Map projection - Designing and Classifying Projections

Design and Construction of Projections The Two-Step Process Creating a map projection always involves two fundamental steps. First, you must choose a model for Earth's shape: either a sphere (simpler) or an ellipsoid (more accurate, since Earth is slightly flattened at the poles). This choice affects the precision of the final map. Second, you must transform geographic coordinates into plane coordinates. Geographic coordinates use latitude (φ) and longitude (λ) to pinpoint locations on Earth's curved surface. A map projection converts these into flat, Cartesian coordinates (x, y) or polar coordinates (r, θ) that can be drawn on paper or displayed on a screen. Developable Surfaces: The Key Concept The central challenge of map projection is that you cannot flatten a sphere perfectly without distorting it. The solution is to use an intermediate surface—one that is developable, meaning it can be unfolded into a flat plane without stretching or tearing. Three surfaces are developable: Cylinders: can be unrolled into a rectangle Cones: can be unrolled into a sector of a circle Planes: already flat, used for azimuthal projections Any projection method projects Earth's features onto one of these surfaces, then unfolds it to create the final map. This is why projection design begins by choosing a projection surface. How the Surface Touches the Globe When you place a developable surface against Earth, it can touch Earth in two ways: Tangent surfaces touch Earth at a single line. For example, a cylinder wrapped around Earth might touch along the equator, or a cone might touch along a single latitude circle. At this contact line (called a standard line), scale is perfectly accurate and distortion is zero. Secant surfaces cut through Earth, making contact along two lines. For example, a secant cone contacts along two latitude circles called standard parallels. Between the standard parallels, scale is generally better than with a tangent surface, making secant surfaces useful for minimizing overall distortion. Aspect: Orientation of the Surface The aspect of a projection describes how the developable surface is oriented relative to Earth's axis: Normal aspect: The surface's axis aligns with Earth's axis. For a cylinder, the contact line is at the equator; for a cone, standard parallels are placed symmetrically around a latitude of interest. Transverse aspect: The surface's axis is perpendicular to Earth's axis. The contact line becomes a meridian (line of longitude). Oblique aspect: The surface is tilted at some intermediate angle, useful for centering a map on a region that is neither on the equator nor a pole. Different aspects allow you to minimize distortion for different map areas. A map of a polar region benefits from polar azimuthal aspect, while a map of a narrow east-west strip might use transverse cylindrical aspect. Standard Lines and the Central Meridian Standard parallels are latitudes where a secant cone or cylinder touches the globe without distortion. When you specify standard parallels, you control where the map is most accurate. For example, the Albers conic projection for the United States often uses standard parallels at 29.5°N and 45.5°N, minimizing distortion across that region. The central meridian (λ₀) is a reference longitude that becomes the origin of the coordinate system in the projection. It marks where x = 0. Choosing a central meridian near the center of your map region helps keep coordinates manageable and centers the map usefully. Classification of Map Projections Projections are classified in two ways: by the type of surface they use, and by the property they preserve. Classification by Surface Type Cylindrical projections: Use a cylinder; meridians are vertical parallel lines, parallels are horizontal lines (though spacing varies). Good for showing the world or equatorial regions. Conic projections: Use a cone; meridians radiate outward from a point, parallels are circular arcs. Good for mid-latitude regions. Azimuthal (planar) projections: Use a plane; directions from the center point are preserved. Good for poles and small regions. Classification by Preserved Property A projection cannot preserve all properties simultaneously—there is always a trade-off. Different projections prioritize different properties: Conformal projections preserve angles and local shapes. The scale may vary with location, but angles within small areas remain accurate. Useful for navigation. Equal-area projections preserve area; regions have correct relative sizes. Shapes may be distorted. Useful for comparing regional statistics. Equidistant projections preserve distances from a center point or along certain lines. Useful for showing distance from a specific location. Retroazimuthal projections preserve direction to a fixed location (rare; useful for radial applications like radio transmission). Cylindrical Projections Cylindrical projections are among the most common, used for world maps, navigation, and regional displays. In normal aspect, a cylinder wraps around Earth at the equator. Scale Behavior The defining characteristic of a cylindrical projection is how north-south and east-west scales vary with latitude (φ): Mercator (Conformal): East-west scale = North-south scale = sec(φ) This means scale increases dramatically toward the poles, causing extreme exaggeration at high latitudes (Greenland appears as large as Africa, though Africa is much larger). Conformal: angles are preserved locally, making it valuable for navigation where compass headings are critical. Equirectangular (Plate Carrée): Both scales equal 1 (constant everywhere) Simple to construct and understand, but distorts shapes at high latitudes Equal spacing in both directions Equal-Area Cylindrical (Lambert): East-west scale = 1; North-south scale = cos(φ) The reciprocal relationship ensures area is preserved Latitudes are compressed toward the poles, making high-latitude regions appear squeezed Several variants exist—Gall-Peters, Behrmann, and others—differing in which latitude they choose as the standard parallel (where scale is true). This choice affects how distortion is distributed across the map. Transverse Cylindrical Transverse Mercator rotates the cylinder 90°, so its contact line runs along a meridian instead of the equator. This is ideal for mapping regions that are taller than they are wide, like countries in narrow east-west bands. It's so effective that many national mapping systems use it as their official projection. Pseudocylindrical Projections Pseudocylindrical projections look superficially like cylindrical projections (parallels are straight horizontal lines) but meridians curve instead of remaining parallel. This curvature allows better properties. Sinusoidal projection is a key example: Preserves area perfectly The length of each parallel is proportional to cos(φ), matching Earth's true surface; this is why meridians must curve Useful for thematic maps (climate, population, etc.) where accurate area representation is essential Conic Projections Conic projections project Earth onto a cone, which is then unrolled into a flat sector. They are particularly useful for mid-latitude regions because distortion can be controlled through choice of standard parallels. Basic Structure In a conic projection: Meridians are equally-spaced straight lines radiating outward from the cone's apex (a single point on the map) Parallels are circular arcs centered on the apex The "opening angle" of the sector depends on which cone is used (a narrower cone produces a wider sector, and vice versa) Standard Parallels Like cylindrical projections, conic projections can be tangent (touching along one latitude) or secant (touching along two standard parallels). With two standard parallels, you can control distortion across a range of latitudes—important for mapping regions like the United States. Three Common Conic Types Equidistant Conic: Preserves distances along meridians (north-south distances are accurate) Doesn't preserve area or angles, so it's a compromise projection Good for reconnaissance maps where distance matters moderately Albers Equal-Area Conic: Preserves area Adjusts spacing of parallels (north-south scale) to achieve equal area Widely used for thematic maps and atlases covering regions like the contiguous United States Lambert Conformal Conic: Preserves angles and local shapes Used extensively for aeronautical and topographic maps Parallels are more tightly spaced away from the standard parallels to maintain angles The State Plane Coordinate System in the United States uses this projection Azimuthal (Planar) Projections Azimuthal projections project Earth onto a plane tangent to the globe. Their defining property is that directions and great-circle paths through the center point are preserved as straight lines. This makes them invaluable for navigation and communications. Core Property In any azimuthal projection, a straight line from the center of the map points in the true direction (bearing) from that center location in the real world. Great circles (shortest paths on Earth) passing through the center appear as straight lines on the map. Perspective Azimuthal Projections Some azimuthal projections can be derived geometrically by imagining a light source at a specific location, projecting Earth's features onto a tangent plane: Gnomonic: Light source at Earth's center Great circles appear as straight lines (useful for great-circle route planning) Extreme distortion away from center; shows only a hemisphere Used for navigation planning Stereographic: Light source at the antipode (opposite point) of the tangent point Conformal (preserves angles and shapes) Finite distortion everywhere, so it can show more than a single hemisphere Used for polar maps and historical atlases Orthographic: Light source at infinite distance; parallel rays Maps a hemisphere as a circle, appearing like a globe viewed from space Distorts distances and areas; useful for visualization rather than measurement Non-Perspective Azimuthal Projections These are constructed mathematically rather than from a geometric viewpoint: Azimuthal Equidistant: Distance from the center point is proportional to true surface distance Perfect for showing distances from a specific location (e.g., distance from your home city) Used in emergency response, radio transmission planning, and seismic applications Directions are also preserved, like all azimuthal projections Lambert Azimuthal Equal-Area: Preserves area Area is proportional to the sine of half the central angle Useful for thematic mapping centered on a point <extrainfo> Polyhedral Projections Polyhedral projections take a different approach: instead of using a single surface, the globe is divided into the faces of a polyhedron (such as a cube, icosahedron, or dodecahedron). Each face is then projected separately. After all faces are mapped, they can be unfolded or arranged in different configurations. This approach is useful for: Digital mapping systems where data is naturally stored on discrete faces Creating hierarchical maps that can be zoomed and subdivided Minimizing distortion across an entire globe by distributing it evenly across many small regions </extrainfo>