Creating a flat map of the Earth requires a mathematical transformation known as a map projection. This technique transfers coordinates from the curved, three-dimensional surface of the globe onto a two-dimensional plane. It is impossible to perfectly flatten a sphere without stretching or compressing its surface. This geometric constraint means every flat map contains some degree of inaccuracy, which cartographers term distortion.
Understanding Map Projection and Inherent Distortion
The mathematical impossibility of creating a perfect flat map was demonstrated by Carl Friedrich Gauss’s Theorema Egregium. This theorem proves that a sphere’s surface cannot be mapped to a plane while simultaneously preserving both distances and angles. Since the surface of a sphere is not “developable”—meaning it cannot be unrolled flat—the conversion to a flat plane must introduce systematic errors.
This necessity for distortion forces cartographers to make a deliberate trade-off when selecting a projection. They must choose which properties of the globe they want to preserve at the expense of distorting others. A map can minimize error for a specific characteristic, such as area or shape, but it cannot preserve all properties perfectly. The choice of projection depends entirely on the map’s intended purpose, prioritizing the property most useful for the task.
The Four Principal Categories of Distortion
When the globe’s surface is transformed into a flat map, four fundamental metric properties are subject to distortion: area, shape, distance, and direction. Cartographers use Tissot’s Indicatrix—imaginary circles placed on the globe that become ellipses on a map—to visually represent the type and magnitude of these distortions. Every map projection must distort at least one of these four properties, and often all four to some degree.
Area (Equivalence)
Distortion of area, or equivalence, refers to the incorrect representation of the relative sizes of landmasses. An equal-area projection preserves the proportionality of sizes, meaning a unit of area on the map corresponds to the same area on the ground everywhere. However, preserving area severely distorts the shapes of landmasses, especially toward the edges or poles. For example, the common Mercator projection makes Greenland appear vastly larger than South America, which is actually eight times its size.
Shape (Conformality)
The distortion of shape, known as conformality, occurs when the local angles and outlines of features are not preserved. A conformal map maintains the proper shape of small areas and the correct angular relationships, a property valued for navigation. In a perfectly conformal map, the grid lines of latitude and longitude intersect at right angles, as they do on the globe. However, preserving shape sacrifices the accurate representation of area, causing landmasses far from the map’s center to be inflated in size.
Distance (Equidistance)
Distance distortion means the measured scale across the map is not consistent, making it impossible to measure true distances accurately everywhere. An equidistant projection is designed to preserve true scale. However, this property can only be maintained from one or two specific points to all other points, or along certain lines, such as meridians. For example, a map centered on a city can show the correct distance from that city to any other location. Measuring the distance between two arbitrary points on an equidistant map will likely yield an incorrect value.
Direction (Azimuth/True Bearing)
Directional distortion relates to the inaccuracy of the true compass bearing between two points. A projection that preserves true direction, or azimuth, is only capable of doing so from a single central point to all other points. The shortest path between two points on a sphere, known as a great circle route, often appears curved on a flat map. This means a straight line drawn on the map does not necessarily represent the correct bearing or the most direct route. This property is important for plotting air and sea navigation routes.
Balancing Distortion in Practical Cartography
Since no single flat map can perfectly preserve all four properties, cartographers must choose a projection that minimizes the distortion most detrimental to the map’s intended use. A map analyzing global population density requires an equal-area projection to prevent misrepresenting demographic data. Conversely, a nautical chart must be conformal to ensure compass bearings and angles are accurate for plotting a course.
For general reference and world maps, where the goal is a visually balanced representation, cartographers often employ “compromise projections.” These projections, such as the Robinson or Winkel Tripel, intentionally distort all four properties slightly rather than preserving one perfectly. By distributing the error across the entire map, compromise projections minimize the visual severity of any single distortion, creating a more universally acceptable view of the world.