A method for determining a rotating datum for full-aperture measurement of off-axis parabolic surfaces

Abstract

This application discloses a method for determining a rotating reference for full-aperture measurement of off-axis parabolic surfaces. The method includes calculating the coordinates of boundary points and the geometric center point, and defining the bisecting center point; rotating the off-axis parabolic surface around the rotation axis by an angle θ with the geometric center point as the rotation center; setting parallel constraints on the tangent planes to determine the functional relationship between the coordinates of the bisecting center point and the rotation angle; setting constraints for equal projection distances so that the distances from the projections of each boundary point on the XY plane to the projection of the bisecting center point are equal on both the off-axis and non-off-axis directions; substituting the coordinates of each point into the rotation transformation formula and the constraints to establish a univariate nonlinear equation to solve for the optimal rotation angle; and establishing a detection area centered on the projection of the bisecting center point after rotation for full-aperture measurement. This method ensures symmetry in the mirror projection after rotation, enabling the establishment of a regular circular or elliptical detection area, avoiding the problems of probe overshooting or incomplete surface shape detection in traditional methods.

Description

Technical Field

[0001] This application relates to the field of optical measurement technology, specifically to a method for determining a rotating reference for measuring the full aperture of an off-axis parabolic surface. Background Technology

[0002] Off-axis parabolic optical elements have wide applications in numerous fields such as astronomical telescopes, laser systems, and space optics. With the continuous development of these fields, the requirements for the precision and quality of off-axis parabolic optical elements are becoming increasingly stringent. In the field of advanced optical manufacturing and inspection technology, surface shape inspection is a crucial step in ensuring the quality of optical elements, and accurate determination of the rotational reference is essential for achieving full-aperture measurement. This not only relates to the processing accuracy of the optical element but also affects the performance and application effect of the entire optical system. A reasonable method for determining the rotational reference can improve inspection efficiency and accuracy, providing strong support for the manufacturing and application of optical elements.

[0003] In traditional off-axis parabolic optical element surface shape inspection, to facilitate processing, the off-axis parabola is usually rotated by a certain angle to ensure a more stable machining posture. The traditional method uses the center of curvature of the off-axis parabola as a reference point, rotating it by an off-axis angle to stabilize it. During surface shape inspection, the mirror's test posture must be consistent with the machining posture to ensure that the machining accuracy corresponds to the test results. The traditional inspection method involves inspecting the mirror on a coordinate measuring machine, using the center of curvature of the mirror as the central dividing point to delineate the projection of the mirror's edge contour onto the XY plane.

[0004] However, this traditional method has significant drawbacks. If the mirror body is rotated off-axis around the X or Y axis with the center of curvature as the center of rotation, the distances from each boundary point to the projection of the mirror's edge contour onto the XY plane become unequal due to the asymmetry of the off-axis parabola. This fails to meet the corresponding symmetry conditions, making it impossible to establish a circular or elliptical detection area for full-aperture measurement using a coordinate measuring machine. If the center of curvature of the mirror body is used as the center and the longer end of the projection is used as the radius for scanning, the probe's movement trajectory will exceed the actual size of the shorter end of the mirror body, causing the probe to fall onto the external fixture or the coordinate measuring machine base, resulting in measurement errors and probe damage. If the shorter end is used as the radius for scanning, the probe's movement trajectory is smaller than the actual size of the mirror body, leading to incomplete surface shape detection and making it difficult to perform accurate full-aperture machining and shaping of the mirror surface subsequently. Summary of the Invention

[0005] To address the technical problems in the prior art, this application provides a method for determining a rotating reference for measuring the full diameter of an off-axis parabolic surface.

[0006] The method for determining a rotating reference for measuring the full diameter of an off-axis parabolic surface provided in this application adopts the following technical solution: A method for determining a rotating reference for full-aperture measurement of off-axis parabolic surfaces includes the following steps: S1. Based on the off-axis amount h, aperture diameter d, and mother mirror curvature radius R of the off-axis parabola, calculate the coordinates of two boundary points A and B in the off-axis direction, the coordinates of two boundary points C and D in the non-off-axis direction, and the coordinates of the geometric center point O of the off-axis parabola based on the parabolic equation; define the bisecting center point E on the off-axis parabola, wherein the bisecting center point E is a feature point to be determined on the off-axis parabola, and its coordinates are determined as a function of the rotation angle θ by the parallel constraint condition of the tangent plane in step S3; S2. Determine the rotation axis and rotation center according to the off-axis direction. With the geometric center point as the rotation center, rotate the off-axis parabola around the rotation axis by an angle θ. After rotation, the boundary points A, B, C, D, the bisecting center point E, and the geometric center point O are transformed into the corresponding rotated points a, b, c, d, e, and o, respectively. The reference projection plane is the XY plane in the coordinate system before rotation. S3. Set a parallel constraint condition for the tangent plane so that the tangent plane at the bisecting center point e after rotation is parallel to the XY plane. Determine the functional relationship between the coordinates of the bisecting center point E before rotation and the rotation angle θ according to the parallel constraint condition for the tangent plane. After rotation, the bisecting center point e becomes the equidistant center point in the projection region of the off-axis parabola on the XY plane after rotation, which is equidistant from the projections of the boundary points on both sides of the off-axis direction. S4. Set equal projection distance constraints so that the projections of the boundary points a, b, c, and d of the rotated off-axis parabola onto the XY plane are equal on both sides in the off-axis direction and equal on both sides in the non-off-axis direction. S5. Substitute the coordinates of each point into the rotation transformation formula and the equal projection distance constraint to establish a univariate nonlinear equation about the rotation angle θ and solve it numerically to obtain the optimal rotation angle and the coordinates of the corresponding bisecting center point E. Establish a detection area with the projection point of the bisecting center point e on the XY plane after rotation as the center, and perform full-diameter measurement on the off-axis parabola.

[0007] In summary, this application has the following beneficial technical effects: 1. By using the geometric center of the off-axis parabola as the rotation center and defining the bisecting center point on the off-axis parabola, and combining the constraints of parallel tangent planes and equal projection distances, the optimal rotation angle is solved. This ensures that the distances from the projections of each boundary point of the off-axis parabola onto the XY plane to the projection of the bisecting center point are equal on both sides in the off-axis and non-off-axis directions after rotation. This allows for the establishment of a circular or elliptical detection area centered on this projection point, achieving full-aperture measurement without omissions. This avoids the problems of projection asymmetry, probe overtravel, or incomplete surface detection caused by using the center of curvature as the rotation center in traditional methods. For example, in the embodiment of this application, the difference in projection distance in the off-axis direction of the traditional method is 2.86 mm, while the difference is 0 mm after using the method of this application, increasing the detection coverage from approximately 89% to 100% of the traditional method. 2. By measuring the actual coordinates of at least three boundary points after rotation, the actual rotation angle and actual rotation center are calculated using least squares fitting. The center coordinates and major and minor axis radii of the detection area are recalculated based on the rotation angle deviation and rotation center offset to generate a compensated detection area. This ensures that the detection area can completely cover the full-aperture projection range of the off-axis parabolic surface after rotation in the XY plane even with mechanical assembly deviations, thus reducing the accuracy requirements of the assembly equipment. 3. By fixing the off-axis parabolic mirror and the adapter fixture together with an adhesive layer to form an integrated structure, the contour support surface of the adapter fixture matches the non-working surface shape of the off-axis parabolic mirror at the optimal rotation angle. This allows the off-axis parabolic mirror to automatically be in the posture corresponding to the optimal rotation angle after being glued and fixed. After measurement, it is transferred to the processing equipment in the same posture for shaping. The consistency between the detection datum and the processing datum is ensured by the cooperation of the bottom positioning pin and the positioning hole, reducing the need for repeated assembly and adjustment. Attached Figure Description

[0008] Figure 1 This is a flowchart illustrating the method for determining the rotating reference for full-aperture measurement of off-axis parabolic surfaces in this application. Figure 2 This is a schematic diagram of the three-dimensional model of the original off-axis parabola in the coordinate system of the parent mirror in the embodiment of this application. The diagram shows the origin of the coordinate system, the positive directions of the X-axis, Y-axis, and Z-axis, as well as the positions of each feature point A, B, C, D, E, and O. Figure 3 This is a schematic diagram of an off-axis parabolic model after rotating it 18.5691° off-axis around the X-axis using the conventional method in this embodiment of the application, with the center of curvature as the base point. Figure 4 This is a schematic diagram of an off-axis parabolic model after rotating around the X-axis by the method of this application with the geometric center point o as the base point at an optimal rotation angle of 19.7149° in an embodiment of this application. Figure 5This is a schematic diagram of the projection of the off-axis parabola after rotation onto the XY plane and the detection area in an embodiment of this application. The projection positions of points a, b, c, d, e, and o after rotation, as well as the major and minor axes of the elliptical detection area, are marked in the figure. Figure 6 This is a schematic diagram of the edge marking trajectory of the off-axis parabolic mirror model used in the actual measurement of a coordinate measuring machine in the embodiments of this application; Figure 7 This is a schematic diagram of the integrated structure of the off-axis parabolic mirror model and the adapter tooling in an embodiment of this application. The positions of the off-axis parabolic mirror, adhesive layer, adapter tooling, contour support surface, and positioning pin are marked in the figure. Figure 8 This is a schematic diagram of adaptive variable density detection path planning in an embodiment of this application; Figure 9 This is a schematic diagram of rotational adjustment error compensation in an embodiment of this application; Figure 10 This is a schematic diagram of iterative correction based on detection data in an embodiment of this application. Detailed Implementation

[0009] The embodiments of this application will now be described in detail with reference to the accompanying drawings. The described embodiments are merely possible technical implementations of this application, and are not limited thereto. Other embodiments that can be obtained by those skilled in the art based on the embodiments of this application without creative effort are all within the protection scope of this application.

[0010] This application mainly adopts precise calculation of the rotation datum to realize the full-diameter measurement of off-axis parabolic surfaces, which avoids the defects of traditional detection methods and achieves the effect of synergistic optimization of full-diameter detection and processing posture. The following is a further detailed description of this application.

[0011] Coordinate system definition: The coordinate system adopted in this application is a right-handed coordinate system, with the origin located at the vertex of the parabolic surface of the mother mirror. The X and Y axes lie in the horizontal plane, and the Z axis is vertically downwards as the positive direction. Under this coordinate system, the equation of the parabolic surface is expressed as: , where R is the radius of curvature of the parent mirror. The off-axis parabolic surface is located above the parent mirror parabolic surface (Z is the negative value region).

[0012] The method for determining the rotational datum for full-diameter measurement of off-axis parabolic surfaces provided in this application includes the following steps: determining the coordinates of the boundary points and geometric center points of the off-axis parabolic surface (step S1); rotating the off-axis parabolic surface (step S2); setting parallel constraints on the tangent planes (step S3); setting constraints on equal projection distances (step S4); solving for the optimal rotation angle and establishing the detection area (step S5); generating an adaptive variable density detection path (step S6); compensating for rotational adjustment errors (step S7); and iterative correction based on detection data (step S8). Through the coordinated operation of these steps, the rotational datum of the off-axis parabolic surface can be accurately determined, thereby achieving full-diameter measurement. This avoids problems such as probe misalignment, measurement errors, or incomplete surface shape detection caused by projection asymmetry in traditional methods. It also reduces processing difficulty and ensures the consistency between the detection datum and the processing datum. Steps S1 to S5 are the basic rotational datum determination process, while steps S6, S7, and S8 are further optimization processes. These three steps can be executed in parallel or selectively. This method can be implemented by a computer program, run on a computing device equipped with a data processing module, and can be integrated into optical inspection software or CNC system. The computing device can be an industrial control computer or host computer that is connected to a coordinate measuring machine.

[0013] Specifically, in the step of determining the coordinates of the boundary points and geometric center point of the off-axis parabola (step S1), it is necessary to consider the off-axis amount of the off-axis parabola. Diameter and the radius of curvature of the mother mirror The calculation is based on the equation of a parabola. In a coordinate system where the Z-axis is vertically downward and positive, the equation of the parabola is: When the off-axis direction is along the Y-axis, the coordinates of the geometric center point are: ,in The boundary points in the off-axis direction are respectively and ,in , The boundary points in the non-off-axis direction are respectively and ,in When the off-axis direction is along the X-axis, the coordinates of the geometric center point are: ,in The boundary points in the non-off-axis direction are respectively and ,in , The boundary points in the off-axis direction are respectively and ,in , The equation of the parabola here is derived based on the geometric properties of an off-axis parabola, and it forms the basis for determining the coordinates of each point. Specifically, boundary point A is the boundary point on the off-axis side closer to the vertex of the parent mirror, and boundary point B is the boundary point on the off-axis side farther from the vertex of the parent mirror.

[0014] In step S1, it is also necessary to define the bisecting center point E on the off-axis parabola. The bisecting center point E is a feature point on the off-axis parabola to be determined, and its coordinates are determined as a function of the rotation angle θ by the parallel constraint condition of the tangent plane in step S3. Its physical meaning is: when the off-axis parabola rotates around the rotation axis with the geometric center point O as the center of rotation, the optimal angle is determined. After rotation, point e, obtained by rotating feature point E, becomes the equidistant center point in the projection region of the rotated off-axis parabola on the XY plane, with equal distances to the projections of the two boundary points in the off-axis direction. In other words, the projection of point e on the XY plane after rotation satisfies the condition that it is equidistant from the projections of the two boundary points in the off-axis direction. The coordinates of the bisecting center point E are determined in step S3 by the parallel constraint condition of the tangent plane, and its coordinates are the rotation angle. The function.

[0015] The process of this step is as follows: First, based on the known off-axis parabolic parameters (off-axis amount)... Diameter Mother mirror curvature radius The coordinates of each feature point are calculated by substituting them into the parabolic equation. Then, the bisecting center point E is defined on the off-axis parabola, laying the conceptual foundation for the subsequent constraint conditions. The technical effect of this step is that it accurately calculates the coordinates of each feature point of the off-axis parabola using the parabolic equation, providing an accurate data basis for subsequent rotation transformations and constraint conditions. At the same time, it clearly defines the physical meaning of the bisecting center point E, giving the constraint conditions in subsequent steps a clear geometric meaning.

[0016] In the step of rotating the off-axis parabola (step S2), the rotation axis and rotation center are determined according to the off-axis direction. With the geometric center point O as the rotation center, the off-axis parabola is rotated about the rotation axis by an angle. When the off-axis direction is along the Y-axis, the rotation axis is the X-axis; when the off-axis direction is along the X-axis, the rotation axis is the Y-axis. After rotation, boundary points A, B, C, D, the bisecting center point E, and the geometric center point O are transformed into corresponding rotated points a, b, c, d, e, and o, respectively. The reference projection plane is the XY plane in the original coordinate system. The purpose of rotation is to make the projection of the off-axis parabola on the projection plane more conducive to full-aperture measurement. Specifically, a is the corresponding point after the rotation of boundary point A, b is the corresponding point after the rotation of boundary point B, c is the corresponding point after the rotation of boundary point C, d is the corresponding point after the rotation of boundary point D, e is the corresponding point after the rotation of the bisecting center point E, and o is the corresponding point after the rotation of the geometric center point O. The coordinates of each point after rotation are calculated using the rotation transformation formula in step S5. All rotation transformations in this application are based on a right-handed coordinate system, and the rotation direction follows the right-hand rule.

[0017] Using the traditional method, rotate the off-axis angle around the X-axis with the center of curvature as the reference point. After rotation, the coordinates of points A', B', C', D', and O' are determined. The condition of equal projection distance is not met, making it impossible to establish a symmetrical detection region. The method of this application uses the geometric center point o as the base point and rotates around the X-axis at the optimal angle. After rotation, the coordinates of points a, b, c, d, o, and e are then... , The condition of equal projection distance is met.

[0018] The process of this step is as follows: First, determine the off-axis direction (along the X-axis or along the Y-axis) and identify the corresponding axis of rotation; then, using the geometric center point O as the center of rotation, rotate the off-axis parabola around the axis of rotation by an angle. The coordinates of each feature point after rotation are calculated. The technical advantage of this step is that by using the geometric center of the mirror as the rotation center rather than the curvature center, the position of the mirror center remains unchanged after rotation, which facilitates the subsequent establishment of a symmetrical detection area, while avoiding the projection asymmetry problem caused by using the curvature center as the rotation center in traditional methods.

[0019] In the step of setting the parallel constraint condition for the tangent plane (step S3), to ensure that the tangent plane at the bisecting center point e after rotation is parallel to the XY plane, the coordinates of the bisecting center point E before rotation and the rotation angle are determined according to this constraint condition. The functional relationship. When the off-axis direction is along the Y-axis, the coordinates of the center point E before rotation satisfy... When the off-axis direction is along the X-axis, the coordinates of the center point E before rotation satisfy... This constraint is to ensure that the rotated off-axis parabola has a specific orientation at the bisecting center point, which facilitates the subsequent establishment of the detection area.

[0020] The E-coordinate is derived from the partial derivative condition of the parabolic equation, and the specific derivation process is as follows. For a parabolic surface... Its partial derivative in the X direction is The partial derivative in the Y direction is Any point on the parabola The normal vector at is (Unnormalized). After rotation, the tangent plane is parallel to the XY plane, meaning that the surface normal vector at that point is parallel to the Z-axis after rotation. When rotating by an angle θ along the Y-axis and about the X-axis, the rotation transformation does not change the X-component of the normal vector, only the Y and Z-components. Let the normal vector at point E be... After rotation, the normal vector becomes To make the rotated normal vector parallel to the Z-axis, the following must be satisfied: ,Right now ; ,Right now Substituting into the equation of the parabola, we get... .therefore .

[0021] When the off-axis direction is along the X-axis, a similar derivation can be obtained. , , ,Right now .

[0022] The process of this step is as follows: Based on the differential geometric properties of the parabola, the coordinates and rotation angle of the bisecting center point E are derived using the partial derivative conditions. The functional relationship is as follows. The technical effect of this step is that, by using the parallel constraint condition of the tangent plane, it ensures that the tangent plane at the bisecting center point e after rotation is parallel to the XY plane, thereby ensuring the symmetry when projecting around this point, and providing a geometric basis for establishing the equal projection distance constraint condition in step S4.

[0023] In the step of setting the equal projection distance constraint (step S4), the distances from the projections of the boundary points a, b, c, and d of the rotated off-axis parabola onto the XY plane to the projection point of the bisector center point e onto the XY plane should be equal on both sides in the off-axis direction and equal on both sides in the non-off-axis direction. When the off-axis direction is along the Y-axis, the equal projection distance constraint in the off-axis direction is applied. Conditions for non-off-axis directions Because rotation around the X-axis does not change the X-coordinate, this constraint is automatically satisfied; when the off-axis direction is along the X-axis, an equal off-axis projection distance constraint is applied. Conditions for non-off-axis directions Automatically satisfied. This constraint is designed to ensure the symmetry of the projection of the rotated off-axis parabola onto the projection plane, thereby enabling the establishment of a regular detection region.

[0024] in, This represents the Euclidean distance between the projections of point a and point e onto the XY plane after rotation, i.e. ,in and Let a and e be the projected coordinates of points a and e on the XY plane, respectively. , , The meaning is similar. The projections of the rotated off-axis parabola onto the XY plane are symmetrically distributed about the projection points.

[0025] The process of this step is as follows: Based on the coordinates of point E determined in step S3, the coordinates of each boundary point and the bisecting center point after rotation are substituted into the projection distance formula to establish a constraint equation for equal projection distances in the off-axis direction. The technical effect of this step is that, through the constraint condition of equal projection distances, it ensures that the edges of the rotated mirror are symmetrically distributed on the projection plane about the bisecting center point, enabling the coordinate measuring machine to establish a regular detection trajectory based on this center point, achieving full-diameter detection without omissions or overstepping boundaries.

[0026] In the step of solving for the optimal rotation angle and establishing the detection area (step S5), the coordinates of each point are substituted into the rotation transformation formula and the constraint of equal projection distance to establish the rotation angle. The univariate nonlinear equation was numerically solved to obtain the optimal rotation angle and the coordinates of the corresponding bisecting center point E. A detection area was established centered on the projection of the rotated bisecting center point e onto the XY plane, and full-diameter measurements were performed on the off-axis paraboloid. The rotation transformation formula is based on a right-hand coordinate system, and the rotation direction follows the right-hand rule. The geometric center point... any point is the center of rotation. Rotate around the X-axis back: Rotate about the Y-axis back: When establishing the detection area, the projected distance in the off-axis direction after rotation is... Equal to the projected distance in the non-off-axis direction At that time, with the projection point as the center, and with Establish a circular detection area with a radius; when At that time, with the projection point as the center of the ellipse, and with and An elliptical detection region is established by setting the larger value of the major axis radius and the smaller value of the minor axis radius, where the major axis direction corresponds to the direction with the larger projection distance.

[0027] Solving for the optimal rotation angle At the same time, the condition of minimizing the tilt angle can also be satisfied. Define the original off-axis angle. ,in Off-axis quantity Let be the radius of curvature of the parent mirror. Construct an optimization model: ; The constraints are: , , , in The maximum permissible rotation angle deviation typically ranges from 2° to 5°. Under the premise of satisfying the constraint of equal projection distance, the rotation angle is... Compared with the original off-axis angle The deviation is minimized, thereby minimizing the tilt angle of the rotated off-axis parabolic surface relative to the XY plane. The physical significance of this optimization model is to minimize the tilt of the mirror surface after rotation, while ensuring the feasibility of full-aperture inspection, thus reducing the difficulty of subsequent processing.

[0028] Specifically, taking the off-axis direction along the Y-axis as an example, after substituting the coordinates of each point determined in steps S1 to S4 into the rotation transformation formula, the expansion form of the univariate nonlinear equation is as follows: Substituting the coordinates of points A, B, and E into the rotation transformation formula about the X-axis, we obtain the Y-coordinate expressions for points a, b, and e after rotation. , , Since rotation around the X-axis does not change the X-coordinate, the constraint of equal projected distance is maintained. Simplified to The coordinates of E Substituting the coordinates of A and B into the above equation and expanding, we obtain a univariate nonlinear equation about θ. This equation can be solved using Newton's iteration method, with the initial value taken as... The convergence criterion is that the angle difference between two adjacent iterations is less than 1. radian.

[0029] The process of this step is as follows: First, substitute the coordinates and constraints of each point determined in steps S1 to S4 into the rotation transformation formula to establish the relationship between the coordinates and constraints of each point. The equation is a univariate nonlinear equation; then, the optimal rotation angle is obtained through numerical solution methods (such as Newton's iteration method, bisection method, or other numerical optimization algorithms, which can be implemented in computing environments such as MATLAB and Python). The coordinates of the corresponding bisecting center point E are determined; finally, the shape and size of the detection area are determined based on the projected distances of each boundary point after rotation. The technical effect of this step is that: by numerically solving, a uniquely determined optimal rotation angle is obtained, so that the off-axis paraboloid after rotation has a symmetrical projection distribution on the projection plane, thereby enabling the establishment of a regular circular or elliptical detection area and achieving full-aperture, complete detection.

[0030] The step of generating the adaptive variable density detection path (step S6) includes: firstly, uniformly selecting sampling positions along the detection area according to the preset initial grid spacing, and calculating the Gaussian curvature at each sampling position after the off-axis parabolic rotation. The formula is ,in and These are the principal radii of curvature in the two mutually perpendicular principal directions at this location, for a parabola. The principal curvatures at each location can be calculated using the first and second fundamental forms at that location. Then, based on the Gaussian curvature... The distribution of curvature is used to divide the detection area into high curvature change areas and low curvature change areas, and a curvature change rate threshold is set. When the rate of change of curvature between adjacent sampling points The time is divided into a high curvature variation zone, when The time is divided into low curvature variation regions, in which The difference in Gaussian curvature between adjacent sampling points. This represents the arc length between adjacent sampling points. Finally, in regions of high curvature variation, the sampling interval is increased. Sparse sampling intervals are used in regions with low curvature variation. ,in A variable-density spiral scan path or a variable-density grid scan path is generated as the detection path. This adaptive variable-density detection path is planned based on the curvature distribution of the rotated off-axis paraboloid. Since the curvature distribution of the rotated mirror is directly related to the rotation angle θ, this path planning is technically coupled with the determination of the rotation reference in steps S1 to S5. The sampling interval is adaptively determined based on the local Gaussian curvature. In areas with high curvature variation, the sampling interval is densified. According to the formula Calculation; sparse sampling interval in the low curvature variation region. According to the formula Calculation. Due to the high curvature variation region ,therefore To achieve encrypted sampling; in the low curvature variation region near ,therefore To achieve sparse sampling, thereby ensuring .in The reference sampling interval is set according to the measurement accuracy of the detection equipment, with a typical range of 0.1 mm to 1 mm. The minimum Gaussian curvature value among all sampling locations within the detection area; This represents the local Gaussian curvature value at the current sampling location. The total number of sampling points along the detection path. Meets the accuracy requirements for surface reconstruction: ,in The area to be detected. The maximum allowable sampling interval, The oversampling coefficients are related to the order of the Zernike polynomial used for surface fitting; the higher the order of the Zernike polynomial, the better. The larger the value, the better when the Zernike polynomial has 36 terms. The typical value is between 3 and 5.

[0031] For parabolic surfaces The principal curvature at each location can be expressed by the first fundamental form coefficient at that location ( , , ) and the second fundamental form coefficient ( , , The calculation yielded the following results: , , , For parabolic surfaces , , Therefore, the Gaussian curvature at each location can be calculated analytically.

[0032] Within the elliptical detection area, the high curvature variation region is distributed in the area near the off-axis boundary points a and b, while the low curvature variation region is distributed in the central region.

[0033] The process of this step is as follows: First, initial sampling positions are uniformly selected within the detection area, and the Gaussian curvature at each position is calculated; then, based on the curvature change rate threshold... The detection area is divided into high curvature variation zones and low curvature variation zones. Finally, the sampling interval of each zone is adaptively determined based on the local curvature to generate a variable density detection path. The technical effect of this step is that, through an adaptive variable density sampling strategy, the sampling density is increased in areas with drastic curvature changes to improve surface shape detection accuracy, while the sampling density is reduced in areas with gentle curvature changes to improve detection efficiency, thus achieving a balance between detection accuracy and detection efficiency.

[0034] The rotation adjustment error compensation step (step S7) includes: rotating the off-axis parabolic surface at the optimal angle. After actual assembly and rotation, the actual coordinates of at least three boundary points after rotation are measured using the detection probe of a coordinate measuring machine. These boundary points include two boundary points in the off-axis direction and at least one boundary point in the non-off-axis direction. Two boundary points in the off-axis direction are chosen because the projected distance in the off-axis direction is most sensitive to the rotation angle and can provide the most effective rotation angle information; at least one boundary point in the non-off-axis direction is chosen to provide information on the offset of the rotation center in the non-off-axis direction. Then, based on the measured boundary point coordinates, the actual rotation angle is calculated using least-squares fitting. and actual center of rotation Construct the objective function : ,in For the first Measured coordinates of the boundary points For the first The original coordinates of the boundary points before rotation Represents Euclidean distance. For Rotate around the center The transformation matrix when rotating about the X-axis: When rotating about the Y-axis: The translation effect of the rotation center is achieved by... First translate to In a coordinate system with the origin as the origin, the longitude is... This is achieved by transforming and then translating back to the original coordinate system. The number of measured boundary points and . This is because the unknown parameters to be solved include the rotation angle. (1 degree of freedom) and center of rotation The coordinate offset (with 2 effective degrees of freedom in the plane of rotation) requires solving at least 3 independent equations. Next, the rotation angle deviation is calculated. and rotation center offset Finally, based on the rotation angle deviation... and rotation center offset Using the rotation transformation formula, Center of rotation The coordinates of each boundary point and the bisecting center point after rotation are recalculated for the rotation angle, and then the center coordinates and major and minor axis radii of the detection area are redefined to generate the compensated detection area, so that the projection range of the compensated detection area completely covers the full-aperture projection range of the off-axis parabolic surface on the XY plane after rotation.

[0035] The process is as follows: First, the actual coordinates of at least three boundary points are measured using a detection probe; then, the actual rotation parameters are calculated using least-squares fitting; next, the rotation angle deviation and rotation center offset are calculated; finally, the center and size of the detection area are recalculated using the actual rotation parameters to generate the compensated detection area. The technical advantage of this step is that, through measured back-calculation and coordinate compensation, the influence of mechanical assembly errors on detection accuracy is eliminated, ensuring the integrity and accuracy of full-caliber detection even in the event of assembly deviations.

[0036] The iterative correction step based on the detection data (step S8) includes: firstly, performing initial surface shape detection on the off-axis paraboloid according to the detection area and detection path to obtain surface shape error distribution data, which can be represented as Zernike polynomial coefficients or direct point cloud error data. Then, based on the surface shape error distribution data, identifying surface shape errors exceeding a preset error threshold. The local area is designated as the key area for correction. The typical value of λ ranges from 0.3λ to 1λ, where λ is the wavelength of the detection light source. Next, a densified detection sub-region is added within the key correction region. The shape of the densified detection sub-region is consistent with the bounding rectangle of the key correction region, and the sampling density of the densified detection sub-region is 1 / 3 of the initial detection sampling density. times, , Typical values ​​range from 2 to 5. Finally, a second precision inspection is performed on the encrypted detection sub-region, and the second inspection data is fused with the first inspection data. During fusion, a weighted average method is used in the overlapping area of ​​the second and first inspection data, and the weight of each data point is inversely proportional to its measurement uncertainty to generate a high-precision full-aperture surface shape error map.

[0037] The process of this step is as follows: First, perform initial surface shape detection to obtain surface shape error distribution data; then, based on the error threshold... The process involves identifying key areas for correction; then, adding encrypted detection sub-regions within these key areas and performing a second, more precise detection; finally, fusing the second detection data with the first detection data to generate a high-precision surface error map. This iterative correction method can be repeated until the surface error converges to the target accuracy range. The technical advantage of this step is that, through an iterative correction strategy, refined second detection is performed on areas with large surface errors based on the first detection, improving the local accuracy and overall reliability of the surface error data, and providing high-precision surface error data support for subsequent deterministic shaping processes.

[0038] This application also includes an integrated structure of an off-axis parabolic mirror and its adapter fixture, and its application in full-aperture measurement and machining. The off-axis parabolic mirror is fixed to the adapter fixture via an adhesive layer to form an integrated structure. The adhesive layer uses optical-grade epoxy resin adhesive, with a cured thickness of 0.1 mm to 0.3 mm. The upper surface of the adapter fixture is machined into a contoured support surface, which is formed by CNC milling. The shape of the contoured support surface is at the optimal rotation angle with the off-axis parabolic surface. The non-working surface of the off-axis parabolic mirror is matched with a surface shape deviation of no more than 0.05 mm. The non-working surface is the side of the off-axis parabolic mirror opposite the aspherical mirror surface. The surface shape of the contour support surface is determined according to the specific structural parameters of the off-axis parabolic mirror (including mirror thickness, material, and machining surface shape data of the non-working surface). Specifically, the actual surface shape data of the non-working surface of the off-axis parabolic mirror is first measured using a coordinate measuring machine. Then, a CNC machining program is generated based on this surface shape data to perform CNC milling on the upper surface of the fitting fixture. The contour support surface ensures that the aspherical mirror surface of the off-axis parabolic mirror is at the optimal rotation angle after gluing and fixing. The corresponding orientation is as follows: the aspherical mirror is an effective reflective working surface of an off-axis paraboloid. The integrated structure is placed on the coordinate measuring machine (CMM) table, and a probe is used to measure the surface shape of the aspherical mirror along the detection area. After measurement, it is transferred to the machining equipment in the same orientation for shaping. This same orientation is achieved by the engagement of at least two locating pins on the bottom of the fitting with corresponding locating holes on the CMM table and the machining equipment table. The diameter of the locating pins is 6mm to 10mm, with a positioning accuracy of ±0.01mm. The line connecting at least two locating pins is parallel or perpendicular to the off-axis direction to constrain the translational and rotational freedom of the integrated structure within the table surface, ensuring the consistency between the detection datum and the machining datum.

[0039] The method of this application will be verified below with specific numerical values.

[0040] The parameters of a certain off-axis parabolic mirror to be tested are as follows: off-axis amount in the Y-axis direction. mm, radius of curvature of the mother mirror mm, diameter mm (diameter radius is) (mm). The original off-axis angle can be calculated from the equation of the parabola. =18.5691°.

[0041] Based on step S1, substitute the coordinates of each feature point into the parabolic equation to calculate the geometric center point. ,in mm; boundary point in the off-axis direction ,in = h - d / 2 = 21.5 - 24 = -2.5mm, = -0.05mm; ,in mm, mm; boundary points in non-off-axis directions ,in mm .

[0042] According to step S3, the coordinates of the bisecting center point E are: In solving for the optimal rotation angle =19.7149°, then substituting it into the equation gives... = 64 tan(19.7149°) = 64 = 22.90mm, mm, that is .

[0043] If the detection is performed directly using the original pose, the center point of the mirror geometry is used. Divide the projection of the mirror edge contour onto the XY plane using the central dividing point as the dividing point, satisfying the following conditions: mm, mm. At this point, a coordinate measuring machine is used to establish a circular detection range with a radius of 24 mm centered at point O, which can complete the full diameter measurement of the off-axis parabolic surface. However, the tilt angle of the aspherical surface is too large at this point, making it difficult to perform fine polishing with a traditional robotic arm in this posture.

[0044] If measured using traditional methods, the mirror body is rotated around the X-axis with the center of curvature as the center of rotation, by an off-axis angle. =18.5691°, the coordinates of each point after rotation are , , , , ,at this time mm, mm, the difference in projection distance is mm, not satisfied It is impossible to use a coordinate measuring machine to establish a circular or elliptical detection area for full-diameter measurement.

[0045] Using the method proposed in this application, the mirror body should be located at the geometric center point of the mirror surface. Using the X-axis as the center of rotation, find the optimal rotation angle. = 19.7149°. The coordinates of the points after rotation are a(0.00, -2.29, -8.35), b(0.00, 48.33, -7.34), c(24.00, 23.02, -7.85), d(-24.00, 23.02, -7.85), o(0.00, 21.50, -3.61), e(0.00, 23.02, -3.60). Taking the center point e as the dividing point, the condition is satisfied. mm, mm. At this point, with the projection point as the center of the ellipse, and... mm is the radius of the major axis and mm is used to construct an elliptical detection region with the minor axis radius. Meanwhile... and The difference is only 1.1458°, indicating a very small off-axis parabolic inclination. The difference in projected distance is... mm, achieving strict symmetry and 100% detection coverage.

[0046] The above values ​​will be verified below. The coordinates of point A will be... Substituting into the rotation transformation formula about the X-axis (with...) Center of rotation = 19.7149°): ; =21.5 + (-2.5 - 21.5)\cos(19.7149°) - (-0.05 - (-3.61))\sin(19.7149°), , mm; =-3.61 + (-2.5 - 21.5)\sin(19.7149°) + (-0.05 - (-3.61))\cos(19.7149°), , mm; Right now , and the aforementioned The slight difference comes from rounding in the intermediate calculations, and the numerical verification is successful.

[0047] Similarly, the coordinates of point E Substitute into the rotation transformation formula: ; = 21.5 + (22.90 - 21.5)\cos(19.7149°) - (-4.10 - (-3.61))\sin(19.7149°), , mm; Right now , and the aforementioned They are basically the same, with minor differences due to rounding.

[0048] Verify projection distance: mm. After similar calculation of point B, we get... mm, satisfying The slight difference from the aforementioned 25.31 mm comes from intermediate calculation precision.

[0049] Furthermore, based on step S6, adaptive detection path planning is performed on this embodiment. The Gaussian curvature distribution at each position after the off-axis parabolic surface rotation is calculated. It is found that the curvature changes significantly near the off-axis boundary points a and b, while the curvature changes less in the central region and near the non-off-axis boundary. A baseline sampling interval is set. mm, threshold of rate of change of curvature Based on the actual curvature distribution, a denser sampling interval is used in the high curvature variation region. mm, using sparse sampling interval in the low curvature variation region. mm, generating a variable density spiral scan path.

[0050] Furthermore, based on step S7, rotational adjustment error compensation is performed on this embodiment. After actual assembly and adjustment, the coordinates of three boundary points (including two boundary points a and b in the off-axis direction and one boundary point c in the non-off-axis direction) are measured and then calculated in reverse. =19.73°, rotation angle deviation = 0.015°, rotation center offset mm. Based on the deviation, the coordinates of each point are recalculated using the rotation transformation formula, and the detection area is compensated and corrected. The center of the compensated detection area is shifted to [location missing]. The major and minor axis radii were adjusted to 25.28 mm and 23.98 mm respectively, still meeting the full-caliber coverage requirements.

[0051] Furthermore, the embodiment is iteratively corrected based on step S8. After the initial detection, a surface shape error exceeding [a certain value] was found near the off-axis boundary. ( A region of nm was designated as the key correction region, and an encrypted detection sub-region was added within it. The sub-region was shaped as the bounding rectangle of the key correction region, and the sampling density was increased by 3 times. After a second precision inspection, a weighted average method is used to fuse the data in the overlapping areas, resulting in a high-precision surface shape error map with a PV value that increases from the initial inspection value. Reduce to RMS value from Reduce to .

[0052] Using a coordinate measuring machine to measure points With the ellipse center, the probe detection stroke elliptical trajectory region was established with a major semi-axis of 23.81 mm and a minor semi-axis of 22.5 mm (considering a probe radius compensation of 1.5 mm and a safety margin of 0.5 mm, the actual detection stroke radius was reduced by about 2 mm from the theoretical value). The measured data showed that its maximum error value deviated from the theoretical result by only 0.13 mm, which met the theoretical and processing requirements.

[0053] The off-axis parabolic mirror and its adapter are fixed together using an adhesive layer to form an integrated structure. The adhesive layer uses optical-grade epoxy resin adhesive, with a cured thickness of approximately 0.2 mm. The upper surface of the adapter has a contoured support surface machined using CNC milling. The shape of the contoured support surface matches the shape of the non-working surface at the bottom of the off-axis parabolic mirror, with a shape deviation not exceeding 0.05 mm. By using this integrated structure for measuring and machining the mirror surface, full-aperture measurement and machining of the off-axis parabolic mirror can be achieved. During inspection, the integrated structure is placed on the worktable of a coordinate measuring machine, and the inspection probe scans the aspherical mirror surface along the elliptical inspection area using an adaptive variable density inspection path. After measurement, the integrated structure is transferred to precision machining equipment (such as computer-controlled optical surface forming (CCOS) equipment) in the same posture for shaping. The same posture is achieved by the cooperation of two positioning pins (8mm in diameter, positioning accuracy ±0.01mm) set at the bottom of the fitting fixture with the corresponding positioning holes on the worktable of the coordinate measuring machine and the worktable of the machining equipment, so as to ensure the consistency between the detection datum and the machining datum.

[0054] The universality of the method in this application will be verified below using the second set of parameters.

[0055] The parameters of a certain off-axis parabolic mirror to be tested are as follows: off-axis amount in the X-axis direction. mm, radius of curvature of the mother mirror mm, diameter mm (diameter radius is) (mm). The original off-axis angle can be calculated from the equation of the parabola. = = arctan(30 / 100) = 16.6992°.

[0056] According to step S1, when the off-axis direction is along the X-axis, substitute the coordinates of each feature point into the parabolic equation to calculate the geometric center point. ,in mm; boundary point in the off-axis direction ,in mm, mm; ,in mm, mm; boundary points in non-off-axis directions ,in: mm; , mm.

[0057] According to step S3, when the off-axis direction is along the X-axis... Solve for the optimal rotation angle. = 17.4523°, then substituting it into the equation gives... = 100 \times \tan(17.4523°) = 100 \times 0.3143 =31.43mm, mm, that is .

[0058] After rotation, all points satisfy mm, mm. An elliptical detection area is constructed with the projection point as the center of the ellipse, and the major axis radius is 21.52 mm and the minor axis radius is 20 mm. and The difference of only 0.7531° verifies that the method of this application is applicable to off-axis parabolic surfaces with different parameter combinations.

[0059] The overall working process of the method described in this application is described below.

[0060] The overall working process of the method described in this application is as follows: Phase 1: Rotational reference calculation. The operator inputs the parameters (off-axis amount) of the off-axis parabola into the calculation equipment. Diameter Mother mirror curvature radius The calculation program automatically executes steps S1 to S5 to calculate the coordinates of each feature point, establish constraints, and solve for the optimal rotation angle. The coordinates of the center point E of the peace section are determined, and the shape and size (circular or elliptical) of the detection area are determined. The output of this stage is the optimal rotation angle. The coordinates of the bisecting center point E and the parameters of the detection area.

[0061] Phase Two: Tooling Preparation and Adjustment. Based on the optimal rotation angle calculated in Phase One. The machining process involves creating a contoured support surface that fits the tooling, ensuring that its surface shape is at the optimal rotation angle to the off-axis parabola. Matching the non-working surface shape. The off-axis parabolic mirror is fixedly connected to the adapter fixture via an adhesive layer to form an integrated structure. The integrated structure is then placed on the coordinate measuring machine's worktable.

[0062] Phase 3: Assembly and Adjustment Error Compensation (Optional). Using a detection probe, measure the actual coordinates of at least three boundary points, then execute step S7 to compensate for rotational assembly and adjustment errors, generating the compensated detection area. If the assembly and adjustment accuracy meets the requirements (i.e....), and (If all are within the allowable range), this step can be skipped and the theoretical detection area can be used directly.

[0063] Phase 4: Detection Path Planning (Optional). Execute step S6 to generate an adaptive variable density detection path based on the detection area (or the compensated detection area) and the local curvature distribution of the off-axis parabola. If adaptive path planning is not required, a uniformly spaced spiral scanning path or a grid scanning path can be used.

[0064] Phase 5: Surface Shape Detection. The coordinate measuring machine controls the detection probe to perform point scanning on the aspherical mirror surface according to the detection area and detection path to acquire surface shape data.

[0065] Phase 6: Iterative Correction (Optional). Execute step S8 to identify key correction areas based on the initial detection results, add encrypted detection sub-regions for secondary precision detection, and fuse the data to generate a high-precision full-aperture surface shape error map.

[0066] Phase 7: Shaping and Processing. The integrated structure is transferred to the processing equipment in the same orientation, and deterministic shaping and processing are performed based on the high-precision surface shape error map. After processing, the integrated structure is placed back into the coordinate measuring machine for surface shape re-measurement, and phases 5 to 7 are repeated until the surface shape accuracy meets the requirements.

[0067] Through the collaborative work of the above seven stages, this application has achieved a complete closed-loop process from rotational datum calculation, tooling preparation, assembly and adjustment compensation, path planning, surface shape detection, iterative correction to shaping and machining, ensuring high precision and high efficiency in the full-diameter measurement and machining of off-axis parabolic surfaces.

[0068] The implementation principle of this embodiment is as follows: By accurately calculating the rotation reference of the off-axis parabolic surface, the projection of the rotated off-axis parabolic surface onto the projection plane is symmetrical, thereby establishing a regular detection area for full-aperture measurement. Simultaneously, the adaptive variable density detection path can rationally arrange the sampling point density according to the curvature changes of the off-axis parabolic surface, improving detection accuracy and efficiency. The rotation and adjustment error compensation step eliminates errors during the rotation and adjustment process, ensuring detection accuracy. The iterative correction step based on detection data allows for refined secondary detection of areas with large surface shape errors, improving the local accuracy and overall reliability of the surface shape error data. The integrated structure of the off-axis parabolic mirror and the adapter fixture ensures the consistency between the detection reference and the processing reference through the conformal support surface and the cooperation of the positioning pin and positioning hole. This method avoids problems such as probe misalignment, measurement errors, or incomplete surface shape detection caused by projection asymmetry in traditional methods, and is suitable for full-aperture measurement of off-axis parabolic surfaces with different parameters.

[0069] The specific embodiments described above do not constitute a limitation on the scope of protection of this application. Any other corresponding changes and modifications made based on the technical concept of this application should be included within the scope of protection of this application.

Claims

1. A method for determining a rotating reference for measuring the full diameter of an off-axis parabolic surface, characterized in that, Includes the following steps: S1. Based on the off-axis amount h, aperture diameter d, and mother mirror curvature radius R of the off-axis parabola, calculate the coordinates of two boundary points A and B in the off-axis direction, the coordinates of two boundary points C and D in the non-off-axis direction, and the coordinates of the geometric center point O of the off-axis parabola based on the parabolic equation; define the bisecting center point E on the off-axis parabola, wherein the bisecting center point E is a feature point to be determined on the off-axis parabola, and its coordinates are determined as a function of the rotation angle θ by the parallel constraint condition of the tangent plane in step S3; S2. Determine the rotation axis and rotation center according to the off-axis direction. With the geometric center point as the rotation center, rotate the off-axis parabola around the rotation axis by an angle θ. After rotation, the boundary points A, B, C, D, the bisecting center point E, and the geometric center point O are transformed into the corresponding rotated points a, b, c, d, e, and o, respectively. The reference projection plane is the XY plane in the coordinate system before rotation. S3. Set a parallel constraint condition for the tangent plane so that the tangent plane at the bisecting center point e after rotation is parallel to the XY plane. Determine the functional relationship between the coordinates of the bisecting center point E before rotation and the rotation angle θ according to the parallel constraint condition for the tangent plane. After rotation, the bisecting center point e becomes the equidistant center point in the projection region of the off-axis parabola on the XY plane after rotation, which is equidistant from the projections of the boundary points on both sides of the off-axis direction. S4. Set equal projection distance constraints so that the projections of the boundary points a, b, c, and d of the rotated off-axis parabola onto the XY plane are equal on both sides in the off-axis direction and equal on both sides in the non-off-axis direction. S5. Substitute the coordinates of each point into the rotation transformation formula and the equal projection distance constraint to establish a univariate nonlinear equation about the rotation angle θ and solve it numerically to obtain the optimal rotation angle and the coordinates of the corresponding bisecting center point E. Establish a detection area with the projection point of the bisecting center point e on the XY plane after rotation as the center, and perform full-diameter measurement on the off-axis parabola.

2. The method according to claim 1, characterized in that, In step S1, the equation of the parabola is expressed in a coordinate system where the Z-axis is vertically downward and positive as follows: ， When the off-axis direction is along the Y-axis, the coordinates of the geometric center point are: ,in The boundary points in the off-axis direction are respectively and ,in , The boundary points in the non-off-axis direction are respectively and ,in ; When the off-axis direction is along the X-axis, the coordinates of the geometric center point are: ,in The boundary points in the non-off-axis direction are respectively and ,in , The boundary points in the off-axis direction are respectively and ,in , .

3. The method according to claim 2, characterized in that, In step S2, when the off-axis direction is along the Y-axis, the rotation axis is the X-axis; when the off-axis direction is along the X-axis, the rotation axis is the Y-axis. In step S3, when the off-axis direction is along the Y-axis, the coordinates of the center point E before rotation satisfy... When the off-axis direction is along the X-axis, the coordinates of the center point E before rotation satisfy... The E-coordinate is derived from the partial derivative condition of the parabolic equation: for a parabolic surface... The absolute value of the partial derivative at point E along the off-axis direction is equal to This ensures that the normal vector at that point is parallel to the Z-axis direction after rotation. In step S4, when the off-axis direction is along the Y-axis, an equal off-axis projection distance constraint is applied. ,in This represents the Euclidean distance between the projections of points a and e onto the XY plane after rotation. This represents the Euclidean distance between the projections of points b and e onto the XY plane after rotation; condition for non-off-axis directions. Because rotation around the X-axis does not change the X-coordinate, this constraint is automatically satisfied; when the off-axis direction is along the X-axis, an equal off-axis projection distance constraint is applied. Conditions for non-off-axis directions Automatic fulfillment.

4. The method according to claim 1, characterized in that, In step S5, the rotation transformation formula is based on a right-hand coordinate system, and the rotation direction follows the right-hand rule, specifically: With geometric center point any point is the center of rotation. After rotating θ around the X-axis: ； With geometric center point any point is the center of rotation. After rotating θ around the Y-axis: ； When establishing the detection area, the projected distance in the off-axis direction after rotation Equal to the projected distance in the non-off-axis direction At that time, with the projection point as the center, and with Establish a circular detection area with a radius; when At that time, with the projection point as the center of the ellipse, and with and An elliptical detection region is established by using the larger value of the major axis radius and the smaller value of the minor axis radius, where the major axis direction corresponds to the direction with the larger projection distance.

5. The method according to claim 1, characterized in that, The method further includes step S6: generating an adaptive variable density detection path based on the shape of the detection area and the local curvature distribution after the off-axis parabolic rotation, specifically including: S61. Select sampling positions uniformly along the detection area according to the preset initial grid spacing, and calculate the Gaussian curvature at each sampling position after the off-axis parabolic surface is rotated. : ； in and These are the principal radii of curvature in the two mutually perpendicular principal directions at that location; for a parabola The principal curvature at each location can be calculated using the first and second fundamental forms at that location; S62, Based on Gaussian curvature The distribution of curvature is used to divide the detection area into high curvature change areas and low curvature change areas, and a curvature change rate threshold is set. When the rate of change of curvature between adjacent sampling points The time is divided into high curvature variation regions, when The time is divided into low curvature variation regions, in which The difference in Gaussian curvature between adjacent sampling points. The arc length between adjacent sampling points; S63. Use denser sampling intervals in areas of high curvature variation. Sparse sampling intervals are used in regions with low curvature variation. ,in , generate a variable density spiral scan path or a variable density raster scan path as the detection path.

6. The method according to claim 5, characterized in that, The sampling interval is adaptively determined based on the local Gaussian curvature, specifically as follows: In regions of high curvature variation, the sampling interval is increased. Calculate using the following formula: ； In the region of low curvature variation, sparse sampling interval Calculate using the following formula: ， in The baseline sampling interval is set according to the measurement accuracy of the detection equipment. The minimum Gaussian curvature value among all sampling locations within the detection area; This represents the local Gaussian curvature value at the current sampling location; The total number of sampling points in the detection path Meets the accuracy requirements for surface reconstruction: ， in The area to be detected. The maximum allowable sampling interval, The oversampling coefficients are related to the order of the Zernike polynomial used for surface fitting; the higher the order of the Zernike polynomial, the better. The larger the value, the better.

7. The method according to claim 1, characterized in that, The method further includes a rotation adjustment error compensation step S7: S71. After the off-axis parabolic surface is actually rotated and adjusted according to the optimal rotation angle θ, the actual coordinates of at least three boundary points after rotation are measured using the detection probe of a coordinate measuring machine. The at least three boundary points include two boundary points in the off-axis direction and at least one boundary point in the non-off-axis direction. S72. Based on the measured boundary point coordinates, calculate the actual rotation angle using least-squares fitting. and actual center of rotation ; S73, Calculate rotation angle deviation and rotation center offset ; S74, Based on the aforementioned rotation angle deviation and rotation center offset Using the rotation transformation formula in step S5, to Center of rotation The coordinates of each boundary point and the bisecting center point after rotation are recalculated for the rotation angle, and then the center coordinates and major and minor axis radii of the detection area are redefined to generate the compensated detection area, so that the projection range of the compensated detection area completely covers the full-aperture projection range of the off-axis parabolic surface on the XY plane after rotation.

8. The method according to claim 7, characterized in that, In step S72, the least squares fitting inverse calculation includes: Construct the objective function : ， in For the first Measured coordinates of the boundary points For the first The original coordinates of the boundary points before rotation Represents Euclidean distance. For Rotate around the center The transformation matrix, when rotated about the X-axis, the transformation matrix for: ； When rotating about the Y-axis, the transformation matrix for: ； The translation effect of the rotation center is achieved by... First translate to In a coordinate system with the origin as the origin, the longitude is... This is achieved by transforming and then translating back to the original coordinate system. The number of measured boundary points and ; By minimizing the objective function Solve and .

9. The method according to claim 1, characterized in that, The method further includes an iterative correction step S8 based on the detection data: S81. Perform initial surface shape detection on the off-axis parabolic surface according to the detection area and detection path to obtain surface shape error distribution data; S82. Based on the surface shape error distribution data, identify whether the surface shape error exceeds a preset error threshold. The local area is designated as the key area for correction; S83. An encrypted detection sub-region is added within the key correction area. The shape of the encrypted detection sub-region is consistent with the bounding rectangle of the key correction area, and the sampling density of the encrypted detection sub-region is the same as the initial detection sampling density. times, ; S84. Perform secondary precision detection on the encrypted detection sub-region, and fuse the secondary detection data with the first detection data. During the fusion, a weighted average method is used in the overlapping area of ​​the secondary detection data and the first detection data. The weight of each data point is inversely proportional to its measurement uncertainty, and a high-precision full-aperture surface shape error map is generated.

10. The method according to claim 1, characterized in that, In step S5, when solving for the optimal rotation angle θ, in addition to satisfying the constraint of equal projection distance, the condition of minimizing tilt angle is also satisfied. Specifically, the original off-axis angle is defined. Where h is the off-axis amount and R is the radius of curvature of the parent mirror; construct the optimization model: ， The constraints are: ， ， ， in To determine the maximum allowable rotation angle deviation; under the premise of satisfying the constraint of equal projection distance, make the rotation angle θ equal to the original off-axis angle. Minimize the deviation, thereby minimizing the tilt angle of the rotated off-axis parabola relative to the XY plane.

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