Map projection--Gaussian-gram gauss–krüger projection, Mercator projection, and UTM projection

Map Projection ( Map Projection )

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Concept:

Map projection is the theory and method of converting any point on the Earth's surface to a map plane using certain mathematical laws.

Since the earth is an irregular pear-shaped sphere that is slightly flattened at the equator, its surface is a non-flattened surface, so the use of any mathematical method for this conversion will produce errors and deformations, in order to reduce the error according to different needs, resulting in a variety of projection methods.

Method:

1 , geometric perspective: Geometric Perspective is a projection method that uses perspective to project the point of the Earth's dignity onto the projection surface. If the earth is scaled down into a transparent globe-like sphere, placing a light source outside its sphere or sphere and sphere, projecting the longitude on the spherical surface onto a projection plane outside the sphere, and converting the spherical longitude into a longitude on the plane.

2 , Mathematical analytic method: The mathematical analytic method is to establish the function relation between the point and the point between the spherical surface and the projection plane, and to determine a projection method of the longitude intersection position by the mathematical method.

Geometry perspective is a relatively primitive projection method, it has great limitation, it is difficult to correct the projection deformation, the accuracy is low, and most of the current map projections adopt mathematical analytic method. Most of the mathematical analytic methods are often based on perspective projection, the development of the spherical and projection surface between the point and point of the function of the relationship, so the two projection methods have a certain connection.

Classification:

1 , by deformation properties: conformal projection, equal product projection and arbitrary projection.

2 , the shape of the graticule when projected on the positive axis:(1) geometric projection: Azimuth projection, cylindrical projection and conic projection, (2) Conditional projection: pseudo-azimuth projection, pseudo-cylindrical projection, pseudo-conic projection, multi-conic projection.

3 , by the projection axis and the axis of the relationship: the positive axis projection (coincident), oblique axis projection (skew), horizontal projection (vertical).

4 , according to the projection surface and the Earth's surface relationship: cutting projection, cutting projection.

Gauss - Crugge Projection ( Gauss-kruger Projection )--conformal transverse tangent elliptical column projection

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By the German mathematician, physicist, astronomer Gauss in the 19th century 20 's formulation, after the German geodesy home Crugge in 1912 to complement the projection formula, so called Gauss -gram gauss–krüger projection.

Gaussian -gram gauss–krüger projection is conformal transverse elliptic column projection. An elliptical column is assumed to cross over a certain line of the Earth's ellipsoid, a meridian tangent to the cylindrical face, called the central Meridian. With the central Meridian as the projection of the symmetrical axis, the thing each 3° or 1°30′ two meridian clip by the difference 6° or 3° strip region by the mathematical law, projection law projection to the cylindrical surface, and then expand into a plane, that is, Gaussian -gram gauss–krüger projection, referred to as Gauss projection. This narrow band of longitude is called the Gaussian -gram Gauss–krüger projection belt.

Gaussian -gram Gauss–krüger projection features:

1, the Central meridian is a straight line, its length is not deformed; the other meridian is the arc of the concave central meridian, and the central meridian is the axis of symmetry;

2, the Equator line is a straight line, but the length of deformation, the other parallels are convex to the equator arc, and the equator as the axis of symmetry;

3, meridians and parallels are still orthogonal after projection;

4, the farther away from the central meridian, the larger the deformation.

If the method of sub-band projection is used, the deformation of the projection edge can not be too large. Different Gaussian -gram Gauss–krüger projection belts are used in various large and medium scale topographic maps in China. The topographic map of which greater than 1:10,000 uses 3° belt,1:25,000 to 1:500,000 of the topographic map adopts the 6° belt.

Gaussian projection

Split-Band projection

Gaussian plane Cartesian coordinate system

Mercator Projection ( Mercator Projection )--conformal positive axis tangent cylindrical projection

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Created by the Dutch map biologist Mercator (G. Mercator) in 1569, it is the most influential, also known as the positive axis conformal cylindrical projection of the map projection method.

Assuming that the earth is enclosed in a hollow cylinder, its reference parallels are in contact with the cylindrical tangent (equator), and then the Imaginary Earth Center has a lamp projecting the shape on the spherical surface onto the cylinder, and then the cylinder expands, which is the map drawn by the Mercator projection on the selected datum parallels .

Mercator Projection Features:

1, no angle deformation, from each point to each direction of the length ratio equal;

2, the longitude are parallel straight lines, and intersect at right angles, the warp spacing is equal, and the weft spacing increases gradually from the datum parallels to the poles.

3, the length and the area deformation is obvious, but the datum latitude is not deformed, the deformation gradually increases from the datum parallels to the polar deformation, but because it has the characteristic which each direction equal expands, maintains the direction and the reciprocal position relations correct.

Maintaining the correct orientation and angle on the map is the merit of the Mercator projection map, which is commonly used as a nautical chart and aerial map, and if the direction remains the same in the straight line between two points on the Mercator projection, it is advantageous for the ship to locate and determine its course in the voyage. It brings great convenience to the seafaring people.

the projection method used by Baidu maps and Google maps is the Mercator projection.

The Mercator map projection

The transverse Mercator map projection.

UTM Projection ( Universal transverse mercatol Projection , Universal Transverse Mercator projection)--conformal transverse cut elliptic column projection

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Http://therucksack.tripod.com/MiBSAR/LandNav/UTM/UTM.htm

UTM is a "conformal transverse cylindrical projection ", the oval column cut the Earth at 80 degrees south latitude, 84 degrees north longitude two, such as High circle, the projection after the two tangent meridians are not deformed, and the central meridian length than 0.9996. The UTM projection was created for global warfare, and the United States completed the calculation of this universal projection system in 1948.

Similar to the Gaussian -gram gauss–krüger projection, the projection angle is not deformed, the central meridian is a straight line, and is a projection of the symmetrical axis, the central meridian of the scale factor of 0.9996 is to ensure that the central meridian around 330km There are two non-distorted standard meridians.

This coordinate grid system and its projection are widely used in topographic maps as reference grids for satellite imagery and natural resource databases and other applications where precise positioning is required.

The map above represents a transverse Mercator projection of the world with a standard meridian at 0°longitude. (Note that because of the very small size of the map, the graticule are shown at 30°resolution.) The Globe wrapped In a cylinder is a conceptual model of what the transverse Mercator projection formula transfers positions on the globe to Positions on a plane (thecylinder can was flattened to a plane surface after it was unwrapped from the globe.) the Thicker red line on the cylinder and the map are the standard line along which scale distortion is zero. As the distortion ellipses on the map indicate, distortion increases with distance from the standard line.

Global coverage of the Universal transverse Mercator (UTM)

and Universal Polar stereographic (UPS) coordinate systems.

In UTM systems, the Earth's surface area between 84 and 80 degrees latitude is divided into North and South longitudinal belts (projection bands) at longitude 6 degrees, starting from 180 degrees longitude and numbering these projections, from 1 to 60 (Beijing is in Band 50). Each band is then divided into 8 degrees of latitude, four quads, four-sided shape from 80 degrees south, with the letter C to X (without I and O) in turn mark ( X line includes the northern hemisphere from 72 degrees north latitude to 84 degree total land area, total 12 degrees). Each quadrilateral is labeled with a combination of numbers and letters, and the reference grid reads upward to the right, and each quadrilateral is divided into a lot of 1000000-metre-long neighborhoods, labeled with a monogram system. In each projection band, a meridian with a center is placed, giving a horizontal axis value of 500000 meters. For the Northern hemisphere, the marker coordinate value is 0, for the southern hemisphere is 10000000 meters, toward the south descending.

The grid rows labeled C through H, J through N, and P through X of the Universal transverse Mercator (UTM) coor Dinate system-depicted by the lettered, horizontal rows in the above image-cover the entire world, save the northern and S Outhern polar regions, which is covered by the Universal polar stereographic (UPS) coordinate system.

The component parts of a UTM grid coordinate data string.

Zone in the Universal transverse Mercator (UTM) coordinate system.

Gauss - Crugge Projection and UTM projection

Http://tian0226.blog.sohu.com/142843049.html

Http://www.cnblogs.com/yaweno/archive/2010/07/23/1783873.html

Http://eternalsep.blog.hexun.com/52122534_d.html

The Gaussian -gram gauss–krüger projection and the UTM projection are all variants of the transverse Mercator projection, and currently some foreign software or supporting software imported from abroad often does not support Gaussian -gram gauss–krüger projections, but supports UTM projections, so the UTM projection is often used as a Gaussian -gram Gauss–krüger projection phenomenon.

From the projection geometry, the Gaussian -Crugge projection is " conformal crosscutting elliptic column projection " , after projection the central meridian remains the same length, that is, the scale factor is 1 ; utm projection is " Isometric horizontal cut cylindrical projection " , cylindrical cut the earth in the south latitude 80 degrees, north latitude 84 degree two, such as High circle, there is no distortion on the two secant lines after projection, and the length ratio on the central meridian is greater than 0.9996 .

from the calculation results, the main difference between the two is the scale factor, Gauss - Crugge projection on the central meridian of the scale coefficient is 1 , utm projection to 0.9996 , Gaussian -Crugge projection with utm projection can be approximated by x[utm]=0.9996*x[Gauss ],y[utm]=0.9996*y[Gauss 500000 m, the conversion must be y value minus 500000 multiply the scaling factor and then add 500000).

	Input coordinates Dimension	Gaussian projection M	UTM projection M	Xutm=0.9996*x Gauss Yutm=0.9996*y Gauss
Latitude value (X)	32	3543600.9	3542183.5	3543600.9*0.9996≈3542183.5
Longitude value (Y)	121	21310996.8	311072.4	(310996.8-500000)*0.9996+500000≈311072.4

From the point of view of the band, the beginning of the difference between the two, Gauss -G Gauss–krüger projection from 0 degrees from the meridian of the 6 degrees from the west to the east, the central longitude of 1 with 3°;utm projection from 180° longitude per deviation 6 degrees from west to east, the central longitude of band 1 is -177°, so the 1 band of the Gaussian -gram Gauss–krüger projection is the 31 band of UTM.

In addition, the East pseudo-offset of both projections (false easting) is 500 km, the Gaussian -gram Gauss–krüger projection North Pseudo-offset (false_northing) is zero, and theutm Northern hemisphere projection North pseudo-offset (false northing) is zero and the southern hemisphere is 10000 km.

Map projection--Gaussian-gram gauss–krüger projection, Mercator projection, and UTM projection