Geodetic Fundamentals 1, Gaussian projection and the calculation of the belt change

This paper introduces the normal projection process of the Cartesian coordinate system from the Earth coordinate on the ellipsoid plane. This paper studies how to transform the geodetic coordinate, the length and direction of the geodetic line and the azimuth angle of the geodetic plane. This paper focuses on the principle and method of Gauss projection, solves the problem of conversion from spherical to planar, and solves the coordinate conversion of adjacent belts.

Overview of Gauss projection

The basic knowledge includes: positive projection, gauss coordinate inverse calculation and the change of band.

Map Projection: The elements of the ellipsoid (including coordinates, direction and length) are projected onto the plane by a certain mathematical law. The specialized discipline that studies this question is called the map projection study.

Ellipsoidal surface is a convex, non-flattened surface, if the elements on this surface (such as a distance, an angle, a graph) projected onto the plane, and the original distance, angle, graphic rendering difference, this difference is called projection deformation.

The ratio of the side length on the projection surface to the corresponding length on the original surface is called the length ratio .

1.1 Classification of Map projections • Classification by deformation properties

(1) Conformal projection, also known as a positive projection. The angle of any two-direction line of a point on the projection plane is equal to the angle of the corresponding two segments on the ellipsoid, that is, the angular deformation is zero. Conformal projections are equal in length in any direction at a point, but differ in length from one location to another.

(2) Equal product projection, the area on the projection plane and the corresponding area on the ellipsoid, that is, the area deformation is equal to zero. (3) An isometric projection, defined as a distance along a particular direction, remains unchanged before and after the projection, that is, a length ratio of 1 along that particular direction. There is no length distortion on this projection, it only has no length distortion in a particular direction.

• Classification by the shape of the projection surface

(1) Azimuth projection: A plane as a projection plane, so that the plane and the spherical surface tangent or phase cut, the spherical surface of the longitude projection to the plane. (2) Cylindrical projection: a cylindrical surface as a projection surface, so that the cylindrical surface and the ball face cut or tangent, the spherical surface of the longitude projection onto the cylindrical surface, and then the cylindrical surface to show as a plane. (3) conic projection: A conical surface as a projection surface, so that the cone surface and the ball face cut or tangent, the longitude on the spherical surface projection to the cone, and then the cone surface to show as a plane. 1.2 China various Map Projections 1) China National Map projection:Oblique axis equal area azimuth projection, oblique isometric azimuth projection, pseudo-azimuth projection, positive axis and other area cut conic projection, positive axis conformal cut conic projection. 2) Projection of China's provincial (district) Map:positive shaft conformal cutting conic projection, positive axis equal area cutting cone projection, positive axis conformal cylindrical projection, Gaussian-gram gauss–krüger projection (broadband). 3) Projection of large scale maps of China:multi-faceted projection(Northern warlords period),conformal cut conic projection(Lambert projection) (before liberation),Gaussian-gram Gauss–krüger projection(after liberation).

From the world's perspective, large and medium-scale topographic maps used by countries in the projection is not uniform, according to incomplete statistics there are more than 10 kinds of, the most commonly used are horizontal isometric elliptic column projection. After the founding of the People's Republic of China, the large and medium scale topographic maps of our country are provided with Gaussian-gram gauss–krüger projections based on Krasovsky ellipsoid elements. China's new 1:1 million topographic map, the use of Bing and middle latitude deformation absolute equivalent of the positive axis conformal conic projection.

1.3 Gaussian projection Overview Control measurement requirements for map projection

1) Conformal projection (also called positive projection)

2) The length and area deformation is not small, and can be used to calculate the number of corrections caused by deformation by simple formula.

3) can be easily carried out according to the sub-band, and can be high-precision, simple, the same calculation formula and the table to the whole.

Basic concepts of Gaussian projectionThe Gaussian projection is a conformal transverse elliptic column projection. Gaussian projection is a Conformal projection。 It is by the German mathematician Gauss (gauss,1777 ~ 1855) proposed, after the German geodesy home Crugge (kruger,1857~1923) to complement the perfect, it is also called "Gauss-gram Gauss–krüger projection", referred to as "Gauss projection".

(1) Gaussian projection principle

The Gaussian projection uses a split-band projection. The ellipsoidal surface is projected by a certain difference band, respectively.

(2) Gaussian projection must meet

Gaussian projection is a positive projection, i.e. conformal projection;

The central meridian is projected into a straight line, and is a projected symmetrical axis;

The length of the central meridian is unchanged after projection.

(3) Characteristics of Gaussian projection

The central meridian is projected into a straight line with the same length.

In addition to the central Meridian, the projection of the remaining meridian is the curve of the concave central meridian, and the central meridian is the axis of symmetry. The length is deformed after the projection.

The equator line is projected to be straight, but has a length distortion.

In addition to the remaining parallels outside the equator, the projection is the curve of the convex equator, with the equator as the axis of symmetry.

Longitude and latitude are still orthogonal after projection.

All lengths of deformed segments, whose length deformation ratio are greater than L.

The farther away from the central meridian, the greater the length deformation.

(4) Division of the projection band

Our country stipulates to carry on the projection 6º and 3º by the difference of the warp. 6º with the self-surrender meridian, according to the 6º by the difference from west to east into 60 bands. 3º Belt from the beginning of 1.5º, according to the 3º by the difference from west to east into 120 bands.

  6 º 3 º :

3 º 6 º Engineering surveying uses Span style= "font-family: Chinese in italics;" >3 º belt, special works can be used 1.5 º band or any band.

Gaussian plane Cartesian coordinate system the Establishment

Because our country is located in the northern hemisphere, the east and west of the 6º belt, each belt and alone constitute a Cartesian coordinate system. therefore:theX value is positive, and the Y value has a positive negative.

triangulation of ellipsoidal surface to Gaussian plane

The main content of the ellipsoid triangle system to the Gaussian projection plane is

The geodetic coordinates of the starting point b l Span style= "font-family: italics _gb2312;" > normalized to Gaussian plane Cartesian coordinates x y x y inverse calculation b l by calculating the meridian Convergence Angle and direction correction of the point, the azimuth angle of the starting edge of the ellipsoid is calculated to the coordinate azimuth of the corresponding edge on the Gaussian plane. by calculating the curvature correction and the direction correction in each direction, the inner angles of each triangle on the ellipsoid are counted to the triangular inner angles of the corresponding straight lines on the Gaussian plane. by calculating the distance correction, the length of the starting edge on the ellipsoid is computed to the straight line length on the Gaussian plane . The main content of the ellipsoid triangle system to the Gaussian projection plane is: When the control network crosses two adjacent projection belts, the coordinate of the plane coordinates should be converted.

1.4 Gauss projection Coordinate inverse formula

second, the ellipsoid element is calculated to the Gaussian projection plane

Including direction modification and distance modification

Iii. Overview of various projection methods

Reference documents:

Baidu Library, 1th Chapter Geodesy Foundation. ppt

Geodetic Fundamentals 1, Gaussian projection and the calculation of the belt change