meridians are curved.
• Polyconic: Parallels are non-concentric circular arcs.
• Azimuthal: Meridians are radiating straight lines; parallels are concentric circles.

Tobler's Parametric Classification

Waldo Tobler introduced a parametric classification based on the mathematical equations used to transform spherical coordinates (latitude, longitude) into map coordinates. He categorizes projections into four classes (A-D) based on variable dependency (Tobler, 1962):

• Class A: The most general class where both map coordinates depend on both latitude and longitude (e.g., Polyconic).
• Class B: A rare category including some complex azimuthal forms.
• Class C: One coordinate depends on both variables, the other only on latitude. This includes Pseudocylindrical and Pseudoconic projections (e.g., Sinusoidal, Bonne).
• Class D: The simplest and most common class where coordinates are separable. This includes standard Cylindrical, Conic, and Azimuthal projections in their normal aspect.

Comparative Overview of Projections

The following table correlates the practical classification used in this App (Snyder) with the scientific classifications of Lee and Tobler.

Use and Properties

In the GeoCanvas App, map projections are organized according to the classification families defined by Snyder. This system groups projections based on the developable surface - the geometric shape (cylinder, cone, or plane) onto which the Earth's surface is projected (Snyder, 1987).

Choosing the correct family is critical because a map cannot preserve all properties simultaneously. As Snyder notes, "the cartographer must choose the characteristic which is to be shown accurately at the expense of others, or a compromise of several characteristics" (Snyder, 1987, p. 3). The selection depends on the geographic extent of the region (e.g., a continent versus a small town) and the specific properties required, such as accurate area, shape, or direction.