Currently commonly used projections include the Mercator projection (normal conformal cylindrical projection), Gauss-Krüger projection (transverse conformal cylindrical projection), UTM projection (transverse conformal secant cylindrical projection), Lambert projection (conformal secant conic projection), etc.
Mercator Projection
The Mercator projection is an "equatorial conformal cylindrical projection" devised by the Dutch cartographer Mercator in 1569. It assumes that the Earth is enclosed in a hollow cylinder, with the standard parallel tangent to the cylinder. Then, imagining a light source at the Earth's center, the spherical features are projected onto the cylinder, which is then unrolled to form a map drawn using the "Mercator projection" at the selected standard parallel.
The Mercator projection has no angular distortion, with equal scale ratios in all directions from each point. Its meridians and parallels are straight parallel lines intersecting at right angles, with equidistant meridians and parallels that gradually increase in spacing from the standard parallel toward the poles. On a Mercator projection map, length and area distortions are evident, but there is no distortion at the standard parallel. Distortion increases from the standard parallel toward the poles, but because it expands equally in all directions, it maintains correct direction and positional relationships.
Preserving correct directions and angles on the map is an advantage of the Mercator projection. Mercator projection maps are commonly used as nautical and aeronautical charts. If sailing along a straight line between two points on a Mercator projection map, the direction remains constant until reaching the destination. Thus, it is advantageous for ship positioning and course determination during navigation, providing great convenience to mariners.
Gauss-Krüger Projection
The Gauss-Krüger projection is a "transverse conformal cylindrical projection." It was devised by the German mathematician, physicist, and astronomer Carl Friedrich Gauss (1777–1855) in the 1820s and later supplemented by the German geodesist Johannes Krüger (1857–1928) in 1912, hence the name. It envisions a cylinder tangent to the central meridian of a projection zone on the sphere. Under the conditions that the central meridian projects as a straight line without length distortion and the equator projects as a straight line, the spherical surface within a certain longitude difference range on both sides of the central meridian is conformally projected onto the cylinder. Then, the cylinder is cut along a meridian passing through the poles and flattened to obtain the Gauss-Krüger projection plane.
After Gauss-Krüger projection, apart from the central meridian and the equator being straight lines, other meridians are curves symmetrical about the central meridian. The Gauss-Krüger projection has no angular distortion, and distortion in length and area is minimal. There is no distortion at the central meridian; distortion gradually increases from the central meridian toward the edges of the projection zone, with the maximum distortion at the ends of the equator within the zone. Due to its high projection accuracy, small distortion, and simple calculation (coordinates are consistent across zones; only one zone needs to be calculated, and others can be applied), it is used in large-scale topographic maps, meeting various military needs and allowing precise measurement and calculation on the map.
Dividing the Earth's ellipsoid into several projection zones by a certain longitude difference is the most effective method to limit length distortion in Gauss projection. When dividing zones, it is necessary to control length distortion so that it does not exceed mapping errors and to minimize the number of zones to reduce conversion calculations. According to this principle, the Earth's ellipsoid is divided along meridians into melon-shaped zones with equal longitude differences for zonal projection. Usually, it is divided into 6-degree zones or 3-degree zones based on a longitude difference of 6 degrees or 3 degrees. Six-degree zones start from the 0-degree meridian and are divided from west to east every 6 degrees of longitude difference, with zone numbers sequentially 1, 2, ..., 60. Three-degree zones are subdivided from six-degree zones; their prime meridians coincide with those of six-degree zones and zone boundary meridians. Starting from the 1.5-degree meridian, they are divided every 3 degrees of longitude difference from west to east, with zone numbers sequentially 1, 2, ..., 120 for three-degree zones. China's longitude ranges from 73° west to 135° east, which can be divided into eleven six-degree zones with central meridians at 75°, 81°, 87°, ..., 117°, 123°, 129°, 135°, or twenty-two three-degree zones.
UTM Projection
The UTM projection, short for "Universal Transverse Mercator projection," is a "transverse conformal secant cylindrical projection." The cylinder secants the Earth at two parallels: 80° south latitude and 84° north latitude. After projection, there is no distortion on the two secant meridians, while the scale factor on the central meridian is 0.9996. The UTM projection was created for global warfare needs, with the United States completing the calculations for this universal projection system in 1948. Similar to the Gauss-Krüger projection, this projection has no angular distortion; the central meridian is a straight line and the axis of symmetry for the projection. The scale factor of the central meridian is set to 0.9996 to ensure two distortion-free standard meridians at approximately 330 km east and west of the central meridian. The zone division method for UTM projection is similar to that of the Gauss-Krüger projection: starting from 180° west longitude, it is divided every 6 degrees of longitude from west to east, dividing the Earth into 60 projection zones. China's satellite imagery data often uses the UTM projection.
Lambert Projection
The Lambert projection, also known as "conformal secant conic projection," was devised by the German mathematician J.H. Lambert in 1772. It envisions a cone tangent to or secant to the sphere, applying conformal conditions to project the Earth's surface onto the cone, which is then developed along a meridian to form the Lambert projection plane. After projection, parallels are concentric arcs, and meridians are radii of concentric circles. The Lambert projection uses two standard parallels for secant projection, compared to a single standard parallel for tangent projection, resulting in smaller and more uniform distortion. The distortion distribution pattern of the Lambert projection is:
- There is no angular distortion; corresponding differential areas before and after projection maintain shape similarity, so it can also be called a conformal projection.
- Iso-distortion lines coincide with parallels, meaning distortion is equal everywhere on the same parallel.
- There is no distortion on the two standard parallels.
- On the same meridian, outside the two standard parallels is positive distortion (scale factor greater than 1), while between the two standard parallels is negative distortion (scale factor less than 1). Therefore, distortion is relatively uniform, and the absolute value of distortion is small.
- On the same parallel, segments of equal longitude difference have equal lengths, and the lengths of meridians between two parallels are equal everywhere.
China's 1:1,000,000 topographic maps use the Lambert projection. Their sheet division principles are consistent with the globally unified international millionth map projection specified by the International Geographical Union. Latitude is divided into zones by 4-degree latitude difference, totaling 15 projection zones from south to north. Each projection zone calculates coordinates independently, with two standard parallels: the first standard parallel is the parallel at the southern end of the map sheet plus 30 minutes latitude, and the second is the parallel at the northern end minus 30 minutes latitude. Thus, coordinate results are identical for all map sheets within the same projection zone, and distortion values for sheets in different zones are nearly equal. Therefore, only one sheet (4-degree latitude difference, 6-degree longitude difference) per projection zone needs its projection calculated. Since it is zoned by 4-degree latitude difference, when splicing maps along parallels, no gaps occur regardless of the number of sheets. However, when splicing along meridians, because the splicing lines are in different upper and lower projection zones, the curvature after projection differs, causing gaps during splicing.
Related Topics
Description of Datum Transformation Methods