Map Projections Explained: Why Every World Map Is Distorted

The geographic coordinate system uses two types of imaginary lines to locate any point on Earth. Parallels (paralelos) are horizontal lines that run east-west around the Earth, measuring the distance in degrees from the Equator (0°) to a specific location. Latitude indicates this distance, ranging from 0° at the Equator to 90° at the poles (North or South). Meridians (meridianos) are vertical lines that run north-south, connecting the poles. Longitude measures the distance in degrees from the Prime Meridian (Greenwich, 0°) to a specific location, extending east or west. Together, latitude and longitude allow precise identification of any point on Earth's surface.

This section introduces the spherical coordinate system used to uniquely locate positions on Earth's curved surface. The system uses the Equator as the horizontal reference circle and the Prime Meridian (through Greenwich) and International Date Line as vertical reference circles. Unlike planar coordinates, measurements are made using angles rather than linear distances. Lines of longitude run pole-to-pole at regular intervals, while lines of latitude form parallel circles around the planet. This angular measurement system allows precise identification of any location despite Earth's three-dimensional curvature.

The geographic coordinate system determines any person's location on Earth using latitude and longitude. Latitudes run horizontally from north to south based on the Equator, while longitudes run vertically from east to west based on the Greenwich Meridian. This system enables identification of any point's coordinates, such as latitude 60°N and longitude 40°E for point A, or latitude 0° and longitude 20°E for point C. GPS technology also operates on this same coordinate system to navigate anywhere globally.

The Earth uses a coordinate system of latitude and longitude to locate any point on its surface. Latitude measures angular distance north or south of the equator (0 degrees), ranging from 0 to 90 degrees at the poles. Longitude measures angular distance east or west of the prime meridian (0 degrees through Greenwich, England), ranging from 0 to 180 degrees. The equator divides Earth into Northern and Southern Hemispheres, while the prime meridian divides it into Eastern and Western Hemispheres. This system enables precise navigation and mapping of locations worldwide.

Latitude and longitude form a coordinate system for identifying absolute locations on Earth, used by GPS and mapping applications. Coordinates are always listed in pairs with latitude first, followed by longitude. The equator (0° latitude) divides Earth into northern and southern hemispheres, while the prime meridian (0° longitude) divides it into eastern and western hemispheres. Latitude lines run parallel to the equator, while longitude lines converge at the poles. To find a location, start with latitude, then follow that line to match with the longitude. Directions (N, S, E, W) indicate the hemisphere, and zero degrees for either coordinate means no directional abbreviation is needed.

Geometry begins with a sphere that can be sectioned by a plane according to the maximum visible circle. This plane projects all curves behind and in front of it, flattening them while maintaining absolute respect between the curve and the plane. When a three-dimensional event is projected onto a frontal plane, it reduces to two dimensions, allowing correct representation of three-dimensional volume on two-dimensional plane. The angle that requires 360° to close on itself becomes 180° when flattened, representing a halving of the angle. This allows imagining two faces of the plane, one in front and one facing an observer behind.

The fold and cut problem generalizes to 3D: flattening polyhedron surfaces so all edges align on a common plane. Recent work shows orientable manifolds (without Möbius strips) can be flattened using disk-packing methods. The ultimate open problem generalizes to D dimensions: bringing all 0D points to a common point, 1D edges to a common line, 2D faces to a common plane, and so on up to D-1 dimensional faces to a common hyperplane. While surface flattening is known, solid 3D object flattening remains unsolved. This represents the frontier of geometric folding research.

This section explores how certain curved surfaces can actually be flat when properly understood. The cylinder is examined as having zero Gaussian curvature despite appearing curved extrinsically—it curves along only one axis while remaining straight along another. The presenter then introduces the Clifford torus, a two-dimensional surface embedded in four-dimensional space that can be made completely flat. This requires four spatial dimensions to avoid self-intersection, similar to the Klein bottle. The section clarifies the distinction between intrinsic curvature (detectable by beings within the surface) and extrinsic curvature (how surfaces bend in higher-dimensional space).

When representing the surface of a 3D object on a 2D surface, some distortion is unavoidable because 3D objects have three dimensions (length, width, and height) while 2D surfaces only have two dimensions (length and width); this dimensional mismatch means that even simple objects like cubes must be distorted to fit onto a flat surface, as demonstrated by the speaker's dice example where the 2-inch cube appears with unequal sides when flattened.

The three essential components of maps are: Directions (to indicate orientation and cardinal points), Symbols (to represent various features using standardized icons), and Scale (to show the relationship between map distance and actual ground distance). Scale is necessary for maps because it allows large areas to be represented on smaller surfaces while maintaining accurate distance relationships. Maps are true representations of the Earth's surface, specifically conformal maps that maintain the true shapes of areas. Maps show special relationships between places, including directions, distances, and spatial connections. Thematic maps are designed to show the distribution of specific features like forests, providing specialized information about particular topics.

Maps maintain different properties to varying degrees: conformity preserves angles and shapes (latitude-longitude lines intersect at 90°), while equivalence maintains correct area relationships. These properties cannot be perfectly achieved simultaneously—the Mercator projection exemplifies this trade-off by preserving direction and shape while severely distorting size. The choice between conformal and equivalent projections depends on analytical goals: navigation requires conformal maps, while area-based studies require equivalent projections. This fundamental trade-off shapes all cartographic decisions.

Maps are classified by what properties they preserve: equidistant maps preserve distances between points, equivalent maps preserve surface areas (showing continents in correct relative sizes), and conformal maps preserve angles (crucial for navigation). The Mercator projection is conformal, allowing navigators to follow straight lines (rhumb lines) for constant compass directions. The Peters projection is equivalent, showing Africa's true size relative to other continents. These properties are mutually exclusive—no map can preserve all three simultaneously. The choice depends on the map's purpose: navigation, area comparison, or distance measurement.

Os mapas possuem quatro principais propriedades: (1) Conforme - as coordenadas se cruzam formando ângulos retos, preservando formas e direções, mas distorcendo distâncias e tamanhos; (2) Equidistante - as distâncias entre pontos e escalas entre continentes permanecem sem deformação, enquanto outros aspectos não são representados com fidelidade; (3) Equivalente - as áreas e tamanhos permanecem fiéis, mas há distorção das formas e distâncias; (4) Afilática - tenta minimizar distorções em todos os aspectos acima citados, sem conservar nenhum deles perfeitamente.

Maps contain three essential components: Scale (the proportional relationship between actual distance and map distance, e.g., 1 cm = 100 km), Direction (using cardinal points like North, East, South, West and intermediate directions like North East, South West), and Symbols (standardized representations for features like water, mountains, and vegetation). Small scale maps show large areas with compressed detail, while large scale maps show smaller areas with more detail.