Chapter 4 Map Projections

A map projection is a systematic method used to transfer the network of latitudes and longitudes from the Earth's spherical surface (or a model of it) onto a flat surface, like a piece of paper. The Earth is a geoid (a sphere-like shape), and a globe is the most accurate model of its shape, correctly showing relative sizes, shapes, directions, and distances on its surface.

The globe is covered by a network of parallels of latitude (horizontal circles) and meridians of longitude (vertical semi-circles), known as the graticule. This graticule is the basis for drawing maps. However, a flat map is a two-dimensional representation of a three-dimensional surface. When the curved graticule from a globe is transferred to a flat surface, the parallels become intersecting straight or curved lines.

Glossary terms introduced in the text:

• Map projection: The system of transforming the spherical Earth surface onto a plane surface using parallels of latitude and meridians of longitude at a chosen scale.
• Lexodrome or Rhumb Line: A straight line on Mercator's projection representing a constant compass bearing, useful for navigation.
• The Great Circle: A circle on the Earth's surface whose plane passes through the Earth's center (e.g., the Equator, or any two opposite meridians). It represents the shortest distance between two points on the surface.
• Homolographic Projection: A projection where areas on the map are proportional to the corresponding areas on the globe (equal-area projection).
• Orthomorphic Projection: A projection that preserves the correct shape of small areas on the Earth's surface.

Need For Map Projection

While a globe provides an accurate representation, it has limitations. Globes are expensive, bulky, cannot be carried everywhere easily, and are not suitable for showing fine details of smaller regions or comparing different areas side-by-side. To study regions in detail or compare them conveniently, accurate large-scale maps on flat paper are needed.

The primary challenge is transferring the curved surface of the globe to a flat sheet. It is impossible to perfectly flatten a spherical surface without stretching, shrinking, or tearing. When the graticule is transferred, some distortion is inevitable (Figure 4.1 illustrates distortions). Properties like shape, size, distance, and direction may not be accurately represented across the entire map.

Diagram illustrating the challenge of transferring the Earth's curved surface onto a flat map, showing how distortion occurs.

Different map projections are developed to minimize specific types of distortion or preserve particular properties that are important for the map's purpose. Map projection is also viewed as the study of various methods developed to transfer the graticule from the globe to a flat sheet.

Elements Of Map Projection

Understanding map projections involves several basic elements:

Reduced Earth

Since maps are drawn at a reduced scale, a model of the Earth, smaller than the actual planet but maintaining its geoidal shape (slightly flattened at the poles, bulging at the equator), is used conceptually or mathematically as the base for projection. This scaled-down model is called the reduced earth. The graticule is transferred from this reduced Earth model onto a flat surface.

Parallels Of Latitude

These are the conceptual circles on the reduced Earth parallel to the Equator, representing angular distance north or south of the Equator (from $0^\circ$ to $90^\circ$ N/S). They lie on planes perpendicular to the Earth's axis. On a globe, their length decreases from the Equator to the poles.

Meridians Of Longitude

These are conceptual semi-circles on the reduced Earth connecting the North and South Poles. They represent angular distance east or west of the Prime Meridian ($0^\circ$). All meridians are equal in length and intersect at the poles, crossing parallels at right angles.

Global Property

When creating a map projection, the aim is to preserve some key properties of the Earth's surface. However, no projection can perfectly maintain all properties simultaneously. The four major properties considered are:

• Distance: Maintaining correct distances between points.
• Shape: Preserving the correct shape of areas.
• Size (Area): Ensuring that the relative areas of regions are accurately represented.
• Direction (Bearing): Showing the correct direction or bearing from one point to another.

Depending on the map's purpose, a cartographer chooses a projection that minimizes distortion in the most important property while accepting distortions in others.

Classification Of Map Projections

Map projections can be classified based on different criteria:

Drawing Techniques

Based on how the projection is conceptually constructed, techniques include:

• Perspective Projections: Derived by projecting the graticule from a globe onto a developable surface using a conceptual light source placed at a specific point (e.g., at the Earth's center, on the surface, or at infinity).
• Non-Perspective Projections: Developed using mathematical calculations or geometric constructions without the concept of a light source.
• Conventional or Mathematical Projections: Derived purely through mathematical formulas, with little relation to a geometric projection model.

Developable Surface

This classification is based on using surfaces that can be flattened into a plane without distortion. A sphere (like the Earth) is a non-developable surface. However, a cylinder, a cone, and a flat plane are developable surfaces. Map projections can be made by conceptually wrapping one of these surfaces around the globe and projecting the graticule onto it.

• Cylindrical Projections: The graticule is projected onto a cylinder wrapped around the globe, typically touching the Equator. When the cylinder is unrolled, it forms a flat map.
• Conical Projections: The graticule is projected onto a cone placed over the globe, usually touching along a parallel of latitude (the standard parallel). When the cone is cut and unrolled, it forms a flat map section.
• Zenithal (or Azimuthal) Projections: The graticule is projected directly onto a flat plane that touches the globe at a single point. This point is often a pole (polar zenithal), the Equator (equatorial zenithal), or a point in between (oblique zenithal), defining the perspective of the projection center.

These projections are further categorized based on the position of the developable surface relative to the globe (normal, oblique, or polar).

Global Properties

As mentioned, a map projection cannot preserve all global properties accurately across the entire map. Classifications based on the property primarily preserved include:

• Equal Area (Homolographic) Projections: Maintain the correct relative sizes or areas of regions. Shapes may be distorted, especially at the edges of the map.
• Orthomorphic (True-Shape) Projections: Preserve the correct shape of small areas. This is achieved by ensuring that the scale is equal in all directions at any given point. However, preserving shape over a large area typically results in significant distortion of area, particularly towards the poles.
• Azimuthal (True-Bearing) Projections: Correctly show the direction or bearing from a central point (the point of tangency or the projection center) to all other points on the map.
• Equi-distant (True Scale) Projections: Preserve correct distances. However, no projection can maintain correct scale everywhere; distances are typically true only along specific lines (e.g., along meridians, along standard parallels) or from a central point, depending on the projection.

Source Of Light

For perspective projections, the location of the conceptual light source is used for classification:

• Gnomonic Projection: Source of light is placed at the Earth's center.
• Stereographic Projection: Source of light is placed at the periphery of the globe, opposite to the point where the plane surface touches the globe.
• Orthographic Projection: Source of light is placed at an infinite distance from the globe, resulting in parallel rays.

Constructing Some Selected Projections

The process of constructing a map projection involves mathematical calculations to determine the positions of parallels and meridians on a plane surface based on the chosen projection type and its parameters (scale, standard parallels, central meridian). Here, we examine the construction and characteristics of a few specific projections.

Conical Projection With One Standard Parallel

In a simple conical projection with one standard parallel, the graticule is projected onto a cone that touches the globe along a specific parallel of latitude (the standard parallel). The standard parallel is represented at its true scale on the map. Other parallels are concentric arcs, and meridians are straight lines radiating from the apex of the cone (which represents the pole) (Figure 4.3 illustrates the concept and construction steps).

Diagram showing the conceptual geometric method to construct a simple conical projection by projecting parallels and meridians onto a cone tangential to the globe along a standard parallel.

Properties:

• Parallels are concentric arcs, equally spaced along meridians.
• Meridians are straight lines converging at the pole (represented as an arc).
• Meridians intersect parallels at right angles.
• Scale is true (accurate) along all meridians. Distances along meridians are correct.
• Scale is true along the standard parallel but becomes increasingly exaggerated away from it (both towards the equator and the pole).
• It is neither an equal-area nor an orthomorphic projection.

Limitations:

• Not suitable for world maps due to extreme distortion in the hemisphere opposite the one where the standard parallel is located.
• Not ideal for representing very large areas even within the same hemisphere due to increasing distortion near the pole and the equator.

Uses:

• Best suited for mapping mid-latitude regions with a limited latitudinal extent but a larger longitudinal extent.
• Useful for depicting long, narrow geographical features or routes running roughly parallel to the standard parallel (e.g., Trans-Siberian Railway, US-Canada border in some sections).

Cylindrical Equal Area Projection

Also known as Lambert's cylindrical equal-area projection, this projection is conceptually created by projecting the graticule onto a cylinder touching the globe at the Equator using parallel rays (Figure 4.4 illustrates the projection).

Diagram illustrating the appearance of a Cylindrical Equal Area Projection, where parallels and meridians are straight lines, and the poles are represented as lines equal in length to the Equator.

Properties:

• All parallels and meridians are straight lines intersecting at right angles.
• Parallels are equally spaced.
• All parallels have the same length, equal to the length of the Equator on the globe. This is the feature that maintains the equal area property, but it causes significant shape distortion away from the Equator.
• Meridians are equally spaced.
• Scale is true only along the Equator. Scale along other parallels is exaggerated.
• It is an equal-area (homolographic) projection; areas are correctly represented.
• It is a non-orthomorphic projection; shapes are distorted, especially towards the poles. The poles are represented as lines equal in length to the Equator, causing severe distortion there.

Limitations:

• Shape distortion increases significantly towards the poles.
• Not suitable for mapping polar regions.

Uses:

• Most suitable for representing areas located between approximately $45^\circ$ N and $45^\circ$ S latitudes.
• Useful for thematic maps showing the distribution of phenomena where the correct relative area is important, such as the distribution of population, crops, or natural resources (e.g., distribution of tropical crops like rice, tea, coffee).

Mercator’s Projection

Developed by the Dutch cartographer Gerardus Mercator in 1569, this is a mathematical projection based on formulas. It is an orthomorphic (true-shape) projection, meaning it preserves the correct shape of small areas (Figure 4.5 illustrates the projection).

Diagram illustrating the appearance of Mercator's Projection, known for straight parallels and meridians and increasing spacing between parallels towards the poles.

Properties:

• All parallels and meridians are straight lines intersecting at right angles.
• Parallels are parallel straight lines, but their spacing increases towards the poles. This increasing spacing compensates for the increasing length of parallels away from the Equator, maintaining the orthomorphic property.
• Meridians are parallel straight lines, equally spaced.
• Scale is true along the Equator. Scale is not true elsewhere, but it is equal in all directions at any given point, preserving shape (orthomorphism).
• A unique property is that a straight line drawn between any two points on a Mercator projection represents a line of constant compass bearing (a Lexodrome or Rhumb line) (Figure 4.6 compares rhumb lines and great circles). This makes it invaluable for navigation.

Diagram illustrating the difference between a Rhumb Line (straight line of constant bearing on Mercator) and a Great Circle (shortest distance curved line on Mercator).

Limitations:

• There is significant exaggeration of scale and area towards the poles. The size of areas at high latitudes is greatly distorted (e.g., Greenland appears disproportionately large compared to countries near the Equator).
• The poles ($90^\circ$ N and S) cannot be shown on this projection as the parallels and meridians extend infinitely.

Uses:

• Extremely useful for navigational purposes, as rhumb lines appear as straight lines, making it easy to plot and follow a constant compass course. Widely used for marine charts.
• Suitable for world maps, especially for showing routes.
• Often used for thematic maps showing phenomena where true direction or shape is important over a large area, though area distortion needs to be considered.

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