Coordinate transformation is the description of the location of spatial entities and the process of transforming from one coordinate system to another.This is achieved by establishing a one-to-one correspondence between the two coordinate systems.It is an essential step to establish the mathematical basis of maps in the surveying and mapping of various scale maps.When two or more coordinates are converted, the dimension space is determined by relative reference to polar coordinates.
There are five kinds of coordinate transformation. Except for translation, the coordinate origin is taken as the reference point, that is, the coordinate origin remains unchanged before and after transformation.
The Chinese and English names andmatrixDescription.[1]
Basic transformation
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Multidimensional view
two-dimensional
one
zero
zero
zero
one
zero
x
y
one
three-dimensional
one
zero
zero
zero
zero
one
zero
zero
zero
zero
one
zero
x
y
z
one
[1]
Doubler
scale
two-dimensional
x
zero
zero
zero
y
zero
zero
zero
one
three-dimensional
x
zero
zero
zero
zero
y
zero
zero
zero
zero
z
zero
zero
zero
zero
one
[1]
rotate
Rotate: Take counterclockwise rotation as an example.Then the transformation matrix from the old coordinate to the new coordinate is
Reflect: The essence of reflection transformation is negative magnification, which is actually called "mirror".
two-dimensional
Based on X axis Based on Y axis
▏1 0 0▕ ▏-1 0 0▕
▏0 -1 0▕ ▏0 1 0▕
▏0 0 1▕ ▏0 0 1▕
three-dimensional
Based on Y-X plane Based on X-Z plane Based on Z-Y plane
▏1 0 0 0▕ ▏1 0 0 0▕ ▏-1 0 0 0▕
▏0 1 0 0▕ ▏0 -1 0 0▕ ▏0 1 0 0▕
▏0 0 -1 0▕ ▏0 0 1 0▕ ▏0 0 1 0▕
▏0 0 0 1▕ ▏0 0 0 1▕ ▏0 0 0 1▕
The coordinate transformation based on any plane can be regarded as the mapping of two linear spaces[1]
Application type
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Plane analytic geometry
In plane geometry, there areCartesian coordinatesTranslation and rotation, as well as mutual transformation between polar coordinates and rectangular coordinates.
In the rectangular coordinate system, the translation of coordinates focuses onRelative coordinateAnd absolute coordinates.The translation of coordinates is determined byAxisCaused by the translation and rotation of.If we can find outChuyuanThe moving distance, moving direction and turning angle of the coordinate (before moving relative to the original coordinate).Then the required coordinates can be found in the new coordinates by performing the same transformation as the original coordinates.
Cartographic mathematics
stayGeographic Information SystemIn, there are two kinds of coordinate transformation, one isMap projectionTransformation, that is, from one map projection to another, the coordinates of each point on the map change;The other is coordinate transformation of the measurement system, that is, fromGeodetic coordinate systemTo map coordinate systemDigitizerCoordinate systemgraph plotterCoordinate conversion between coordinate systems or display coordinate systems.
measure
During the construction of the project, due to the use of different coordinate systems, coordinate transformation between different coordinate systems is required.
