Measurement apparatus and method for rotation angle of geometric axis of object

By using cameras and computer analysis images through the measurement angle of object geometric axis, calculate the angle between the reference object and the geometric axis to be tested, solve the problem of measurement range and distance in existing technology, and achieve a wider range Measurement range and longer measurement distance.

WO2025016085A1PCT designated stage expired Publication Date: 2025-01-23SHANGHAI MIDU MEASUREMENT TECHNOLOGY CO LTD
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Patent Information

Application Number
PCT/CN2024/097411
Authority / Receiving Office
WO · WO
Patent Type
Applications
Current Assignee / Owner
Priority Date
2023-07-14
Filing Date
2024-06-05
Publication Date
2025-01-23

AI Technical Summary

Technical Problem

In the prior art, the method of measuring the rotation angle of the object geometric axis in the object cannot meet special needs due to its dependence on an automatic total station.

Method used

By setting the first feature point and the second feature point between the reference object and the object to be measured, and installing the first camera on the reference object and the second camera on the object to be measured, the computer is used to calculate the reference based on the image acquired by the camera. The angle between the object and the geometric axis of the object to be measured realizes the measurement of the rotation angle of the geometric axis of the object to be measured.

Benefits of technology

This method expands the measurement range and distance, with an incident angle range of ±45 degrees and a maximum distance of 600 meters, exceeding the limitations of traditional automatic total stations.

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    Figure CN2024097411_23012025_PF_FP_ABST
Patent Text Reader

Abstract

A measurement apparatus and method for a rotation angle of the geometric axis of an object. The measurement apparatus comprises: a reference object, which is placed opposite an object to be measured; a first feature point, which is located on the reference object; a second feature point, which is located on the object to be measured; a first camera, which is located on the reference object, wherein the second feature point is within a photographing range of the first camera; a second camera, which is located on the object to be measured, wherein the first feature point is within a photographing range of the second camera; and a computer, which is connected to the first camera and the second camera, and is configured to calculate, according to images acquired by the first camera and the second camera, a transverse included angle and a longitudinal included angle between the geometric axis of the reference object and the geometric axis of the object to be measured.
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Description

Device and method for measuring the rotation angle of an object's geometric axis

[0001] This disclosure claims priority to the Chinese patent application filed with the China Patent Office on July 14, 2023, with application number 202310864067.9. The entire contents of the above application are incorporated by reference into this disclosure. Technical Field

[0002] The present application relates to the technical field of measuring the rotation angle of a geometric axis of an object, for example, to a device and method for measuring the rotation angle of a geometric axis of an object. Background Art

[0003] The method for measuring the rotation angle of the geometric axis of an object in the related art is usually implemented using an automatic total station and a laser target provided with a front screen and a rear screen. The laser target is located on the object to be measured. The total station emits a beam of red laser or a beam of infrared light. The light beam forms a light spot 1 through the front screen and forms a light spot 2 on the rear screen. By accurately measuring the relative coordinates of the light spot 1 and the light spot 2 on the laser target and combining the distance between the front screen and the rear screen, the angle between the light beam and the axis of the laser target is calculated, thereby obtaining the rotation angle of the geometric axis of the object to be measured.

[0004] Since this method completely relies on the automatic total station, the beam incident angle range of the automatic total station is ±12 degrees, and the maximum distance between the automatic total station and the laser target is 150 meters, the measurement range and distance are relatively small and cannot meet special needs.

[0005] Summary of the Invention

[0006] The present application provides a device and method for measuring the rotation angle of a geometric axis of an object, which can increase the measurement range and distance.

[0007] The technical solution adopted in this application is to provide a device for measuring the rotation angle of a geometric axis of an object, comprising:

[0008] The reference object is placed opposite to the object to be measured;

[0009] A first feature point is located on the reference object;

[0010] A second characteristic point is located on the object to be measured;

[0011] a first camera, located on the reference object, wherein the second feature point is within a shooting range of the first camera;

[0012] a second camera, located on the object to be measured, wherein the first feature point is within a shooting range of the second camera;

[0013] A computer is connected to the first camera and the second camera, and is configured to calculate a horizontal angle and a vertical angle between the geometric axis of the reference object and the geometric axis of the object to be measured based on images acquired by the first camera and the second camera.

[0014] The technical solution adopted in this application is to provide a method for measuring the rotation angle of the geometric axis of an object, comprising:

[0015] Setting a reference object so that the reference object is placed opposite to the object to be measured;

[0016] Setting a first characteristic point on the reference object and setting a second characteristic point on the object to be measured;

[0017] Setting a first camera on the reference object, wherein the second feature point is within a shooting range of the first camera;

[0018] A second camera is set on the object to be measured, and the first feature point is within the shooting range of the second camera;

[0019] The first camera and the second camera respectively acquire images, and calculate the horizontal angle and the vertical angle between the geometric axis of the reference object and the geometric axis of the object to be measured based on the acquired images. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] FIG1 is a flow chart of a method for measuring the rotation angle of a geometric axis of an object according to a first embodiment of the present application;

[0021] 2 is a flow chart of calculating the angle between the geometric axis of the reference object and the geometric axis of the object to be measured in the first embodiment of the present application;

[0022] FIG3 is a schematic diagram of a device for measuring the rotation angle of a geometric axis of an object according to a second embodiment of the present application. DETAILED DESCRIPTION

[0023] The first embodiment of the present application relates to a method for measuring the rotation angle of a geometric axis of an object, as shown in FIG1 , comprising:

[0024] Step 1: Setting a reference object so that the reference object and the object to be measured are placed opposite to each other;

[0025] Step 2: setting a first feature point on the reference object and a second feature point on the object to be measured; wherein the first feature point and the second feature point can be any one of a pattern, a light source, and a structural part with a clear outline.

[0026] Step 3: Set up a first camera on the reference object, and the second feature point is within the shooting range of the first camera; wherein, the lateral angle δ1 between the first camera axis and the geometric axis of the reference object, the vertical angle σ1 between the first camera axis and the geometric axis of the reference object, the lateral distance H1 from the first feature point to the first camera axis, and the vertical distance V1 from the first feature point to the first camera axis can be calibrated by conventional precise measurement. The internal parameters of the first camera are: The first distortion coefficient k 11 and the second distortion coefficient k 12 Calibration can also be performed through precise measurement.

[0027] Step 4: Set a second camera on the object to be measured, and the first feature point is within the shooting range of the second camera; wherein, the lateral angle δ2 between the second camera axis and the geometric axis of the object to be measured, the vertical angle σ2 between the second camera axis and the geometric axis of the object to be measured, the lateral distance H2 from the second feature point to the second camera axis, and the vertical distance V2 from the second feature point to the second camera axis can be calibrated by conventional precise measurement. The internal parameters of the second camera are: The first distortion coefficient k 21 and the second distortion coefficient k 22 Calibration can also be performed through precise measurement.

[0028] Step 5: The first camera and the second camera acquire images respectively, and calculate the horizontal angle and vertical angle between the geometric axis of the reference object and the geometric axis of the object to be measured based on the acquired images. As shown in Figure 2, this step includes:

[0029] Step 5.1, calculating the horizontal angle α1 and the vertical angle β1 of the second feature point with respect to the axis of the first camera based on the image acquired by the first camera, including:

[0030] Obtaining pixel coordinates of the second feature point in the image acquired by the first camera;

[0031] According to the internal parameters of the first camera The first distortion coefficient k 11 and the second distortion coefficient k 12 The pixel coordinates of the second feature point in the image acquired by the first camera are converted into physical coordinates on the axis of the first camera; the calculation formula is: Wherein, u2 and v2 are the horizontal physical coordinates and vertical physical coordinates of the second feature point on the axis of the first camera; x2 and y2 are the horizontal pixel coordinates and vertical pixel coordinates of the second feature point in the image acquired by the first camera;

[0032] By α1=tan-1 (x2) calculating the lateral angle α1 of the second feature point with respect to the first camera axis;

[0033] By β1=tan -1 (y2) Calculate the vertical angle β1 of the second feature point with respect to the first camera axis.

[0034] Step 5.2, calculating the horizontal angle α2 and the vertical angle β2 of the first feature point with respect to the axis of the second camera based on the image acquired by the second camera, including:

[0035] Obtaining pixel coordinates of the first feature point in the image acquired by the second camera;

[0036] According to the internal parameters of the second camera The first distortion coefficient k 21 and the second distortion coefficient k 22 The pixel coordinates of the first feature point in the image acquired by the second camera are converted into physical coordinates on the axis of the second camera; the calculation formula is: Wherein, u1 and v1 are the horizontal physical coordinates and vertical physical coordinates of the first feature point on the axis of the second camera; x1 and y1 are the horizontal pixel coordinates and vertical pixel coordinates of the first feature point in the image acquired by the second camera;

[0037] By α2=tan -1 (x1) calculating the lateral angle α2 of the first feature point with respect to the second camera axis;

[0038] By β2=tan -1 (y1) Calculate the vertical angle β2 of the first feature point with respect to the second camera axis.

[0039] Step 5.3: Calculate the transverse angle α′ between the first camera axis and the second camera axis based on the transverse angle α1 of the second feature point with respect to the first camera axis, the transverse angle α2 of the first feature point with respect to the second camera axis, the transverse distance H1 between the first feature point and the first camera axis, the transverse distance H2 between the second feature point and the second camera axis, and the axial distance D of the second feature point in the first camera. An example calculation formula is: The axial distance D of the second feature point from the first camera can be calibrated by precise measurement.

[0040] Step 5.4: Calculate the vertical angle β′ between the first camera axis and the second camera axis based on the vertical angle β1 of the second feature point and the first camera axis, the vertical angle β2 of the first feature point and the second camera axis, the vertical distance V1 from the first feature point to the first camera axis, the vertical distance V2 from the second feature point to the second camera axis, and the axial distance D of the second feature point in the first camera. An example calculation method is:

[0041] Step 5.5, calculate the lateral angle α between the geometric axis of the reference object and the geometric axis of the object to be measured based on the lateral angle α′, the lateral angle δ1 between the first camera axis and the geometric axis of the reference object, and the lateral angle δ2 between the second camera axis and the geometric axis of the object to be measured. The example calculation method is: α=α′+δ1-δ2.

[0042] Step 5.6, calculate the vertical angle β between the geometric axis of the reference object and the geometric axis of the object to be measured based on the vertical angle β′, the vertical angle σ1 between the first camera axis and the geometric axis of the reference object, and the vertical angle σ2 between the second camera axis and the geometric axis of the object to be measured. The example calculation method is: β=β′+σ1-σ2.

[0043] A second embodiment of the present application relates to a device for measuring the rotation angle of a geometric axis of an object, as shown in FIG3 , comprising:

[0044] Reference object 1 is placed opposite to object to be tested 2;

[0045] A first feature point 3, located on the reference object 1;

[0046] The second characteristic point 4 is located on the object to be measured 2;

[0047] a first camera 5 , located on the reference object 1 , and the second feature point 4 is within the shooting range of the first camera 5 ;

[0048] A second camera 6 is located on the object to be measured 2, and the first feature point 3 is within the shooting range of the second camera 6;

[0049] The computer 7 is connected to the first camera 5 and the second camera 6 , and is configured to calculate the angle between the first camera axis and the second camera axis based on images acquired by the first camera 5 and the second camera 6 .

[0050] The computer 7 includes:

[0051] a first image analysis module, configured to calculate a lateral angle α1 and a vertical angle β1 of the second feature point with respect to the first camera axis based on the image acquired by the first camera;

[0052] a second image analysis module, configured to calculate a transverse angle α2 and a vertical angle β2 of the first feature point with respect to the axis of the second camera based on the image acquired by the second camera;

[0053] a first angle calculation module, configured to calculate a transverse angle α′ between the first camera axis and the second camera axis based on a transverse angle α1 of the second feature point on the first camera axis, a transverse angle α2 of the first feature point on the second camera axis, a transverse distance H1 from the first feature point to the first camera axis, a transverse distance H2 from the second feature point to the second camera axis, and an axial distance D of the second feature point on the first camera;

[0054] a second angle calculation module, configured to calculate a vertical angle β′ between the first camera axis and the second camera axis based on a vertical angle β1 of the second feature point on the first camera axis, a vertical angle β2 of the first feature point on the second camera axis, a vertical distance V1 from the first feature point to the first camera axis, a vertical distance V2 from the second feature point to the second camera axis, and an axial distance D of the second feature point on the first camera;

[0055] a third angle calculation module, configured to calculate the transverse angle α between the geometric axis of the reference object and the geometric axis of the object to be measured based on the transverse angle α′, the transverse angle δ1 between the first camera axis and the geometric axis of the reference object, and the transverse angle δ2 between the second camera axis and the geometric axis of the object to be measured;

[0056] The fourth angle calculation module is configured to calculate the vertical angle β between the geometric axis of the reference object and the geometric axis of the object to be measured based on the vertical angle β′, the vertical angle σ1 between the first camera axis and the geometric axis of the reference object, and the vertical angle σ2 between the second camera axis and the geometric axis of the object to be measured.

[0057] The first image analysis module includes:

[0058] a first acquiring unit, configured to acquire pixel coordinates of the second feature point in the image acquired by the first camera;

[0059] A first conversion unit is configured to convert the first camera's internal parameters The first distortion coefficient k 11 and the second distortion coefficient k 12 The pixel coordinates of the second feature point in the image acquired by the first camera are converted into physical coordinates on the axis of the first camera; the calculation formula is: Wherein, u2 and v2 are the horizontal physical coordinates and vertical physical coordinates of the second feature point on the axis of the first camera; x2 and y2 are the horizontal pixel coordinates and vertical pixel coordinates of the second feature point in the image acquired by the first camera;

[0060] The first lateral angle calculation unit is set to pass α1 = tan -1 (x2) calculating the lateral angle α1 of the second feature point with respect to the first camera axis;

[0061] The first vertical angle calculation unit is set to be β1=tan -1 (y2) Calculate the vertical angle β1 of the second feature point with respect to the first camera axis.

[0062] The second image analysis module includes:

[0063] a second acquiring unit, configured to acquire pixel coordinates of the first feature point in the image acquired by the second camera;

[0064] The second conversion unit is configured to convert the internal parameters of the second camera into The first distortion coefficient k 21 and the second distortion coefficient k 22 The pixel coordinates of the first feature point in the image acquired by the second camera are converted into physical coordinates on the axis of the second camera; the calculation formula is: Wherein, u1 and v1 are the horizontal physical coordinates and vertical physical coordinates of the first feature point on the axis of the second camera; x1 and y1 are the horizontal pixel coordinates and vertical pixel coordinates of the first feature point in the image acquired by the second camera;

[0065] The second lateral angle calculation unit is set to pass α2=tan -1 (x1) calculating the lateral angle α2 of the first feature point with respect to the second camera axis;

[0066] The second vertical angle calculation unit is set to be β2=tan -1 (y1) Calculate the vertical angle β2 of the first feature point with respect to the second camera axis.

[0067] The calculation method adopted by the first angle calculation module is: The calculation method adopted by the second angle calculation module is:

[0068] The calculation method adopted by the third angle calculation module is: α=α′+δ1-δ2; the calculation method adopted by the fourth angle calculation module is β=β′+σ1-σ2.

[0069] It can be seen that the present application adds a first camera and a first feature point to a reference object, adds a second camera and a second feature point to the object to be measured, uses the two cameras to image each other, obtains the angle between the feature point and the camera axis, and then determines the angle between the axis of the object to be measured and the axis of the reference object based on the obtained angle, thereby realizing the measurement of the rotation angle of the geometric axis of the object to be measured. The incident angle range of this method is ±45 degrees, and the maximum distance between the two can reach 600 meters. Compared with the traditional automatic total station, the measurement range and distance are increased.

Claims

1. A device for measuring the rotation angle of a geometric axis of an object, comprising: The reference object is placed opposite to the object to be tested; A first feature point is located on the reference object; A second characteristic point is located on the object to be measured; A first camera is located on the reference object, and the second feature point is within the shooting range of the first camera; A second camera is located on the object to be measured, and the first feature point is within the shooting range of the second camera; The computer is connected to the first camera and the second camera, and is configured to calculate the horizontal angle and the vertical angle between the geometric axis of the reference object and the geometric axis of the object to be measured according to the images acquired by the first camera and the second camera.

2. The device for measuring the rotation angle of a geometric axis of an object according to claim 1, wherein: The computer comprises: A first image analysis module, configured to calculate a lateral angle α1 and a vertical angle β1 of the second feature point on the axis of the first camera according to the image acquired by the first camera; A second image analysis module, configured to calculate a lateral angle α2 and a vertical angle β2 of the first feature point on the axis of the second camera according to the image acquired by the second camera; a first angle calculation module, configured to calculate a transverse angle α′ between the first camera axis and the second camera axis according to a transverse angle α1 of the second feature point on the first camera axis, a transverse angle α2 of the first feature point on the second camera axis, a transverse distance H1 from the first feature point to the first camera axis, a transverse distance H2 from the second feature point to the second camera axis, and an axial distance D of the second feature point on the first camera; a second angle calculation module, configured to calculate a vertical angle β′ between the first camera axis and the second camera axis according to a vertical angle β1 of the second feature point on the first camera axis, a vertical angle β2 of the first feature point on the second camera axis, a vertical distance V1 from the first feature point to the first camera axis, a vertical distance V2 from the second feature point to the second camera axis, and an axial distance D of the second feature point on the first camera; A third angle calculation module is configured to calculate a transverse angle α between the reference object geometric axis and the object to be measured geometric axis according to the transverse angle α′, a transverse angle δ1 between the first camera axis and the reference object geometric axis, and a transverse angle δ2 between the second camera axis and the object to be measured geometric axis; The fourth angle calculation module is configured to calculate the vertical angle β′, the first camera axis and the The vertical angle β between the geometric axis of the reference object and the geometric axis of the object to be measured is calculated based on the vertical angle σ1 of the geometric axis of the reference object and the vertical angle σ2 between the second camera axis and the geometric axis of the object to be measured.

3. The device for measuring the rotation angle of a geometric axis of an object according to claim 2, wherein: The first image analysis module comprises: A first acquisition unit, configured to acquire pixel coordinates of the second feature point in the image acquired by the first camera; A first conversion unit is configured to convert the first camera into The first distortion coefficient k 11 and the second distortion coefficient k 12 The pixel coordinates of the second feature point in the image acquired by the first camera are converted into physical coordinates on the axis of the first camera; the calculation formula is: Wherein, u2 and v2 are the horizontal physical coordinates and vertical physical coordinates of the second feature point on the axis of the first camera; x2 and y2 are the horizontal pixel coordinates and vertical pixel coordinates of the second feature point in the image acquired by the first camera; The first lateral angle calculation unit is set to pass α1 = tan -1 (x2) calculating the lateral angle α1 of the second feature point on the first camera axis; The first vertical angle calculation unit is set to pass β1 = tan -1 (y2) Calculate the vertical angle β1 of the second feature point on the first camera axis.

4. The device for measuring the rotation angle of a geometric axis of an object according to claim 2, wherein: The second image analysis module comprises: A second acquisition unit, configured to acquire pixel coordinates of the first feature point in the image acquired by the second camera; A second conversion unit is configured to convert the internal parameters of the second camera into The first distortion coefficient k 21 and the second distortion coefficient k 22 The pixel coordinates of the first feature point in the image acquired by the second camera are The coordinates are converted into physical coordinates on the axis of the second camera; the calculation formula is: Wherein, u1 and v1 are the horizontal physical coordinates and vertical physical coordinates of the first feature point on the axis of the second camera; x1 and y1 are the horizontal pixel coordinates and vertical pixel coordinates of the first feature point in the image acquired by the second camera; The second lateral angle calculation unit is set to pass α2 = tan -1 (x1) calculating the lateral angle α2 of the first feature point on the axis of the second camera; The second vertical angle calculation unit is set to pass β2 = tan -1 (y1) Calculate the vertical angle β2 of the first feature point on the second camera axis.

5. The device for measuring the rotation angle of a geometric axis of an object according to claim 2, wherein: The calculation method adopted by the first angle calculation module is: The calculation method adopted by the second angle calculation module is:

6. The device for measuring the rotation angle of a geometric axis of an object according to claim 2, wherein: The calculation method adopted by the third angle calculation module is: α=α′+δ1-δ2; the calculation method adopted by the fourth angle calculation module is β=β′+σ1-σ2.

7. A method for measuring the rotation angle of a geometric axis of an object, comprising: Setting a reference object so that the reference object is placed opposite to the object to be measured; Setting a first characteristic point on the reference object and setting a second characteristic point on the object to be measured; Setting a first camera on the reference object, wherein the second feature point is within a shooting range of the first camera; A second camera is arranged on the object to be measured, and the first feature point is within the shooting range of the second camera; The first camera and the second camera respectively acquire images, and calculate the horizontal angle and the vertical angle between the geometric axis of the reference object and the geometric axis of the object to be measured according to the acquired images.

8. The method for measuring the rotation angle of a geometric axis of an object according to claim 7, wherein: The step of calculating the horizontal angle and the vertical angle between the geometric axis of the reference object and the geometric axis of the object to be measured according to the acquired image comprises: Calculating a lateral angle α1 and a vertical angle β1 of the second feature point on the axis of the first camera according to the image acquired by the first camera; Calculating a lateral angle α2 and a vertical angle β2 of the first feature point on the axis of the second camera according to the image acquired by the second camera; Calculate the transverse angle α′ between the first camera axis and the second camera axis according to the transverse angle α1 of the second feature point on the first camera axis, the transverse angle α2 of the first feature point on the second camera axis, the transverse distance H1 from the first feature point to the first camera axis, the transverse distance H2 from the second feature point to the second camera axis, and the axial distance D of the second feature point on the first camera; Calculate a vertical angle β′ between the first camera axis and the second camera axis according to a vertical angle β1 of the second feature point on the first camera axis, a vertical angle β2 of the first feature point on the second camera axis, a vertical distance V1 from the first feature point to the first camera axis, a vertical distance V2 from the second feature point to the second camera axis, and an axial distance D of the second feature point on the first camera; Calculate the lateral angle α between the geometric axis of the reference object and the geometric axis of the object to be measured according to the lateral angle α′, the lateral angle δ1 between the first camera axis and the geometric axis of the reference object, and the lateral angle δ2 between the second camera axis and the geometric axis of the object to be measured; The vertical angle β between the reference object geometric axis and the object to be measured is calculated according to the vertical angle β′, the vertical angle σ1 between the first camera axis and the reference object geometric axis, and the vertical angle σ2 between the second camera axis and the object to be measured geometric axis.

9. The method for measuring the rotation angle of a geometric axis of an object according to claim 7, wherein: The step of calculating the lateral angle α1 and the vertical angle β1 of the second feature point on the axis of the first camera according to the image acquired by the first camera includes: Obtaining pixel coordinates of the second feature point in the image acquired by the first camera; According to the internal parameters of the first camera The first distortion coefficient k 11 and the second distortion coefficient k 12 The pixel coordinates of the second feature point in the image acquired by the first camera are converted into pixel coordinates of the second feature point in the image acquired by the first camera. The physical coordinates of the machine axis; the calculation formula is: Wherein, u2 and v2 are the horizontal physical coordinates and vertical physical coordinates of the second feature point on the axis of the first camera; x2 and y2 are the horizontal pixel coordinates and vertical pixel coordinates of the second feature point in the image acquired by the first camera; By α1 = tan -1 (x2) calculating the lateral angle α1 of the second feature point on the first camera axis; By β1 = tan -1 (y2) calculating a vertical angle β1 of the second feature point on the first camera axis; The step of calculating the lateral angle α2 and the vertical angle β2 of the first feature point on the axis of the second camera according to the image acquired by the second camera includes: Obtaining pixel coordinates of the first feature point in the image acquired by the second camera; According to the internal parameters of the second camera The first distortion coefficient k 21 and the second distortion coefficient k 22 The pixel coordinates of the first feature point in the image acquired by the second camera are converted into physical coordinates on the axis of the second camera; the calculation formula is: Wherein, u1 and v1 are the horizontal physical coordinates and vertical physical coordinates of the first feature point on the axis of the second camera; x1 and y1 are the horizontal pixel coordinates and vertical pixel coordinates of the first feature point in the image acquired by the second camera; By α2 = tan -1 (x1) calculating the lateral angle α2 of the first feature point on the axis of the second camera; By β2 = tan -1 (y1) Calculate the vertical angle β2 of the first feature point on the second camera axis.

10. The method for measuring the rotation angle of a geometric axis of an object according to claim 7, wherein: The lateral included angle α′ between the first camera axis and the second camera axis is Calculated; The vertical angle β′ between the first camera axis and the second camera axis is Calculated; The lateral angle α between the geometric axis of the reference object and the geometric axis of the object to be measured is calculated by α=α′+δ1-δ2; The vertical angle β between the geometric axis of the reference object and the geometric axis of the object to be measured is calculated by β=β′+σ1-σ2.

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