Map Projections

Map Projection

A map projection is a systematic method of transferring the geographic coordinates (latitude and longitude) of locations from the curved surface of the Earth (or another celestial body) onto a flat surface, such as a map. Since the Earth is a sphere (or more accurately, an oblate spheroid) and a map is flat, it is impossible to represent the Earth's surface on a map without some form of distortion.

Map projections are essentially mathematical transformations that attempt to minimize these distortions, but they can never eliminate them entirely. Different projections prioritize preserving certain properties over others (e.g., preserving area, shape, distance, or direction).

Key Concept: Projections are a necessary compromise to represent a curved surface on a flat one.

Need For Map Projection

The fundamental need for map projections arises from the inherent geometrical problem of representing a three-dimensional curved surface (the Earth) onto a two-dimensional flat surface (a map). It is mathematically impossible to do this without introducing some degree of distortion.

Reasons for the Need:

• Flat Representation: Maps are typically printed, displayed on screens, or used in atlases as flat surfaces. Representing the spherical Earth directly on a flat plane is impossible without deformation.
• Minimizing Distortion: While eliminating distortion entirely is impossible, map projections provide systematic ways to transfer locations from the globe to the map, controlling the type and location of distortion. Different projections are chosen based on what property is most important to preserve for a particular map's purpose.
• Preserving Properties: Users might need maps that accurately represent:
• Area: For comparing the sizes of countries or regions (Equal-Area Projections).
• Shape: For accurately depicting coastlines and landforms (Conformal Projections).
• Distance: For measuring distances accurately from a central point or along specific routes (Equidistant Projections).
• Direction: For navigation and maintaining accurate bearings (Conformal or Azimuthal Projections).
• Navigation and Travel: Projections like Mercator's are vital for maritime navigation as they allow sailors to plot courses of constant compass bearing (rhumb lines) as straight lines.
• Thematic Mapping: For displaying data on population, climate, or resources, projections help visualize spatial patterns, often prioritizing either area or shape depending on the data's nature.

In essence, map projections are a crucial tool that bridges the gap between the Earth's spherical reality and the practical need for flat maps, allowing us to visualize and understand our world, albeit with controlled distortions.

Elements Of Map Projection

Several key elements and concepts are involved in the process and understanding of map projections.

Reduced Earth

Definition: A map projection begins with the concept of a 'reduced Earth'. This is essentially a scaled-down version of the Earth's spherical or spheroidal surface. The scale chosen for this reduction is fundamental to all subsequent steps in creating the map.

Importance: The choice of scale determines the level of detail that can be represented and the type of distortion that will occur. A larger scale shows a smaller area with more detail, while a smaller scale shows a larger area with less detail.

Parallels Of Latitude

Definition: These are imaginary horizontal circles drawn on the Earth's surface, parallel to the Equator, measuring north-south position.

Representation in Projections: On a map projection, parallels of latitude are transferred from the globe to the flat surface. Their representation varies significantly depending on the projection:

• Equidistant Cylindrical Projection: Parallels are shown as equally spaced horizontal lines.
• Mercator Projection: Parallels are horizontal lines, but they are spaced further apart as you move away from the Equator towards the poles, exaggerating area at higher latitudes.
• Conic Projections: Parallels are represented as curved arcs, usually parts of circles concentric with the standard parallel.
• Azimuthal Projections: Parallels are shown as concentric circles or arcs centered on a pole.

Distortion: The spacing and shape of parallels on a map projection can be distorted, affecting the representation of area and distance.

Meridians Of Longitude

Definition: These are imaginary semi-circular lines running from the North Pole to the South Pole, measuring east-west position relative to the Prime Meridian.

Representation in Projections: Meridians are also transferred from the globe to the map, and their depiction is projection-dependent:

• Cylindrical Projections: Meridians are typically shown as parallel vertical lines, equally spaced at the Equator.
• Conic Projections: Meridians are represented as straight lines radiating outwards from the apex of the cone.
• Azimuthal Projections: Meridians are shown as straight lines radiating outwards from the center of the map (representing a pole).

Distortion: The convergence and spacing of meridians are often distorted, affecting the representation of shape and area, especially away from the projection's center or standard lines.

Global Property

Definition: Global properties refer to the specific geographical characteristics that a map projection aims to preserve, or distort the least, when transferring the Earth's curved surface to a flat map. Since all projections distort something, cartographers choose projections based on which property is most critical for the map's intended use.

Key Global Properties and Corresponding Projection Types:

• Conformality (Shape): Preserves local shapes and angles. Meridians and parallels intersect at right angles, and shapes of small features are maintained. (Example: Mercator Projection). Used for navigation charts.
• Equivalence (Area): Preserves the relative area of features. Areas on the map are proportional to areas on the ground, though shapes may be distorted. (Example: Cylindrical Equal Area Projection). Used for thematic maps showing distribution of resources or population.
• Equidistance (Distance): Preserves true distances from one or two fixed points on the map to all other points, or along specific lines (e.g., meridians or parallels). (Example: Azimuthal Equidistant Projection). Used for airline route planning or showing distances from a central location.
• Azimuthality (Direction): Preserves true directions (azimuths) from a central point to all other points on the map. (Example: Azimuthal Equidistant Projection). Useful for navigation and representing polar regions.

Compromise Projections: Many commonly used projections, like the Robinson or Winkel Tripel, do not perfectly preserve any single property but try to minimize all types of distortion, offering a visually pleasing compromise for world maps.

Classification Of Map Projections

Map projections can be classified in several ways, based on different criteria such as the technique used to create them, the developable surface employed, the properties they preserve, or the source of illumination imagined.

Drawing Techniques

This classification relates to the conceptual method by which the projection is derived:

• Orthographic Projection: Imagines projecting the Earth's surface onto a flat plane tangent to or intersecting the globe, using parallel rays of light from an infinite distance. Shapes and areas are distorted, especially away from the center.
• Stereographic Projection: Projects the Earth's surface from a point on the surface (opposite the point of tangency) onto a tangent plane. It preserves angles locally (conformal) but distorts area and distance away from the point of projection.
• Gnomonic Projection: Projects the Earth's surface from the center of the globe onto a tangent plane. It has the unique property that all great circles (shortest routes) are represented as straight lines, but shapes, areas, and distances are severely distorted away from the central point.

Developable Surface

This classification is based on the geometric shape that is imagined to be wrapped around the globe and then flattened to create the map. The common developable surfaces are a cylinder, a cone, and a plane.

• Cylindrical Projections: Imagine wrapping a cylinder around the globe and projecting the Earth's features onto it.
• Properties: Can be designed to be equal-area, conformal, or equidistant.
• Examples: Mercator Projection (conformal), Cylindrical Equal Area Projection.
• Conical Projections: Imagine placing a cone over the globe and projecting the features onto it. The cone is then slit and flattened.
• Properties: Often good for preserving shape and distance along standard parallels.
• Examples: Albers Conic Equal-Area Projection, Lambert Conformal Conic Projection. Used for mid-latitude regions.
• Azimuthal (or Planar) Projections: Imagine placing a plane tangent to or intersecting the globe at a point, and projecting the features onto it.
• Properties: Can preserve direction and distance from the center point.
• Examples: Azimuthal Equidistant Projection, Gnomonic Projection. Often used for polar regions or hemispheres.

Global Properties

This classification focuses on which geographical property is preserved or distorted least by the projection:

• Conformal (Orthomorphic) Projections: Preserve local shapes and angles. Meridians and parallels intersect at right angles, and the shape of small features is maintained. Area and distance are distorted, especially at larger scales.
• Equal-Area (Equivalent) Projections: Preserve the relative area of features. Areas on the map are proportional to areas on the Earth, but shapes and distances are distorted.
• Equidistant Projections: Preserve true distances along specific lines (e.g., from the center point to all other points, or along meridians/parallels) but distort other properties.
• Azimuthal (Directional) Projections: Preserve true directions from a central point to all other points.
• Compromise Projections: Do not perfectly preserve any single property but attempt to minimize overall distortion, providing a balance between accuracy in shape, area, distance, and direction.

Source Of Light

This classification is based on the imagined location of a light source used to project the features of a globe onto a developable surface:

• Orthographic: Light source is infinitely far away, casting parallel rays.
• Stereographic: Light source is on the surface of the globe, diametrically opposite to the point of projection on the plane.
• Gnomonic: Light source is at the center of the globe.

These techniques primarily relate to the geometric method used to create the projection on the developable surface.

Constructing Some Selected Projections

Map projections are constructed using specific mathematical formulas and geometric principles. Here are conceptual explanations of constructing a few common types:

Conical Projection With One Standard Parallel

Concept: Imagine wrapping a cone around the globe, so it touches the Earth along a single line of latitude (the standard parallel). Then, project the Earth's features onto this cone from the globe's center. Finally, slit the cone along a line and flatten it into a map.

Construction Steps (Conceptual):

1. Standard Parallel: Select a parallel of latitude (e.g., 45° N) that passes through the central part of the area to be mapped. This parallel is shown without distortion (true to scale).
2. Cone Placement: Imagine a cone that touches the globe along this standard parallel.
3. Projection: From the center of the Earth, project the parallels of latitude onto the cone. Parallels closer to the standard parallel will be more accurate, while those further away will be more distorted. Parallels are represented as arcs of circles.
4. Meridians: Meridians are projected as straight lines radiating from a point near the apex of the cone (representing the pole). The angle between adjacent meridians on the map is the same as the angle between them on the globe.
5. Flattening: Slit the cone along a meridian (usually the central meridian, which is drawn straight) and unroll it into a flat surface.

Properties: Generally good for preserving shape and distance along the standard parallel. Distortion increases away from the standard parallel. Suitable for mapping mid-latitude areas with large east-west extent (like the USA or Europe).

Cylindrical Equal Area Projection

Concept: Imagine wrapping a cylinder around the Earth, tangent to the Equator (or intersecting it). Project the Earth's features onto this cylinder from the center of the Earth. Then, slit the cylinder along a meridian and flatten it.

Construction Steps (Conceptual):

1. Cylinder Placement: Imagine a cylinder tangent to the Equator.
2. Projection: Project the grid of parallels and meridians from the Earth's center onto the cylinder.
3. Parallels: Parallels are represented as straight horizontal lines, but their spacing increases towards the poles to compensate for area distortion.
4. Meridians: Meridians are represented as straight vertical lines, equally spaced across the map.
5. Flattening: Slit the cylinder along a meridian (usually the central meridian) and unroll it.

Properties: Preserves area accurately, meaning the relative size of landmasses is correct. However, shapes and angles are distorted, especially at higher latitudes. The distortion of shape and distance increases significantly towards the poles.

Common Type: Cylindrical Equal Area Projection (also known as the Gall-Peters Projection when meridians are parallel and parallels are equally spaced, though this is a variation).

Mercator’s Projection

Concept: Developed by Gerardus Mercator in 1569, this is a cylindrical projection where the cylinder is tangent to the Earth at the Equator. It is designed to be conformal, meaning it preserves local shapes and angles.

Construction Principles (Conceptual):

1. Cylinder Tangent at Equator: Imagine a cylinder wrapped around the Earth, touching it at the Equator.
2. Projection Technique: Unlike a simple gnomonic projection onto the cylinder, Mercator's projection uses a mathematical method to expand the spacing of parallels of latitude as you move away from the Equator towards the poles. This expansion is designed to compensate for the foreshortening of meridians and preserve angles.
3. Parallels: Drawn as horizontal lines, but their spacing increases proportionally with latitude.
4. Meridians: Drawn as parallel vertical lines, equally spaced across the map.
5. Result: Meridians and parallels intersect at right angles, preserving local shapes and directions (rhumb lines are straight lines).

Properties:

• Conformal: Excellent for preserving shape and direction.
• Distortion: Area and distance are severely distorted, especially at higher latitudes. Greenland, for example, appears larger than South America on a Mercator map, while in reality, South America is much larger.

Uses: Widely used for maritime navigation charts because navigators can plot a course of constant compass bearing (a rhumb line) as a straight line on the map.