Coordinate Axis translation and Rotation

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• 1. Coordinate System on the plane

 

1. Coordinate System on the plane

Geographic coordinates are spherical coordinates. Because the earth's surface is a non-expandable surface, that is to say, the points on the surface cannot be directly expressed on the plane. Therefore, the map projection method must be used to establish the functional relationship between the earth's surface and the points on the plane, make any point on the earth surface composed of geographical coordinates(PHI, λ)A definite point must correspond to a point on the plane. The position of any point on the plane can be expressed in polar or Cartesian coordinates.

1. Establishment of the Cartesian coordinate system of the plane

Select a point on the planeOIs the Cartesian coordinate origin.OTwo Vertical axesX' oxAndY' OyCreate a plane Cartesian coordinate system, as shown in Figure 5.

In the Cartesian coordinate systemOx, oyThe direction is positive,Ox, oyThe direction is negative, so a known point in the coordinate systemPAnd its location can be determined byOxAndOyThe vertical line length of the axis is uniquely determined, that isX = ap,Y = BP, Usually recordedP (x, y).

1. 2. Creation of Polar Coordinate

Figure 4-5: Cartesian coordinate system and Polar Coordinate System

5, SetO'Is the polar coordinate origin,O 'oIs a polar axis,PIs a point in the coordinate systemO 'PIt is called a Polar distance and is signed.PThat isP = O 'P.∠ Oo 'PIs the polar angle, with a symbolDelta, ThenListen oo 'P = delta. Polar AngleDeltaIt is calculated from the polar axis in a clockwise direction.

A certain relationship can be established between the Polar Coordinate and the Cartesian coordinate of the plane. Figure 5 shows that the X-axis and polar-axis of the Cartesian coordinates overlap, and the distance between the two coordinate system OriginsOO'UseQIndicates that:

X = Q-P Cos Delta

Y = P Sin Delta

2. Translation of the Cartesian coordinate system and rotation of the 2.1 Coordinate System

As shown in figure 1, coordinate systemXoyCoordinate SystemX' o 'y'The corresponding coordinate axes are parallel to each other and have the same forward. Coordinate SystemX' o 'y'By Coordinate SystemXoyParallel movement. SetPPoint in Coordinate SystemXoyThe coordinates are(X, y), InX' o 'y'The coordinates are(X', y '), And(A, B)YesO'In Coordinate SystemXoyCoordinates in, so:

X = x' +

Y = y' + B

The above formula is the coordinate relationship between a point before and after the coordinate system translation.

Figure 1: coordinate translation

2.2 coordinate system rotation

See figure 2, as shown in the coordinate system.XoyCoordinate SystemX' o 'y'And the angle between the two axes isθ, Coordinate systemX' o 'y'By Coordinate SystemXoyClockwise rotation with O as the centerθ.

X = x' cos θ + y' sin θ

Y = y' cos θ-x' sin θ

The above formula is the relationship between the coordinates of a point in the Cartesian coordinate system after the angle θ of rotation.

Figure 2: coordinate rotation

2.3 coordinate system translation and Rotation

As shown in figure 3, coordinate systemX' o 'y'In the Coordinate SystemXoyThe coordinates are a, B,XAxis andX'The angle of the axis is θ. Coordinate SystemX' o 'y'It was originally coincident with the coordinate system xoy, and then because o 'translates the distance between A and B, and the coordinate system's two coordinate axesO 'X'AndO 'y'RelativeOxAndOyClockwise rotationθ.

Introduce an auxiliary coordinate system between two coordinate systemsX "o 'y"To make its two-AxisO 'x"AndO 'y"AndOx,OyParallel.

InX "o 'y"There is a bit of P in the system, and its coordinate is(X ", Y "), The coordinate system translation formula and the coordinate system rotation formula are available:

X = x "+

Y = y "+ B

Therefore

X "= x' cos θ + y' sin θ

Y "= y' cos θ-x' sin θ

That is

X = x' cos θ + y' sin θ +

Y "= y' cos θ-x' sin θ + B

The above formula is the relationship between the coordinate system translation and a coordinate point in the new and old coordinate systems after rotation.

Figure 3: coordinate translation and RotationBasic questions about map projection

3. Concept of map projection

In mathematics, a project is a ing between two point sets. Similarly, in cartography, map projection refers to the one-to-one correspondence between points on the earth surface and those on the projection plane. The basic problem of map projection is to use certain mathematical rules to express the latitude and longitude network on the earth's surface to a plane. All geographic information systems must consider map projection. The use of map projection ensures the connection and integrity of spatial information in the region. During the establishment of various geographic information systems, selecting an appropriate map projection system is the first consideration. Because the earth's elliptical surface is a curved surface, while a map is usually drawn on a plan paper, the first thing to do is to develop a curved surface into a plane. However, the spherical surface is an invisible surface, that is, when it is directly converted into a plane, it is impossible not to crack or fold. It is obviously impractical to draw a map with a broken or folding plane. Therefore, a special method must be used to expand the surface to make it a plane without cracking or folding.

 

Source: http://learn.gxtc.edu.cn/NCourse/GIS/gis_html/GIS2/ch3/4.htm

 

Recently I have been studying the rotation and translation of coordinate axes. There are two online formulas: clockwise and counterclockwise, but I have not made it clear on the Internet. Finally, I found this article, which is comprehensive and clear and reprinted for backup.