Mercator's projection, rhumb lines and Web Mercator

Detail

Who made it and what kind of projection it is

• Map projection means a way of moving the network of latitudes and longitudes from the round globe onto a flat sheet. Every projection distorts something: area, shape, distance or direction.
• Gerardus Mercator made this projection in 1569. The projection was first designed and used by him in that year [3].
• Nationality: he was Flemish (from Flanders, in today's Belgium), and his birth name was Gerard de Kremer. (NCERT calls him "Dutch", which is wrong.)
• It falls into three groups at once:

Basis of grouping	Mercator's class	What it means
Method of making	Mathematical (conventional)	Drawn from a formula or table. It is not a real shadow cast on a cylinder.
Developable surface	Cylindrical	The globe is pictured inside a cylinder that touches it at the equator. The cylinder is then unrolled.
Property kept	Orthomorphic (conformal)	Shapes of small areas and angles at each point stay true.

• Orthomorphic / conformal: a projection that keeps local angles and shapes correct. It keeps angles, scale and shape correct for small areas [3].
• NCERT error: NCERT calls it "azimuthal". It is not azimuthal. An azimuthal (zenithal) projection is drawn on a flat plane touching the globe at one point. Mercator is cylindrical and conformal.

How it keeps shape: the secant rule

• Step 1: parallels are stretched. On a globe, a parallel gets shorter as you go towards the poles. On Mercator, every parallel is drawn as long as the equator. So a parallel at latitude φ is stretched by sec φ (secant of latitude, which is 1 ÷ cos φ).
• Step 2: the gap between parallels is widened by the same amount. Mercator also spaces the parallels farther apart poleward, by the same factor, sec φ.
• Result: at every point, the scale along the meridian equals the scale along the parallel. Small shapes are not squashed or pulled in only one direction. So local shapes stay true.
• Cost: a small area is enlarged in both directions, so its area grows by sec² φ.
• At 30°: length × 1.154, area about × 1.33
• At 60°: length × 2, area about × 4
• Near the poles, sec φ grows without limit, so the poles can never be drawn.

Construction (NCERT example)

• Problem: draw Mercator's projection at 1:250,000,000 with a 15° interval.
• Radius of the reduced globe: R = 250,000,000 ÷ 250,000,000 = 1 inch.
• Length of the equator: 2πR = 2 × 22/7 × 1 = 6.28 inches.
• Meridian spacing: 6.28 ÷ 24 (since 360° ÷ 15°) ≈ 0.26 inch. Meridians are equally spaced vertical lines.
• Parallel spacing: not equal. It is read from a table (distance from the equator):

Latitude	Distance from the equator (inches)
15°	0.265
30°	0.549
45°	0.881
60°	1.317
75°	2.027

• The gaps keep growing: 0.265 → 0.284 → 0.332 → 0.436 → 0.710 inch. The 90° parallel would be infinitely far away, so the drawing stops at around 75°–80°.
• Steps:
• draw the equator, 6.28 inches long
• divide it into 24 parts and draw the meridians at right angles to it
• mark the parallels on both sides of the equator using the table
• draw the parallels as straight lines as long as the equator.

Properties

• Parallels and meridians are straight lines that cross at right angles. The graticule (the grid of latitudes and longitudes) is rectangular.
• Every parallel is as long as the equator. So the 30° parallel is 1.154 times its true length on the globe. (On the globe it is cos 30° = 0.866 of the equator.)
• Parallels get farther apart towards the poles, while meridians stay evenly spaced [3].
• Shapes of small areas are correct, but areas get more and more enlarged poleward.
• Rhumb lines are straight lines (see below).

Limitations

• Huge enlargement of high latitudes. Area is magnified about 4 times at 60°. Features near the poles, such as Greenland, are made to look much bigger [3].
• Greenland vs Africa vs the USA:

Landmass	True area (approx.)	How it looks on Mercator
Greenland	about 2.2 million sq km	about as big as Africa or the USA
USA	about 9.8 million sq km	about the same size as Greenland
Africa	about 30 million sq km	about the same size as Greenland

• In reality, Africa is about 14 times as large as Greenland [3].

• NCERT error: NCERT says Greenland is 1/10th of the USA. It is actually about two-ninths (2.2 ÷ 9.8 ≈ 0.22).

• The poles cannot be shown. At 90°, the stretch factor sec φ becomes infinite.

• Great circles look curved and do not look like the shortest route, which misleads a reader who measures distance with a ruler.
• Because of these problems, it is a poor choice for comparing the sizes of countries near the poles [3]. Other projections, such as Robinson and Winkel Tripel, are often chosen for general world maps because they show area more fairly [3].

Rhumb line (loxodrome) and great circle

• Rhumb line (loxodrome): a line on the Earth that crosses every meridian at the same angle. A ship following it keeps one constant compass bearing (for example, always N 60° E).
• Why it is straight on Mercator:
• meridians are drawn as parallel vertical lines
• the projection keeps angles true (conformal)

• so a line that meets every meridian at the same angle must be a straight line.

• Great circle: a circle on the globe whose plane passes through the Earth's centre (for example, the equator or any pair of opposite meridians). An arc of a great circle is the shortest route between two points on the globe.

• Rhumb line vs great circle:

Feature	Rhumb line	Great circle route
Bearing	stays the same all the way	keeps changing
On Mercator	straight line	curve that bends towards the pole
Length	longer (except along the equator or a meridian)	shortest
Ease of steering	easy, since there is one compass setting	hard, since the course must keep changing

• Practice in long voyages (NCERT Fig. 4.6): navigators combine the two.
• Plot the great circle (the shortest path).
• Break it into short stretches.
• Sail each stretch as a rhumb line with a fixed bearing.

Uses

• Navigation: sea and air routes, because a straight line on the chart gives a fixed compass bearing.
• Atlas world maps of distribution patterns where direction and shape matter more than area:
• drainage patterns
• ocean currents
• winds
• temperature

• rainfall.

• Works best for areas near the equator (for example, India, the East Indies and equatorial Africa), where distortion is small.

Web Mercator (beyond NCERT)

• Web Mercator (also written EPSG:3857, earlier EPSG:900913, and popularly "Google Maps" projection) is a simplified, spherical form of Mercator [2].
• EPSG code: an ID number from a public list of map projections and coordinate systems, which software uses to recognise a projection.

• Spherical: the formulas treat the Earth as a perfect sphere, not as a slightly flattened ellipsoid. This makes the maths simpler.

• Adoption: Google Maps took it up around 2005. Today most online maps use it, including Google Maps, Bing Maps and OpenStreetMap [2]. A variation of Mercator is used by popular services like Google Maps and ArcGIS Online [3].

• NASA use: in April 2014, NASA's GIBS (Global Imagery Browse Services) began offering satellite imagery in Web Mercator. This let it be laid directly over Google Maps, Bing Maps and OpenStreetMap. It sits alongside other projections in GIBS: Geographic (EPSG:4326), Arctic Polar Stereographic (EPSG:3413) and Antarctic Polar Stereographic (EPSG:3031) [2].
• Why web maps chose it:
• it keeps local shapes, so a street corner or building looks right at any zoom
• its north stays up and its grid is square, so the world can be cut into square tiles at each zoom level and joined without seams [2]

• the result is smooth, seamless panning and zooming.

• Problem: it repeats Mercator's high-latitude enlargement. Greenland, Canada, Russia and Antarctica look far bigger than they are, while tropical countries look smaller.

• India and Africa look small next to Greenland.

• This shapes how people picture country sizes, and is a common example of "map bias".

• Polar areas are shown with other projections. For example, NASA GIBS uses polar stereographic projections for the Arctic and Antarctic [2].

Prelims Hooks

• Mercator's projection (1569, Gerardus Mercator, a Flemish cartographer, not Dutch) is cylindrical, mathematical and orthomorphic (conformal), not azimuthal.
• On Mercator, both the parallel length and the spacing of parallels increase by sec φ, so area grows by sec² φ: about × 4 at 60°.
• The 30° parallel on Mercator is 1.154 times its globe length, because every parallel is as long as the equator.
• On Mercator, a straight line = rhumb line (loxodrome), a line of constant compass bearing that cuts all meridians at the same angle. It is not the shortest route.
• The shortest route between two points on the globe is the great circle. On Mercator it appears curved, bending towards the pole.
• Poles cannot be shown on Mercator, because scale becomes infinite at 90°.
• True areas: Greenland about 2.2 million sq km < USA about 9.8 million < Africa about 30 million. Africa is about 14 times Greenland [3], although the two look alike on Mercator.
• Trap: "Greenland is 1/10th of the USA" is wrong. It is about 2/9.
• Web Mercator = EPSG:3857 (also EPSG:900913), used by Google Maps, Bing Maps and OpenStreetMap [2]. It keeps local shape but not area.
• Matching: Mercator → navigation charts, winds, ocean currents. Robinson / Winkel Tripel → general world maps with fairer area [3].

Mains Points

• Why the most-used world map shows sizes wrongly: the secant stretch makes high-latitude lands (Greenland, Canada, Russia) look larger than tropical ones (India, Africa, Brazil). This affects how students and the public judge the size and importance of regions. For general world maps on population, resources or development, equal-area or compromise projections (Robinson, Winkel Tripel) are better [3].
• Choose the projection by purpose:
• Mercator for navigation and for flow maps (winds, currents), where direction matters
• equal-area maps for comparing distributions (forest cover, crop area, population density)

• great-circle planning plus rhumb-line steering for long sea and air routes.

• Geospatial relevance: Web Mercator underpins online maps and satellite-imagery services [2]. Its tile system makes zooming smooth. However, area and distance worked out on it at high latitudes (for example, Arctic shipping routes or Himalayan glacier extent) must be corrected or redone in an equal-area system.

• Cartographic literacy: a map is a model, not a mirror. Knowing a projection's trade-off (shape vs area vs distance) is basic to reading thematic maps, atlas plates and GIS (Geographic Information System) outputs correctly.