Geodetic notes

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GEODETIC NOTES

Scale variations in Web Mercator: effects of Latitude and implications for metric grid

Maps based on the Web Mercator projection, used by OpenStreetMap and most online mapping systems, have a fundamental characteristic: the local scale depends on latitude. At the Equator, the scale is correct, while it progressively increases toward the poles.

The scale factor is given by:

k = 1 / cos(latitude)

This means that an object represented on the map appears progressively enlarged relative to its actual size as latitude increases. For example, at 60° latitude, the scale is approximately twice that at the Equator.

However, the Mercator projection is conformal: at every single point on the map, the scale is identical in all directions. Consequently, there is no difference between North–South and East–West scales at the same point. A very small square on the ground will therefore continue to appear as a square on the map, although it is represented at a different scale from its actual size.

Scale Variation Within a Map

Scale variations become relevant when considering relatively large areas.

Within a single tile or viewing window, the scale factor can change between the southern and northern edges because latitude changes.

Conversely, along an East–West direction at constant latitude, the scale factor remains constant, so there is no scale variation along that direction. This does not mean that the scale is equal to the real-world scale; it means that the scale factor remains constant along that parallel.

For typical latitudes of Northern Italy (approximately 45°N), the scale variation within a single standard tile is very small at high zoom levels:

• Zoom 15: about 0.014%
• Zoom 13: about 0.054%
• Zoom 11: about 0.22%
• Zoom 10: about 0.44%
• Zoom 9: about 0.87%

These values are indicative and refer to a latitude of approximately 45°N and to a single standard tile. The actual variation depends on both latitude and the North–South extent of the map area being considered.

Only at lower zoom levels, such as 7–8, can the variation within a single tile exceed a few percentage points and become relevant when displaying or comparing very long distances.

Map Markers, Pixels and Metric Grids

From a practical point of view, map markers such as points, symbols and icons are not directly affected by the scale variation of the projection when they are drawn using screen coordinates.

However, when markers are used for measurements or to construct metric grids, it is necessary to consider that the conversion between pixels and metres depends on latitude.

Therefore, a metric grid overlaid on a Web Mercator map is not geometrically uniform over large geographic areas. For a single tile or a limited viewing window, however, the scale variation is generally small at zoom levels 9–10 and at intermediate latitudes. A metric grid can therefore be considered approximately uniform for visualization purposes.

This approximation applies only to the graphical representation of the grid and does not replace geodetic calculations when accurate measurements are required.

Measurement Accuracy in MapNavigator

MapNavigator does not need to apply a latitude-dependent scale correction when performing geographic measurements because distances and other metric quantities are not calculated from the pixel dimensions of the Web Mercator map.

Distances are calculated directly from geographic coordinates using the Vincenty geodesic algorithm, with the Haversine formula used when necessary.

Areas are calculated using geodetic algorithms based on an ellipsoidal or spherical surface, depending on the method used.

Angles and azimuths are calculated directly from geographic coordinates using appropriate geodetic algorithms.

These calculations operate directly on geographic coordinates (latitude/longitude) and use the WGS84 ellipsoid or, where applicable, an appropriate spherical model.

As a result, the distortion introduced by the Web Mercator projection affects map visualization but does not affect the accuracy of the geodetic calculations performed by MapNavigator.

Distance, area and angle measurements therefore require no manual correction based on latitude or map zoom level.