Rapid back calculation method for ellipsoid expansion parameters of relatively independent coordinate system

Abstract

The invention discloses a rapid back calculation method for ellipsoid expansion parameters of a relatively independent coordinate system, which is used for establishing a relatively independent coordinate system of a city in a high-altitude area and establishing a relatively independent coordinate system of large engineering construction projects (airports, large reservoirs, expressways, high-speed rails and the like). In order to meet the requirement that the projection deformation of a plane coordinate system is less than 2.5 cm / km (the requirement of high-speed rail or rail transit is less than 1.0 cm / km), an ellipsoid expansion method is generally adopted to realize the establishment of a relative independent coordinate system from a national geodetic coordinate system. The ellipsoid expansion method can meet the requirement of projection deformation, the principle is simple, calculation is convenient, and strict conversion between a national geodetic coordinate system and a relatively independent coordinate system can be rapidly achieved; the invention discloses a rapid back calculation method for ellipsoid expansion parameters of a relatively independent coordinate system, and aims to rapidly calculate the accurate ellipsoid expansion parameters of the relatively independent coordinate system through national geodetic coordinates before ellipsoid expansion and homonymous coordinate results of the relatively independent coordinate system after ellipsoid expansion.

Description

Technical Field

[0001] This invention relates to the field of coordinate transformation in urban and large-scale engineering construction projects, specifically to a rapid inverse calculation method for the ellipsoid expansion parameters of a relatively independent coordinate system. Background Technology

[0002] For establishing relatively independent coordinate systems in cities at high altitudes, and for large-scale engineering projects (airports, large reservoirs, highways, high-speed railways, etc.), to meet the requirement that the projection distortion of the plane coordinate system be less than 2.5 cm / km (less than 1.0 cm / km for high-speed railways or urban rail transit), the ellipsoidal dilatation method is typically used to establish the national geodetic coordinate system to a relatively independent coordinate system. The ellipsoidal dilatation method satisfies the requirements for projection distortion, is simple in principle, convenient to calculate, and can quickly achieve a rigorous transformation between the national geodetic coordinate system and the relatively independent coordinate system.

[0003] The ellipsoid parameters of the relatively independent coordinate system established using the ellipsoid dilatation method are generally provided by the design unit or department of the relatively independent coordinate system, and these parameters are not provided to the user unit or department. Therefore, the user unit or department cannot know the strict conversion relationship between the national geodetic coordinate system results and the relatively independent coordinate system results.

[0004] When using the survey results in actual engineering construction, to achieve a rigorous conversion between the National Geodetic Coordinate System (NGS) and the relatively independent coordinate system, the survey results must be handed over to the design unit or department for conversion. Otherwise, an approximate conversion relationship can only be obtained manually, such as using the Bursa seven-parameter model to calculate the conversion relationship between the NGS and the relatively independent coordinate system results. However, the conversion residual can be several centimeters or even decimeters, making a rigorous conversion impossible. Alternatively, the conversion relationship can be obtained through manual traversal, but this involves a large amount of calculation, as a change of one second in the central meridian can cause an error of nearly 30 meters in the projected coordinates, requiring extensive computational work.

[0005] Currently, existing technologies lack methods for inverse calculation of ellipsoid expansion parameters in an independent coordinate system. This invention is the first to propose an inverse calculation method for ellipsoid expansion parameters in an independent coordinate system. Summary of the Invention

[0006] The technical problem this invention aims to solve is to quickly calculate the ellipsoid expansion parameters of a given ellipsoid, based on both the original geodetic coordinate system results (hereinafter referred to as source coordinates) and the results of the relatively independent coordinate system results (hereinafter referred to as target coordinates) after ellipsoid expansion. The source and target coordinates are coordinate points of the same name. Assume that n coordinate points Ki (i = 1, 2, ..., n) of the same name are known. Generally, the semi-major axis and central meridian of the ellipsoid in the source coordinate results are known, and its Gaussian projection coordinates are ; the target coordinate results are established from the source coordinate results using the ellipsoid expansion method, and its Gaussian projection coordinates are , but the semi-major axis and central meridian of the ellipsoid in the target coordinate results are unknown. This invention aims to calculate the ellipsoid expansion parameters of the target coordinate results using the above coordinate points of the same name: the central meridian, the semi-major axis of the ellipsoid, and the projected height.

[0007] To achieve the above technical objectives and effects, this invention is implemented through the following technical solution: a fast inverse calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system, the specific steps of which are as follows:

[0008] S1: Calculate the geodetic coordinates and spatial rectangular coordinates of the source coordinates; based on the ellipsoid parameters of the source coordinates, use the Gaussian inverse calculation formula to project the source coordinates into Gaussian coordinates P. i (x,y) can be converted to geodetic coordinates P. i (B,L), and using the geodetic coordinate transformation formula to convert spatial rectangular coordinates, the geodetic coordinates P are transformed. i (B,L) is converted to spatial rectangular coordinates P. i (X,Y,Z);

[0009] S2: Set the initial values ​​of the target coordinate ellipsoid parameters, the semi-major axis a of the ellipsoid. 2i =a1, Central Meridian L 2i =L1;

[0010] S3: Calculate the approximate central meridian L″ of the target coordinates. 2i ;

[0011] S4: Calculate the approximate semi-major axis a' of the ellipsoid for the target coordinates. 2i ;

[0012] S5: Determine the approximate central meridian L″ of the target coordinates 2i and L' 2i absolute value of difference L ε and the semi-major axis a″ of the approximate ellipsoid 2i and a' 2i The absolute value of the difference a ε Whether it is less than the tolerance requirement, see formulas (6) and (7):

[0013] L ε >|L″ 2i -L' 2i| (6)

[0014] a ε >|a″ 2i -a' 2i | (7)

[0015] If the tolerance requirements can be met simultaneously, then the semi-major axis of the target ellipsoid, a2 = a″. 2i Central Meridian L2 = L″ 2i Then end the traversal loop and proceed to S6; if the tolerance requirement cannot be met, then calculate L″. 2i and a″ 2i Assign values ​​to L' respectively 2i and a' 2i Then, repeat steps S3, S4, and S5 until the tolerance requirement is met, and then proceed to step S6.

[0016] S6: Judgment Does it meet the tolerance requirement? If the tolerance requirement is met, proceed to S9; if... If the tolerance requirement is not met, proceed to S7;

[0017] S7: Determine if the number of identical points is greater than 3. If the number of identical points is less than 3, end the calculation and proceed to S9; if it is greater than or equal to 3, proceed to S8.

[0018] S8: Based on the Bursa seven-parameter model, using the spatial rectangular coordinates P of the source coordinates i (X,Y,Z) and the approximate spatial rectangular coordinates T″ of the target coordinates i (X,Y,Z), calculate its seven Boolean parameters;

[0019] S9: According to the ellipsoidal expansion formula, the projection height (compensation surface elevation) H2 is calculated as shown in formula (8):

[0020]

[0021] In the formula B m The latitude is the average latitude.

[0022] Furthermore, S3 specifically includes the following steps:

[0023] S3.1: Based on the ellipsoidal parameters of the target coordinates, use the Gaussian inverse calculation formula to calculate the Gaussian projection coordinates T of the target coordinates. i (x,y) can be converted to approximate geodetic coordinates T' i (B,L);

[0024] S3.2: Calculate the longitude difference ΔL between the source coordinates' geodetic coordinates and the target coordinates' approximate geodetic coordinates. i See formula (1):

[0025] ΔL i =P i (L)-T' i (L) (1)

[0026] S3.3: Based on the central meridian L' of the target coordinates 2i And the average longitude difference, to calculate the approximate central meridian L″ of the target coordinates. 2i See formula (2):

[0027] L″ 2i =L' 2i +ΔL i (2)

[0028] Furthermore, S4 specifically includes the following steps:

[0029] S4.1: The approximate central meridian L' calculated as described in step S3. 2i Using the Gaussian inverse calculation formula, the Gaussian projection coordinates T of the target coordinates are obtained. i (x,y), inversely calculated into approximate geodetic coordinates T″ i (B,L);

[0030] S4.2: Using the geodetic coordinate transformation formula for spatial rectangular coordinates, the approximate geodetic coordinates T″ are transformed. i (B,L) is converted to approximate spatial rectangular coordinates T″. i (X,Y,Z);

[0031] S4.3: Calculate the approximate spatial rectangular coordinates T″ of the target coordinates i (Z) and the spatial rectangular coordinates P of the source coordinates i The ratio of (Z) is given by formula (3):

[0032] k i =T″ i (Z) / P i (Z) (3)

[0033] S4.4: Calculate the spatial rectangular coordinates P of the source coordinates i (X,Y,Z) and the approximate spatial rectangular coordinates T″ of the target coordinates i The sum of squares of positional errors for (X,Y,Z) is given by formula (4):

[0034]

[0035] S4.5: Based on the semi-major axis of the target ellipsoid and k calculated in S4.3 i Calculate the approximate semi-major axis a″ of the ellipsoid for the target coordinates. 2i See formula (5):

[0036] a″ 2i =k i *a' 2i (5)

[0037] Furthermore, in the above calculation process, if the average latitude B m Unknown, the average latitude B can be included throughout the calculation process. m Set to 0°;

[0038] Furthermore, in the above calculation process, there is at least one point with the same name;

[0039] The beneficial effects of this invention are:

[0040] The present invention provides a fast inverse calculation method for the expansion parameters of a relatively independent coordinate system ellipsoid. This method involves setting initial values ​​for the target coordinate ellipsoid parameters, such as the semi-major axis a. 2i =a1, Central Meridian L 2i =L1; and calculate the approximate central meridian and the approximate semi-major axis of the ellipsoid of the target coordinates, and then determine the approximate central meridian L″ of the target coordinates. 2i and L' 2i absolute value of difference L ε and the semi-major axis a″ of the approximate ellipsoid 2i and a' 2i The absolute value of the difference a ε Whether the limit requirement is met simultaneously determines whether iterative calculation needs to continue; that is, the initial value of the ellipsoid expansion parameter is included in the back calculation of the ellipsoid expansion parameter, and then through iterative calculation, the accurate ellipsoid expansion parameter is gradually approximated by the back calculation.

[0041] The present invention proposes a novel inverse calculation method that results in a smaller error after transformation compared to calculation using the Bursa seven-parameter model and manual traversal. This reduces the number of iterations and eliminates the need for extensive calculations to achieve a rigorous transformation of coordinate results. Attached Figure Description

[0042] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart of an embodiment of the present invention; Detailed Implementation

[0044] Example 1

[0045] In this embodiment, given a known control point with the same name and average latitude, the ellipsoid parameters of the relatively independent coordinate system are calculated. In a city's relatively independent coordinate system established based on the 2000 National Geodetic Coordinate System, to meet the requirement of projection deformation less than 2.5 cm / km, the city's relatively independent coordinate system was established using the ellipsoid dilatation method, employing an arbitrary central meridian and elevation compensation surface. During specific project construction, the results-using unit obtained Gaussian projection coordinate results for both the 2000 National Geodetic Coordinate System (source coordinates) and the relatively independent coordinate system (target coordinates). Now, it is necessary to calculate the ellipsoid dilatation parameters of the relatively independent coordinate system results: the central meridian and the semi-major axis of the ellipsoid. The ellipsoid of the source coordinates is the CGCS2000 ellipsoid, with a semi-major axis of 6378137.0 m and a central meridian of 102°. The source coordinate results are P1 (2921880.054, 518465.380), and the target coordinate results are T1 (2922867.775, 443619.999), with an average latitude of 25°. The detailed calculation steps are as follows:

[0046] S1: Calculate the geodetic coordinates and spatial rectangular coordinates of the source coordinates.

[0047] Based on the ellipsoid parameters of the source coordinates, the Gaussian inverse calculation formula is used to convert the source coordinates' Gaussian projection coordinates P1 (2921880.054, 518465.380) into geodetic coordinates P1 (26.40646054266961, 102.18507804992068). Then, using the geodetic coordinate transformation formula, the geodetic coordinates P1 are converted into spatial rectangular coordinates. i (B,L) is converted to spatial rectangular coordinates P. i (-1206569.965072307,5587652.0605667951,2819466.2742165462).

[0048] S2: Set the initial values ​​of the target coordinate ellipsoid parameters, the semi-major axis a of the ellipsoid. 2i =a1=6378137.0, Central Meridian L 2i =L1=102°;

[0049] S3: Calculate the approximate central meridian L″ of the target coordinates 2i

[0050] The following is the central meridian L″ 2i Specific calculation steps:

[0051] S3.1 Based on the ellipsoidal parameters of the target coordinates, the Gaussian inverse calculation formula is used to project the target coordinates into Gaussian coordinates T. i (x,y) can be converted to approximate geodetic coordinates T′ i(26.414378724637956,101.43487104947165).

[0052] S3.2 Calculate the longitude difference ΔL between the geodetic coordinates of the source coordinates and the approximate geodetic coordinates of the target coordinates. i See formula (1).

[0053] ΔL i =P i (L)-T' i (L) (1)

[0054] ΔL i =0.75020700044902355

[0055] S3.3 Based on the central meridian L' of the target coordinates 2i And the average longitude difference, to calculate the approximate central meridian L″ of the target coordinates. 2i See formula (2):

[0056] L' 2i =L' 2i +ΔL i (2)

[0057] L′ 2i =102.75020700044902355

[0058] S4: Calculate the approximate semi-major axis a' of the ellipsoid for the target coordinates. 2i

[0059] The following is the approximate semi-major axis a' of the ellipsoid used to calculate the target coordinates. 2i Specific steps:

[0060] S4.1 Approximate Central Meridian L' calculated according to step S3 2i Using the Gaussian inverse calculation formula, the Gaussian projection coordinates T of the target coordinates are obtained. i (x,y), inversely calculated into approximate geodetic coordinates T″ i (26.414378724637956,102.18507804992068);

[0061] S4.2 Using the geodetic coordinate transformation formula for spatial rectangular coordinates, the approximate geodetic coordinates T″ are transformed. i (B,L) is converted to approximate spatial rectangular coordinates T″. i (X,Y,2820252.0018334854);

[0062] S4.3 Calculate the approximate spatial rectangular coordinates T″ of the target coordinates i (Z) and the spatial rectangular coordinates P of the source coordinatesi The ratio of (Z) is given in formula (3);

[0063] k i =T″ i (Z) / P i (Z) (3)

[0064] k i =1.0002786795586542

[0065] S4.4 Calculate the spatial rectangular coordinates P of the source coordinates i (X,Y,Z) and the approximate spatial rectangular coordinates T″ of the target coordinates i The sum of squares of positional errors of (X,Y,Z) is given by formula (4);

[0066]

[0067] S4.5: Based on the semi-major axis of the target ellipsoid and k calculated in S4.3 i Calculate the approximate semi-major axis a″ of the ellipsoid for the target coordinates. 2i See formula (5);

[0068] a″ 2i =k i *a' 2i (5)

[0069] a″ 2i =6379914.4564041961

[0070] S5: Determine the approximate central meridian L″ of the target coordinates 2i and L' 2i The absolute value of the difference, and the semi-major axis a″ of the approximate ellipsoid 2i and a' 2i Whether the absolute value of the difference is less than the limit requirement is shown in formulas (6) and (7):

[0071] |L″ 2i -L' 2i | <L ε (6)

[0072] |a″ 2i -a' 2i | ε (7)

[0073] If the tolerance requirement (L) can be met simultaneously ε The tolerance limit is set to 0.00000001°, a ε If the tolerance is set to 0.001m, then the semi-major axis of the target ellipsoid is a2 = a″. 2i Central Meridian L2 = L″​2i Then end the traversal loop and proceed to the next step; if the tolerance requirement cannot be met, then calculate L″. 2i and a″ 2i Assign values ​​to L' respectively 2i and a' 2i Then repeat steps S3, S4 and S5 until the tolerance requirement is met, and then proceed to the next step.

[0074] S6: Judgment Does it meet the tolerance requirement of 0.001m? If the tolerance requirement is met, proceed to S9; if... If the tolerance requirement is not met, proceed to S7; in this embodiment... Therefore, the system will redirect to S7.

[0075] S7: Determine if the number of identical points is greater than 3. If the number of identical points is less than 3, end the calculation and proceed to S9; if it is greater than or equal to 3, proceed to S8; in this embodiment, the number of identical points is 1, so jump directly to S9.

[0076] S8: Based on the Bursa seven-parameter model, using the spatial rectangular coordinates P of the source coordinates i (X,Y,Z) and the approximate spatial rectangular coordinates T″ of the target coordinates i (X,Y,Z), calculate its Boolean 7 parameters; when the number of corresponding points is greater than or equal to 3, then after S7, proceed to S8 to use this step for calculation. The method for calculating its Boolean 7 parameters is a method of the prior art, and will not be described in detail in this embodiment.

[0077] S9: According to the ellipsoidal expansion formula, the calculation of the projected height (compensation surface elevation) H2 is shown in formula (7):

[0078]

[0079] H2 = 1900m.

[0080] Example 2

[0081] In this embodiment, when the average latitude is unknown in Example 1, the average latitude B can be used. m Set the angle to 0°, then repeat the steps in Example 1 to calculate the projection height; since the steps are the same, they will not be repeated.

[0082] Example 3

[0083] In this embodiment, given the results of multiple control points with the same name, the ellipsoid parameters of the relatively independent coordinate system are calculated. In this embodiment, it is assumed that the results of multiple control points with the same name are already known. Steps S1-S6 are the same as those in Embodiment 1, and will not be repeated here. Only in S7 is the number of points determined. If it is greater than or equal to 3, the Bursa-7 parameter needs to be calculated. The method for calculating the Bursa-7 parameter is a prior art method, and will not be explained in detail in this embodiment.

[0084] The preferred embodiments of the present invention disclosed above are only for the purpose of illustrating the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the invention to the specific implementation described.

Claims

1. A fast inverse calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system, characterized in that, include: S1: Calculates the geodetic coordinates and spatial rectangular coordinates of the source coordinates; S2: Set the initial values ​​of the target coordinate ellipsoid parameters, including the semi-major axis a. 2i =a1, Central Meridian L 2i =L1; S3: Calculate the approximate central meridian L″ of the target coordinates 2i ; S4: Calculate the approximate semi-major axis a' of the ellipsoid for the target coordinates. 2i ; S5: Determine the approximate central meridian L″ of the target coordinates 2i and L' 2i absolute value of difference L ε and the semi-major axis a″ of the approximate ellipsoid 2i and a' 2i The absolute value of the difference a ε Relationship with tolerance requirements; If the tolerance requirements can be met simultaneously, then the semi-major axis of the target ellipsoid, a2 = a″. 2i Central Meridian L2 = L″ 2i Then end the traversal loop and proceed to S6; if the tolerance requirement cannot be met, then calculate L″. 2i and a″ 2i Assign values ​​to L' respectively 2i and a' 2i Then, repeat steps S3, S4, and S5 until the tolerance requirement is met, and then proceed to step S6. S6: Judgment Does it meet the tolerance requirement? If the tolerance requirement is met, proceed to S9; if... If the tolerance requirement is not met, proceed to S7; S7: Determine if the number of points with the same name is greater than 3. If it is less than 3, end the calculation; otherwise, jump to step S8. S8: Based on the Bursa seven-parameter model, using the spatial rectangular coordinates P of the source coordinates i (X,Y,Z) and the approximate spatial rectangular coordinates T″ of the target coordinates i (X,Y,Z), calculate its seven Boolean parameters; S9: Calculate the projected height H2 and B according to the ellipsoidal expansion formula. m When the value is unknown, treat it as 0°:

2. The fast inverse calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system as described in claim 1, characterized in that: In S3: S3.1: Based on the ellipsoidal parameters of the target coordinates, use the Gaussian inverse calculation formula to calculate the Gaussian projection coordinates T of the target coordinates. i (x,y) can be converted to approximate geodetic coordinates T' i (B,L); S3.2: Calculate the longitude difference ΔL between the source coordinates' geodetic coordinates and the target coordinates' approximate geodetic coordinates. i : ΔL i =P i (L)-T′ i (L) S3.3: Based on the central meridian L' of the target coordinates 2i And the average longitude difference, to calculate the approximate central meridian L″ of the target coordinates. 2i : L″ 2i =L′ 2i +ΔL i 3. The fast inverse calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system as described in claim 1, characterized in that: In S4, S4.1: The approximate central meridian L' calculated as described in step S3. 2i Using the Gaussian inverse calculation formula, the Gaussian projection coordinates T of the target coordinates are obtained. i (x,y), inversely calculated into approximate geodetic coordinates T″ i (B,L); S4.2: Using the geodetic coordinate transformation formula for spatial rectangular coordinates, the approximate geodetic coordinates T″ are transformed. i (B,L) is converted to approximate spatial rectangular coordinates T″. i (X,Y,Z); S4.3: Calculate the approximate spatial rectangular coordinates T″ of the target coordinates i (Z) and the spatial rectangular coordinates P of the source coordinates i The ratio of (Z): k i =T″ i (Z) / P i (Z) S4.4: Calculate the spatial rectangular coordinates P of the source coordinates i (X,Y,Z) and the approximate spatial rectangular coordinates T″ of the target coordinates i Sum of squared positional errors for (X,Y,Z): S4.5: Based on the semi-major axis of the target ellipsoid and k calculated in S4.3 i Calculate the approximate semi-major axis a″ of the ellipsoid for the target coordinates. 2i : a″ 2i =k i *a′ 2i 4. The fast inverse calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system as described in claim 1, characterized in that: The geodetic coordinates and spatial rectangular coordinates of the source coordinates in S1 are: Based on the ellipsoid parameters of the source coordinates, the Gaussian projection coordinates P of the source coordinates are obtained using the Gaussian inverse calculation formula. i (x,y) can be converted to geodetic coordinates P. i (B,L), and using the geodetic coordinate transformation formula to convert spatial rectangular coordinates, the geodetic coordinates P are transformed. i (B,L) is converted to spatial rectangular coordinates P. i (X,Y,Z).

5. The fast inverse calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system as described in claim 1, characterized in that: In S5, the approximate central meridian L″ is used to determine the target coordinates. 2i and L' 2i The absolute value of the difference, and the semi-major axis a″ of the approximate ellipsoid 2i and a' 2i Whether the absolute value of the difference is less than the limit requirement is shown in formulas (6) and (7); L ε >|L″ 2i -L' 2i | a ε >|a″ 2i -a′ 2i | 6. The application of the fast back calculation method for the ellipsoidal expansion parameters of a relatively independent coordinate system as described in any one of claims 1-5 in urban and large-scale engineering projects.

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