Projection (cartography) - Design and Families of Projections

Design and Construction of Map Projections Introduction A map projection is a systematic method for representing Earth's three-dimensional curved surface on a flat two-dimensional plane. Since Earth is approximately spherical, and a sphere cannot be flattened without distortion, every map projection involves a trade-off: some properties (like area, shape, direction, or distance) must be sacrificed to preserve others. Understanding how projections are designed and classified is essential for choosing the right projection for any mapping task. Step 1: Choosing an Earth Model The first decision in projection design is selecting a model of Earth's shape. You can choose between: Spherical model: Treats Earth as a perfect sphere with radius $R$. This is simpler and works well for small-scale maps (those showing large regions). Ellipsoidal model: Accounts for Earth's actual shape—slightly flattened at the poles and bulging at the equator. This is more accurate for large-scale maps (those showing smaller regions in detail). The choice affects how much shape information is lost during projection. A spherical model is easier to work with mathematically but less realistic; an ellipsoidal model is more accurate but more complex to compute. Step 2: Transforming Geographic Coordinates Once you've chosen an Earth model, the next step converts the geographic coordinates (latitude $\phi$ and longitude $\lambda$) into flat map coordinates. The two common coordinate systems are: Cartesian coordinates $(x, y)$: standard rectangular grid coordinates on the plane Polar coordinates $(r, \theta)$: distance from a central point and direction from that point Different projections use different transformation formulas. For example, some projections might preserve the angle $\theta$ as the longitude $\lambda$, while others might transform it. Similarly, the radial distance $r$ might depend on the latitude $\phi$ in different ways depending on which metric properties the projection preserves. Developable Surfaces A key concept in projection design is the developable surface—a surface that can be unfolded onto a plane without stretching or distortion. The three types are: Plane: A flat surface, already in 2D Cylinder: Can be unrolled into a rectangle Cone: Can be unfolded into a sector of a circle The sphere and ellipsoid, by contrast, are non-developable surfaces. You cannot unfold them without stretching some parts. This is why projections must make compromises on the properties they preserve. Projection designers place one of these developable surfaces in contact with (or intersecting) the Earth model, then project Earth's features onto that surface, and finally unfold it to create a flat map. Aspect of the Projection The aspect specifies how the developable surface is oriented relative to Earth's axis. There are three possibilities: Normal aspect: The surface's axis aligns with Earth's axis. For a cylinder, this produces the most familiar cylindrical projections with meridians as vertical lines and parallels as horizontal lines. Transverse aspect: The surface's axis is perpendicular to Earth's axis, rotated 90°. Oblique aspect: The surface's axis is tilted at some intermediate angle. The same projection formula can produce very different maps depending on its aspect. A projection might be excellent for showing one region in transverse aspect but poorly suited for that region in normal aspect. Standard Lines and Distortion When a developable surface touches or cuts the Earth model, the lines of contact are called standard lines. At these lines, the projection is distortion-free (the map scale equals the Earth scale). Away from standard lines, distortion increases. Tangent case: The surface touches at exactly one line (e.g., a cylinder tangent to the equator). Distortion is minimal near the tangent line and increases as you move away. Secant case: The surface intersects along two lines (e.g., a cone cutting through two parallels). Distortion is minimal between these two lines and increases both toward and away from them. A standard parallel is a standard line that is a latitude (a parallel of latitude). For instance, a conic projection might have two standard parallels at 30°N and 60°N, with minimal distortion between them. Classification of Map Projections Map projections are classified along two main dimensions: their mathematical construction and the metric property they preserve. By Projection Surface The surface-based classification is the most common way to organize projections: Cylindrical: The developable surface is a cylinder Conic: The developable surface is a cone Planar (or Azimuthal): The developable surface is a plane Each category has normal, transverse, and oblique variants, so there are actually nine basic combinations. Additionally, pseudo variants (pseudocylindrical, pseudoconic) and specialized families (polyconic, retroazimuthal) exist for projections that don't fit strictly into these three categories. By Preserved Metric Property The second classification describes what the projection preserves: Azimuthal (also called equidirectional): Preserves direction from a central point. All great circles through that point appear as straight lines. Conformal: Preserves local angles and shapes. This is essential for navigation maps (like nautical charts) but requires severe scale distortion, especially toward the poles. Equal-area (also called equiareal or equivalent): Preserves area. Regions appear with correct relative sizes, though shapes are distorted. Equidistant: Preserves distance from a central point or along certain lines (typically meridians). The Fundamental Trade-off A crucial fact: no projection can simultaneously be both conformal and equal-area. Because the sphere is non-developable, you cannot have zero distortion everywhere. Any projection that preserves angles locally must allow area to vary. This is not a limitation of mapmakers' skill—it's a mathematical impossibility. Cartographers must choose which property matters most for their purpose. Cylindrical Projections Cylindrical projections are perhaps the most familiar, since they produce rectangular maps with a familiar appearance. Normal Cylindrical Projections In a normal cylindrical projection: Meridians become equally spaced vertical straight lines Parallels become horizontal straight lines The projection preserves the latitude-longitude grid structure The key to cylindrical projections is controlling the spacing of parallels (latitude lines). Since meridians are equally spaced, they always have the same scale, and the east-west stretch is determined by latitude alone. Scale Distortion in Cylindrical Projections At latitude $\phi$, the circumference of the parallel circle on Earth is $2\pi R \cos\phi$ (smaller at higher latitudes due to the $\cos\phi$ factor). On the map, if meridians are equally spaced at their equatorial spacing, the parallel must be stretched to match. This produces an east-west stretch factor of $\sec\phi$ (the secant of the latitude). The north-south scale varies depending on the projection type, as we'll see below. Mercator (Conformal Cylindrical) The Mercator projection makes the projection conformal by applying equal stretch in both directions. North-south stretch equals east-west stretch: $\sec\phi$. Formula: The distance from the equator to latitude $\phi$ is proportional to $\int0^\phi \sec u \, du = \ln\left|\tan\left(\frac{\pi}{4} + \frac{\phi}{2}\right)\right|$ Key properties: Local angles and shapes are preserved (conformal) Meridians and parallels meet at right angles Great circles are not straight lines; instead, rhumb lines (constant bearing paths) are straight Severe distortion toward the poles (Greenland appears enormous, Antarctica is pushed off the map) Essential for navigation but poor for visualizing global patterns Equirectangular (Plate Carrée) The equirectangular projection uses a north-south stretch of $1$, meaning latitude spacing is preserved exactly as it is on Earth. Formula: $x = \lambda$, $y = \phi$ (after scaling to map coordinates) Key properties: Simple: it's just a grid of equally-spaced meridians and parallels Not conformal (angles are distorted) Not equal-area (areas are distorted more toward poles) Minimal distortion near the equator, increasing toward the poles Less commonly used today but still appears in simple atlases Equal-Area Cylindrical The equal-area cylindrical (also called Gall-Peters when using specific latitudes) preserves area by compressing north-south: North-south scale = $\cos\phi$ (the reciprocal of east-west stretch) Key properties: Areas are preserved perfectly Shapes are severely distorted (tall, thin regions) The poles appear as lines rather than points Good for thematic maps where accurate area representation is critical Transverse Cylindrical Projections A transverse cylindrical projection rotates the cylinder so its axis is perpendicular to Earth's axis. Instead of the equator, a chosen meridian becomes the tangent or secant line. Key concept: The properties (conformal, equal-area, etc.) are preserved along the chosen meridian rather than the equator. This makes transverse projections ideal for mapping regions that extend north-south, such as a single country or narrow strip. The most famous example is Transverse Mercator, used for many national mapping systems and as the basis for UTM (Universal Transverse Mercator) coordinates. Transverse Mercator minimizes distortion for a north-south-trending region. Oblique Cylindrical Projections An oblique cylindrical projection aligns the cylinder with a great circle that is neither the equator nor a meridian—typically chosen for best accuracy over a specific region that runs diagonally across Earth's surface. These are less common than normal or transverse variants but are useful when mapping a region that doesn't align neatly with the equator or a meridian. Pseudocylindrical Projections Some projections have cylindrical-like properties (parallels are horizontal, equally spaced) but violate the strict rule that all meridians are straight lines. These are called pseudocylindrical projections. Sinusoidal Projection The sinusoidal projection is one of the most important pseudocylindrical projections. Design principle: The length of each parallel equals the cosine of its latitude times the equatorial circumference. In other words, parallels are shortened toward the poles, just as they are on Earth. Key properties: Equal-area (preserves area perfectly) Meridians curve inward, following a sinusoidal (sine-wave) shape—hence the name Less extreme shape distortion than cylindrical equal-area, since parallels are not stretched Uninterrupted sinusoidal maps show minimal distortion near the equator Commonly used for thematic maps of Africa and South America The projection can be interrupted (split at multiple meridians and rearranged) to reduce distortion over specific regions of interest. Conic Projections Conic projections use a cone as the developable surface and are especially suited for mid-latitude regions. General Properties of Conic Projections In a normal conic projection: Meridians radiate outward from a single point (the cone's apex) Parallels are circular arcs centered on the apex The scale of parallels is determined by the cone's geometry Conic projections naturally minimize distortion in mid-latitudes and are widely used for mapping countries and regions between the tropics and the poles. Standard Parallels Two standard parallels are typically chosen where the cone intersects (secant) or touches (tangent) the globe. The spacing of parallels between these lines is adjusted based on which property the projection preserves: In tangent cones, one standard line is used (minimum distortion there) In secant cones, two standard lines are used (minimum distortion between them) This flexibility is one reason conic projections are popular for regional maps. Equidistant Conic The equidistant conic keeps the spacing of meridians constant along each meridian. This preserves distance along meridians. Key properties: Distances along meridians are accurate Moderate shape and area distortion A good compromise when neither conformal nor equal-area properties are critical Useful for navigation and regional mapping Albers Conic The Albers conic (also called Albers equal-area conic) adjusts parallel spacing to preserve area. Key properties: Equal-area (perfect area preservation) Moderate shape distortion Two standard parallels reduce distortion compared to a tangent cone Excellent for thematic mapping of mid-latitude countries and regions Widely used by government mapping agencies (e.g., USGS for the United States) Lambert Conformal Conic The Lambert conformal conic adjusts spacing to make the projection conformal (preserving local angles and shapes). Key properties: Conformal locally Two standard parallels Excellent for navigation charts and aviation maps Minimal distortion between the standard parallels Widely used for tactical military maps and aeronautical charts Pseudoconic Projections <extrainfo> Bonne Projection The Bonne projection is an equal-area projection with a configurable standard parallel. It combines properties of conic and pseudocylindrical projections. Key properties: Equal-area One central meridian and one standard parallel can be chosen for minimal distortion Parallels are circular arcs (like conic), but some meridians curve (like pseudocylindrical) Minimal shape distortion near the standard parallel Used historically for regional atlases Polyconic Family Polyconic projections use multiple cones, each tangent at a different parallel. The American polyconic projection uses this principle. Key properties: Each parallel is based on a cone tangent at that parallel The central meridian is straight; others curve Can show entire continents with moderate distortion Less commonly used today but historically important in U.S. mapping </extrainfo> Azimuthal (Planar) Projections Azimuthal projections project Earth onto a plane tangent to (or secant to) a chosen point, typically one of the poles or a point of interest. They are distinguished by their radial symmetry: directions and distances from the central point have special significance. Core Property: Azimuthality All azimuthal projections preserve direction from a central point. If you draw a straight line from the map's center to any other point, that line shows the true compass direction to that point. Additionally, all great circles passing through the central point appear as straight lines. This makes azimuthal projections ideal for: Navigation from a specific location Showing regional relationships from a central viewpoint Plotting radio transmission patterns or flight routes Gnomonic Projection The gnomonic projection is constructed using a perspective point at Earth's center. Any point on Earth is projected along a line through Earth's center onto the plane. Mathematical form: For a point at angular distance $d$ from the center, the radial distance on the map is $r(d) = c \tan\left(\frac{d}{R}\right)$, where $c$ is a scale constant. Key properties: All great circles appear as straight lines (unique to gnomonic among azimuthals) Extreme distortion: the projection extends to infinity at 90° from center Only shows one hemisphere (90° from the center) Useful for navigation great-circle routes (plot a straight line to follow the geodesic) Not conformal, not equal-area, not equidistant Orthographic Projection The orthographic projection is constructed as if viewing Earth from infinitely far away. Parallel rays (not converging at a point) project each location orthogonally onto the plane. Mathematical form: $r(d) = c \sin\left(\frac{d}{R}\right)$ Key properties: Shows up to one hemisphere (extends only to 90° from center with finite extent) Looks like a photograph of Earth from space Severe distortion toward the edges (areas near the edge appear compressed) Not conformal, not equal-area, not equidistant Useful for visualization and illustration rather than navigation Stereographic Projection The stereographic projection is constructed using the antipode of the tangent point (the point on the opposite side of Earth) as the perspective point. Mathematical form: $r(d) = c \tan\left(\frac{d}{2R}\right)$ Key properties: Conformal (preserves angles locally) Can show much more than one hemisphere if necessary Distorts areas severely (enlarges regions away from center) All small circles on Earth appear as circles on the map (useful for properties like magnetic declination zones) Common for polar maps since conformal projection is useful for navigation Azimuthal Equidistant Projection The azimuthal equidistant projection preserves distance from the central point. Mathematical form: $r(d) = c \, d$ (linear relationship between angular distance and map distance) Key properties: Distances from the central point are accurate All straight lines from the center show correct directions and distances Useful for showing distances from a specific location (e.g., a capital city) Not conformal, not equal-area Can show the entire globe but with increasing area distortion away from center Lambert Azimuthal Equal-Area Projection The Lambert azimuthal equal-area projection preserves area while maintaining azimuthal properties. Mathematical form: $r(d) = c \sin\left(\frac{d}{2R}\right)$ Key properties: Equal-area (preserves relative sizes of regions) Conformal near the center but with increasing shape distortion away from it Useful for thematic maps of countries or continents Less common than other azimuthals but important for area-preserving regional maps Compromise Projections Compromise projections do not strictly preserve any single metric property. Instead, they balance multiple types of distortion to produce a visually pleasing and reasonably accurate map of the world. Examples include the Robinson projection and the Winkel Tripel projection—both popular for world maps in atlases. These projections sacrifice some accuracy in any single dimension to avoid extreme distortion in all dimensions. Key principle: Compromise projections are ideal for general-reference world maps where no single metric property is critical and the map must serve many purposes equally well. Summary: Choosing a Projection When selecting a projection for a mapping task, ask: What is the geographic extent? Conic for mid-latitudes, azimuthal for poles or centered on a point, cylindrical for global or equatorial regions. What metric property is most important? Conformal for navigation, equal-area for thematic maps, equidistant for distance queries, azimuthal for direction from a point. What is the aspect? Normal for equatorial regions, transverse for north-south strips, oblique for diagonal regions. What standard parallels or points minimize distortion for my region? No single projection is "best"—the best projection depends entirely on your purpose and region.