How the normal Mercator is made
The normal or direct Mercator is a mathematically modified cylindrical projection. Its cylinder is aligned with the Earth's axis and is tangent to the reduced Earth at the equator.
The simple cylindrical starting point
Imagine a cylinder wrapped around the reduced Earth and touching it at the equator. Rays from the centre project the graticule onto the cylinder. When the cylinder is opened, meridians become equally spaced parallel straight lines and parallels become straight lines crossing them at right angles.
Mercator's modification
The simple perspective result stretches shapes incorrectly because scale changes at different rates in the north and south and east and west directions. Gerardus Mercator adjusted the spacing of the parallels mathematically until scale changed equally in every direction at each point. This made the projection conformal.
| Feature | Normal Mercator | Reason |
|---|---|---|
| Projection family | Cylindrical | The receiving surface is a cylinder |
| Construction | Non-perspective | The simple projection is mathematically modified |
| Contact | Tangent at the equator | The cylinder touches the reduced Earth there |
| Conformality | Yes | Directional scales change equally |
Why the poles cannot appear
The geographic poles lie on the axis of the normal cylinder. Mercator spacing grows without limit as latitude approaches 90 degrees, so neither pole can be represented at a finite position on the chart.
The Mercator graticule
A Mercator chart is recognised by a rectangular graticule. Longitude spacing stays equal, while latitude spacing increases towards both poles.
Meridians
Every meridian is a straight line. All meridians are parallel and equally spaced, so chart convergency is zero everywhere. Equal changes of longitude occupy equal chart widths at every latitude.
Parallels
Every parallel is a straight line at right angles to every meridian. The space between successive parallels increases with latitude. Mercator made this increase match the east and west scale expansion, preserving conformality.
Scale in outline
Scale is correct at the equator and expands away from it in proportion to the secant of latitude. It remains within about 1 percent of equatorial scale only to roughly 8 degrees north or south, often treated as about 480 to 500 nautical miles. Full calculation is covered in Chapter 10.
Conformality, shape and area
Mercator preserves local angles and small shapes, but high-latitude scale expansion severely exaggerates area and the apparent width of large regions.
Why the chart is conformal
Meridians and parallels cross at 90 degrees, meeting the first condition for conformality. Mercator's adjusted latitude spacing makes north and south scale expand at the same rate as east and west scale, meeting the second condition. Bearings measured over small areas are therefore correct.
Small shapes and large regions
Small shapes remain recognisable because local angles are correct. Large features extend across areas where scale changes, so their overall shape and area become distorted. Land masses at high latitude appear too wide compared with their north and south extent.
| Property | Mercator result | Operational meaning |
|---|---|---|
| Local angles | Correct | Tracks and bearings can be measured |
| Small shapes | Reasonably correct | Local map reading remains practical |
| Large shapes | Distorted, especially at high latitude | Do not judge the shortest route by appearance alone |
| Areas | Not equal-area | Relative land areas look misleading |
Scale-expansion examples
At 60 degrees latitude, Mercator scale is twice its equatorial value, so the chart length representing 3,000 nautical miles is twice the equatorial chart length. Oxford notes that Africa is about 18 times the area of Greenland although they can look comparable on a world Mercator. Scandinavia can appear similar to India even though its land area is only about one third of India's.
Chart convergence and rhumb lines
Parallel meridians give the Mercator its defining operational property: a straight line cuts every meridian at the same angle and is therefore a rhumb line.
Chart convergence
Chart convergence is the angle between meridians on a chart, or the change in direction of a straight line between them. On a normal Mercator, the meridians are parallel, so chart convergence is zero everywhere.
Earth convergency is different
Earth convergency is also zero at the equator, but increases towards the poles. Mercator chart convergence is therefore correct only at the equator. Elsewhere it remains zero while Earth convergency is non-zero.
| Quantity | At the equator | Away from the equator |
|---|---|---|
| Mercator chart convergence | Zero | Zero |
| Earth convergency | Zero | Increases with latitude and longitude difference |
| Agreement | Correct | Mercator convergence is less than Earth convergency |
Rhumb lines
A rhumb line crosses all meridians at a constant angle. A straight line on a Mercator does exactly this, so every straight line on the chart represents a rhumb line. This is true everywhere the chart exists.
Operational uses
The Mercator is well suited to plotting constant-direction tracks, especially at low latitudes. It is also convenient for plotting radio bearings because those bearings can be converted between great-circle and rhumb-line directions before drawing. Traditional long-route plotting uses its simple rectangular longitude framework, although scale and great-circle curvature must be respected.
Great circles on Mercator
Most great circles appear as curves on a Mercator. Their curves are concave to the equator, so the great-circle route lies on the poleward side of the corresponding rhumb line.
The two straight exceptions
The equator is both a great circle and a rhumb line, so it is straight. Every meridian is also both a great semicircle and a line of constant direction, so meridians are straight. All other great circles curve.
Concave to the equator
In the Northern Hemisphere a great circle bows north of its rhumb line. In the Southern Hemisphere it bows south. In both cases it is nearer the relevant pole and its Mercator curve is concave towards the equator.
Conversion angle calculation
For an approximate London to Los Angeles problem, take change of longitude as 120 degrees and mean latitude as 45 degrees. Conversion angle = one half × 120 × sin 45 degrees = about 42 degrees. If the westbound rhumb-line track is 257 degrees true, the initial great-circle track at London is about 299 degrees true. In the reverse direction, the rhumb-line reciprocal is 077 degrees and the initial great-circle track at Los Angeles is about 035 degrees true.
Routes crossing the equator
A single arithmetic mean latitude can wrongly give zero conversion angle for a route with one endpoint north and the other south. Divide such a route at the equator and calculate each hemisphere segment separately.
Summary and route recognition
Mercator questions are often solved by recognising the graticule and recalling which route is straight, which route curves and where scale is trustworthy.
| Mercator property | Rule to remember |
|---|---|
| Projection | Normal cylindrical, conformal and non-perspective |
| Scale | Correct at the equator, expands with secant latitude |
| Near-constant scale band | Within about 1 percent to 8 degrees north and south |
| Meridians | Straight, parallel and equally spaced |
| Parallels | Straight and parallel, with spacing increasing poleward |
| Chart convergence | Zero everywhere, correct only at the equator |
| Rhumb lines | Straight everywhere |
| Great circles | Curves concave to the equator, except the equator and meridians |
| Shapes | Good locally, distorted over large high-latitude areas |
| Poles | Cannot be shown |
Worked directional checks
From Turin at 45 degrees north, 008 degrees east to Khartoum at 15 degrees north, 032 degrees east, the rhumb-line track is 145 degrees true. Change of longitude is 24 degrees and mean latitude is 30 degrees, giving conversion angle 6 degrees. The initial great-circle track is therefore 139 degrees true. The return initial great-circle track from Khartoum is 331 degrees true.
From Durban at 30 degrees south, 032 degrees east to Perth at 30 degrees south, 116 degrees east, the eastbound rhumb line is 090 degrees. Conversion angle is one half × 84 × sin 30 degrees = 21 degrees. The return great-circle track from Perth to Durban is 249 degrees true.
Where the Mercator is useful
Its simple longitude framework, straight rhumb lines and correct local angles make it useful for plotting, low-latitude navigation and radio-navigation work. Its increasing poleward scale, extreme area distortion and inability to show the poles make it unsuitable as a single high-latitude world-navigation solution.