Big picture
A spherical surface cannot be flattened without distortion. Map projections provide systematic ways to transfer the earth's latitude–longitude grid onto a plane. The choice determines which properties are preserved and which are altered, so it affects how distances, areas and shapes appear.
An equal-area world map helps compare countries' sizes, while a navigation chart may preserve local angles. Neither can preserve every property simultaneously. This chapter explains projection principles, major families and practical constructions, with careful distinction between idealised geometric models and real cartographic methods.
Learning goals
Explain why projection distortion is unavoidable.
Compare equal-area, conformal and equidistant properties.
Distinguish cylindrical, conical and zenithal projection families.
Explain standard parallels, aspect and tangent or secant arrangements.
Construct simple cylindrical and one-standard-parallel conical graticules.
Choose projections appropriate to region and purpose.
Core ideas
1. Globe, graticule and projection
The graticule is the network of meridians and parallels. On a globe, parallels are complete circles except at the poles, meridians converge at the poles, and the equator is a great circle. On a map, the graticule changes according to mathematical rules.
A projection is not always created literally by shining a light through a transparent globe. Some geometric descriptions are useful teaching models, while many projections are mathematical transformations. For example, Mercator cannot be accurately described as simply the shadow cast from a central lamp onto a cylinder. Explain the geometry shown in the practical without assuming every named projection is a physical perspective projection.
2. Properties and distortion
Equal-area projections preserve area relationships. Conformal projections preserve local angles and small shapes, but not the sizes or outlines of very large regions without distortion. Equidistant projections preserve distances only along specified lines or from specified points, not between every possible pair of points.
Desired property | What is preserved | Typical caution |
|---|---|---|
Equal-area | Relative areas | Shapes may stretch |
Conformal | Local angles | Area inflation can be large |
Equidistant | Selected distances | Other distances remain distorted |
True direction in a specified sense | Bearings from selected points or special routes | Does not guarantee all other properties |
For example, a conformal map can show Greenland disproportionately large relative to tropical lands. Local shape preservation does not establish equal area. Every projection should be judged against its intended purpose.
3. Projection surfaces and aspect
Cylindrical, conical and zenithal or azimuthal families use a cylinder, cone or plane as a conceptual development surface. A normal aspect aligns the projection symmetrically with the earth's axis in the usual family-specific arrangement; transverse and oblique aspects change that orientation.
A tangent surface touches the model globe along a line or at a point, while a secant surface cuts it, commonly producing two standard lines for a cone or cylinder. Standard parallels are latitudes along which scale is true in the relevant construction. For example, choosing two suitable standard parallels can distribute distortion across a mid-latitude region more effectively than placing all the region far from one true-scale line.
4. Cylindrical graticules
In a normal cylindrical projection, meridians commonly appear as straight parallel vertical lines, and parallels as straight horizontal lines. Their spacing depends on the projection. On a simple equidistant cylindrical grid, both longitude and latitude intervals have uniform spacing for equal angular increments.
A normal Mercator map has straight meridians and parallels but increasing parallel spacing towards high latitudes. It is conformal and represents rhumb lines, routes of constant compass bearing, as straight lines. These are not generally the shortest great-circle routes. Mercator cannot show the poles at finite distance. For example, using its world map to compare high-latitude and tropical country areas produces misleading impressions.
5. Simple cylindrical construction
Use a reduced globe with radius R. For an equatorial simple cylindrical construction, total width for 360° is 2πR, and pole-to-pole height for 180° is πR. Equal longitude and latitude intervals are laid out with linear spacing R × angle in radians.
Worked example: let reduced radius R = 4 cm and interval = 30°. Since 30° = π/6 radians, interval spacing = 4 × π/6 = 2.09 cm approximately. Full width = 2π × 4 = 25.13 cm and height = π × 4 = 12.57 cm. Draw the rectangle, mark equal intervals, label the equator centrally and place meridians vertically.
This construction has true scale along the equator and preserves meridional distance spacing in its simple form, but area and shape distortions grow away from the equator. It is not Mercator merely because its grid is rectangular.
6. Conical projections
A normal conical projection shows meridians as straight lines converging towards an apex and parallels as concentric circular arcs. It is well suited to many mid-latitude regions extending east–west. The standard parallel or parallels determine where scale is true.
In a simple conical projection with one standard parallel, meridional distances are spaced consistently along the straight meridians, while distortion grows away from the standard latitude. The projected angular separation between meridians is smaller than their true longitude separation by a factor depending on the standard latitude. For example, the 30° standard-parallel case uses sin 30° = 0.5, so a 20° longitude interval becomes a 10° angle at the cone apex.
7. One-standard-parallel conical construction
For reduced globe radius R and standard latitude φ, distance from cone apex to the standard-parallel arc is R cot φ. Meridional distance from that parallel to another latitude changes by R times the latitude difference in radians. Moving northwards in a normal northern cone decreases the arc radius.
Worked example: R = 4 cm and φ = 30°. Apex radius = 4 × cot 30° = 4√3 = 6.93 cm approximately. Meridional spacing for 10° latitude = 4 × π/18 = 0.70 cm. The 40° parallel therefore has radius about 6.93 − 0.70 = 6.23 cm using unrounded calculation before the final rounding. A 20° longitude interval is drawn as 20 × sin 30° = 10° at the apex. Draw concentric arcs, straight meridians, and label the standard parallel and central meridian.
8. Zenithal and perspective examples
In a normal polar zenithal projection, parallels appear as concentric circles and meridians as radiating straight lines. It is useful for polar regions and for directions or distances from a selected centre, depending on the exact projection.
Gnomonic projection uses a central perspective and represents great-circle paths as straight lines. Stereographic uses a perspective point opposite the tangent point and is conformal. Orthographic uses parallel projection rays, producing a globe-like view of a hemisphere. These properties are not interchangeable. For example, a globe-like appearance on an orthographic map does not mean all surface distances can be measured accurately from it.
9. Choosing a projection
Choose by region, extent, orientation and purpose. Equal-area maps suit comparisons of crop area or population density when areas are interpreted carefully. Conformal maps suit uses needing local angular relationships. Conical families are often useful for broad mid-latitude east–west regions, while polar azimuthal maps suit polar settings.
A route-planning question must distinguish constant bearing from shortest path. A thematic world comparison should avoid a projection that grossly inflates the areas being compared. For example, a map for Indian regional analysis should distribute distortion appropriately across its chosen extent rather than simply copy the world's most familiar wall map.
10. Practical checks and limitations
Label all parallels and meridians, state the reduced radius or scale, identify the standard parallel and mark the central meridian. Use construction values consistently and round only after calculation, since repeated rounding can distort spacing.
In a source, inspect the graticule before identifying a family. Straight parallel meridians suggest a normal cylindrical layout; converging straight meridians and arc parallels suggest conical; radiating meridians around a centre suggest polar azimuthal. These are clues within the stated aspect, not universal tests for every oblique transformation. Recognising distortion helps students read maps critically rather than assuming every familiar outline has equal geographical authority.
A reliable approach to questions
Identify the purpose and region before selecting a projection.
State the preserved property and the unavoidable distortion.
Write construction formulas, convert degrees to radians where required and calculate.
Draw the graticule systematically before adding labels.
Check standard lines, spacing, dimensions and final rounding.
Connections that matter
Maps and scale supplies reduced-globe and map-distance relationships.
Latitude and longitude defines the graticule transferred to the plane.
Mathematics supplies trigonometry, circular arcs and radian conversion.
Common mistakes
Calling every rectangular graticule Mercator: spacing and mathematical properties differ.
Assuming equidistant means every distance is true: preservation is restricted.
Equating rhumb lines with all shortest routes: great circles generally differ.
Mixing degrees with radians in arc-length formulas: convert first.
Revision checklist
I can explain why distortion cannot be eliminated everywhere.
I can compare projection families and properties.
I can construct a simple cylindrical grid.
I can calculate a one-standard-parallel cone's basic dimensions.
I can select a projection with an explicit reason.