Definition of projection
The surface of the earth's ellipsoid is a curved surface, and maps are usually drawn on plane drawings, so the first step in drawing is to develop the curved surface into a plane. However, the sphere is a non-developable surface, and when it is developed directly into a plane, it will break or wrinkle. It is obviously not practical to use this kind of plane with cracks or wrinkles to draw maps, so special methods must be used to expand the curved surface to make it a plane without cracks or wrinkles, so the theory of map projection appears.
Its basic principle is: because the position of a point on the spherical surface is represented by Use latitude and longitude, the actual projection is to draw the intersection points of some longitude and latitude lines on the plane first, and then connect the points of the same longitude and latitude into longitude lines, and connect the points of the same latitude into latitude lines to form a longitude and latitude network. With the longitude and latitude network, the points on the spherical surface can be plotted at the corresponding positions on the plane according to their longitude and latitude, as shown in the following figure:
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| Figure: Transferring points on a sphere to a plane by latitude and longitude |
The essence of projection
Many analysis techniques and Spatial Data are designed for two-dimensional coordinates or plane coordinates, which need to store spatial coordinates in the way of plane map projection, so we often need to use map projection to convert three-dimensional GCS into two-dimensional plane coordinates. The so-called map projection is to transform the longitude and latitude coordinates (λ, φ) into plane coordinates (X, Y) through specific mathematical equations. Map projection is used to reduce the distortion that always occurs when a 3D Coordinate Transformation is converted to a 2D coordinate.
The use of map projection ensures that Spatial Info can maintain the geographical contact and integrity after the transformation from GCS to plane coordinates, which is the basic requirement of Map Cartography and the basic premise of spatial operation and Spatial Analysis. Therefore, map projection is very important for the application of geographic data in GIS.
The method of projection
There are two projection methods, namely, geometric perspective and mathematical analysis.
Geometric perspective is a projection method that uses the perspective relationship between objects to project the points on the earth's surface onto the projection plane. For example, take the plane, cylindrical surface and conical surface as the bearing surface, and transfer the curved surface (ellipsoidal surface of the earth) to the plane (map), as shown in the following figure:
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| Figure: Schematic diagram of perspective projection method |
Mathematical analysis is a method to establish the corresponding relationship between the longitude and latitude network on the ellipsoidal surface of the earth and the corresponding longitude and latitude network on the plane. In essence, it is to directly determine the functional relationship between the GCS of a point on the sphere and the rectangular coordinates of the corresponding point on the plane. At present, most map projections adopt this method.
- Geometric perspective
- Mathematical analytic method
Projection deformation
The ellipsoid of the earth is an undevelopable surface, while the map is a plane, because the expansion of such a sphere into a plane will inevitably lead to the rupture or overlap of some parts, so that the features and landforms located in this part become discontinuous and incomplete. From the point of view of practical application, the rupture or overlap must be evenly stretched or compressed. To eliminate cracks and wrinkles. When stretched and compressed, these parts of the map lose their resemblance to the corresponding parts of the earth, and this change is due to the deformation of the projection. The change of the geometric characteristics of the longitude and latitude network caused by the projection of the spherical surface to the plane is called the map projection deformation.
Map deformation includes length deformation, angle deformation, area deformation and shape deformation.
Length distortion is the difference between the length ratio and 1, and the length ratio is the ratio of the length of a tiny line segment on the projection plane to the length of the corresponding tiny line segment on the ellipsoid plane. Length deformation is used to reflect the degree of change of line segment after projection. It is the most basic deformation existing on all projections, and it is the area and angle deformation caused by it.
Angular deformation refers to the difference between the angle between any two directional lines on the projection plane and the angle between the corresponding two directional lines on the ellipsoid. Angular deformation is a specific sign of shape deformation.
Area distortion is the difference between the area ratio and 1, and the area ratio is the ratio of a tiny area on the projection plane to the corresponding tiny area on the ellipsoid. Area deformation is a quantitative index to measure the size of map projection deformation.
Shape distortion refers to the type of mismatch between the shape of a contour on a map and the shape of the corresponding ground contour.
- Length deformation
- Angular deformation
- Area deformation
- Shape deformation
Related topics
Description of reference frame conversion method