A method for evaluating a conical surface topography error

Abstract

The application discloses a kind of conical surface topography error evaluation methods, comprising the following steps: using coordinate measuring system obtains all measurement point coordinate data and unit normal of conical surface;According to the average curvature of measurement point cloud that measurement point coordinate and its unit normal solve;Optimizing mixed distance weight factor, feature matching is carried out to measurement point cloud and initial conical surface, determine feature matching after measurement point cloud;The bidirectional matching algorithm based on measurement point cloud and reconstructed conical surface is used to carry out precision matching to feature matching after measured point cloud, and the distribution of measurement point cloud to conical surface distance, root mean square value, peak and valley value are output.The application solves the interference problem of initial value or initial state of traditional least square fitting algorithm and point cloud matching method to result, effectively avoids the influence of conical surface axial and vertex coordinate and other geometric parameter initial value to conical surface fitting final result, improves the evaluation precision of conical surface topography error, and calculation efficiency is high.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of computer-aided design and manufacturing, and particularly relates to a conical surface topography error evaluation method. BACKGROUND

[0002] With the development of modern science and technology, the fields of space exploration, aerospace, ground remote sensing, chip manufacturing and the like have higher requirements for light weight, miniaturization and ultra-precision of optoelectromechanical systems. The quadric surface element is widely used due to its characteristics of reducing the residual error and the number of elements of the optoelectromechanical system. With the increasing demand for high-quality and ultra-precision elements, quadric surface topography error evaluation is crucial for the high-precision manufacturing and development of optoelectromechanical system elements. As one of the quadric surfaces, the conical surface is widely used in optoelectromechanical systems. The critical quality evaluation of the conical surface is no longer satisfied with the conicity error of the past, but has higher standards, i.e. the conical surface topography error.

[0003] The coordinate measuring system can keep the probe continuously dotting and scanning on the curved surface during the measurement process, and can obtain the coordinates and normal information of the whole conical surface. Then, the measured point cloud is registered with the point cloud of the conical surface model, so as to evaluate the topography error. However, the existing conical surface topography error evaluation method has the problems of low evaluation accuracy, low calculation efficiency and poor measurement repeatability. The patent "Elliptical conical surface and ellipsoidal surface parameter extraction method based on Levenberg-Marquardt method" translates the measured point cloud, so that the vertex of the measured point cloud coincides with the coordinate origin, and then solves the included angle according to the rotation transformation matrix to obtain the elliptical conical surface equation eliminating the offset. Then, the Levenberg-Marquardt method is used to fit the measured point cloud through the surface model equation. However, it is difficult to accurately obtain the vertex coordinates during measurement, and a large error will be caused when the offset is eliminated to determine the surface equation, which reduces the accuracy of the topography error evaluation. The patent "Conical surface fitting method and device, computer equipment and storage medium" estimates the mean value of the conical surface axial, conical surface vertex angle, vertex coordinates and other geometric parameters according to the conical surface measured point cloud. The calculation process lacks optimization of the topography residual error. The extraction of the conical surface topography error is affected by the initial state of the measured point cloud and the initial value setting, and the conical surface topography error is large.

[0004] In general, the existing conical surface error evaluation method has two problems: on the one hand, the method lacks feature matching between the measurement point cloud and the conical surface, and when the measurement point cloud deviates from the conical surface model greatly, the topography evaluation result will be affected; on the other hand, when the target function needs to be optimized, numerical iteration is needed for solving, and the selection of the initial value of the conical surface vertex angle, the vertex coordinate and the conical surface axial direction will greatly affect the accuracy and robustness of the calculation result. As for the conventional feature matching method between the measurement point cloud and the model, the selection of the weight factor of the target function depends on the human setting, and the algorithm has strong dependence on experience, low universality, and it is difficult to provide a good initial state for the precise matching between the measurement point cloud and the conical surface model. SUMMARY

[0005] The present application is to solve the above problems, and aims to provide a conical surface topography error evaluation method, so as to solve the problem of interference of the initial value or initial state of the traditional least square fitting algorithm and the point cloud matching method on the result, effectively avoid the influence of the initial value of the conical surface axial direction and the vertex coordinate on the final result of the conical surface fitting, and improve the evaluation accuracy of the conical surface topography error.

[0006] The present application is realized by the following technical scheme:

[0007] A conical surface topography error evaluation method, comprising the following steps:

[0008] Step 1, obtaining all measurement point coordinate data of the conical surface by using a coordinate measurement system and a unit normal ;

[0009] Step 2, solving the average curvature of the measurement point cloud according to the measurement point coordinates and the unit normal;

[0010] Step 3, optimizing the mixed distance weight factor, performing feature matching between the measurement point cloud and the initial conical surface , and determining the feature-matched measurement point cloud; wherein is the vertex coordinate , the conical surface axial direction , and the initial vertex angle ;

[0011] Step 4, performing precise matching on the feature-matched measurement point cloud by using a bidirectional matching algorithm based on the measurement point cloud and the reconstructed conical surface , and outputting the distribution, root mean square value and peak-to-valley value of the distance from the measurement point cloud to the conical surface.

[0012] Further optimization, the step 2 comprises the following specific steps:

[0013] Step 2-1, establishing a data point Local coordinate system as the origin of the coordinate system Unit normal at data point coordinate axes coordinate axes Choose any option provided it conforms to the right-hand orthogonal rule;

[0014] Step 2-2, perform local coordinate transformation, and... The neighborhood data points are transformed from the original coordinate system to the local coordinate system. After the coordinate transformation... The neighborhood data points are fitted with a local quadratic surface using the least squares method. The equation of the fitted quadratic surface is as follows: ;

[0015] Steps 2-3, calculate points mean curvature at The average curvature set of the measured point cloud is obtained. .

[0016] Further optimization, step 3 includes the following specific steps:

[0017] Step 3-1, Define measurement points and target point The mixing distance is ,in , ;against Algorithm Calculate distance weight factors The measurement point cloud position below , To the cone surface distance set ;

[0018] Step 3-2: Select error peak and valley values The minimum corresponding measurement point cloud is used as the initial measurement point cloud for subsequent fine matching. .

[0019] Further optimization, in step 3-1, the algorithm... The specific steps include the following:

[0020] Step 3-1a, for each Algorithm Calculate the cone surface Above and The point with the smallest mixing distance and mean curvature ;

[0021] Step 3-1b, Solve for the measured points After rigid body transformation and corresponding target point The optimization problem of minimum mixed distance is solved as follows:

[0022] ;

[0023] wherein is the total number of measured point cloud points, and are a rotation matrix and a translation vector, respectively; let

[0024] ;

[0025] SVD decomposition is performed on , wherein is a diagonal matrix with all singular values being non-negative, and the solution is , :

[0026] ;

[0027] ;

[0028] ;

[0029] Step 3-1c, check whether the iterative convergence condition is met; if the absolute value of the difference between the root mean square value of the mixed distance of the current measured point position to the target point cloud and the calculated value of the last iteration is less than a given threshold value, terminate the iteration and output the position of the measured point cloud after feature matching ; otherwise, update the position of the measured point cloud and continue to perform step 3-1a;

[0030] Step 3-1d, calculate the projection and distance of the measured point cloud to the conical surface with an apex angle of .

[0031] Further optimization, in step 3-1d, the algorithm is used to calculate the projection and distance of the measured point cloud to the conical surface with an apex angle of .

[0032] Further optimization, in step 3-1a, the algorithm is used to calculate the point with the minimum mixed distance on the conical surface and the average curvature . The specific steps include:

[0033] (a) Let , , and calculate using the following formula: ​

[0034] ;

[0035] (b) Set , , where , , ; according to the range of , calculate the corresponding conical surface point and the average curvature .

[0036] Further optimization, in step (b), the calculation according to the range of , calculate the corresponding conical surface point and the average curvature includes the following specific steps:

[0037] (b.1) If , , according to the conical surface equation directly calculate ;

[0038] (b.2) If , calculate all positive roots of the cubic equation , calculate the corresponding conical surface point and the mixed distance, take the corresponding point of the minimum mixed distance as , record the corresponding ;

[0039] (b.3) If , calculate all positive roots of the cubic equation , and calculate , calculate the corresponding conical surface point and the mixed distance of and , take the corresponding point of the minimum mixed distance as , record the corresponding ;

[0040] (b.4) Calculate its average curvature .

[0041] Further optimization, in step 4, based on the bidirectional matching algorithm of the measured point cloud and the reconstructed conical surface includes the following specific steps:

[0042] Step 4-1, calculate the best approximation conical surface of the measured point cloud in the current state, the conical surface vertex coordinates , conical surface axis​ , the vertex angle According to The least square fitting method of the distance to the cone surface is obtained:

[0043] ;

[0044] Wherein, ;

[0045] Step 4-2, using Calculate the point cloud The distance to the cone surface The projection point set , using the SVD method to solve the measured point cloud After rigid body change and the corresponding target point cloud The distance minimum optimization problem,

[0046] ;

[0047] Update the measured point cloud position Continue to find The projection point to the cone surface, until the convergence condition of the problem is met, at this time record the measured point cloud position , the distance set of the point cloud To the cone surface , calculate the root mean square value of , the peak and valley value. Further optimization, in step 4-2, when calculating the root mean square value of , the peak and valley value, the following steps are further included:

[0048] If the ratio of the root mean square value of the distance from the current measured point position to the target cone surface to the calculated value of the last iteration is less than a given threshold value, the iteration is terminated, and the position of the fine matched measured point is output; Otherwise, return to step 4-1.

[0049] Further optimization, the algorithm Includes the following specific steps:

[0050] (a) Set the cone surface axial , then the distance

[0051] Of point To the cone surface :

[0052] ;

[0053] (b) Set , and solve the projection point By solving the following equations:

[0054] ​​ ;

[0055] Available:

[0056]

[0057]

[0058] (c) traversing each point in the set, the projection of the measured point cloud on the cone is recorded as , and the distance set of the measured point cloud to the cone is recorded as .

[0059] Compared with the prior art, the present application has the following advantages and beneficial effects:

[0060] 1. The present application provides a topography error evaluation method based on coarse-fine combination of measured point cloud and cone model bidirectional matching, which can solve the problem of initial value or initial state interference of traditional least square fitting algorithm and point cloud matching method, effectively avoid the influence of initial value of cone axial and vertex coordinates and other geometric parameters on the final result of cone fitting, and improve the evaluation precision of cone topography error.

[0061] 2. In the feature matching stage, the hybrid distance of the measured point cloud to the cone point cloud model is taken as the objective function, and the weight factor in the objective function is screened, which improves the initial state of the measured point cloud before fine matching; in the fine matching stage, the projection distance of the measured point cloud to the cone is calculated through bidirectional matching between the measured point cloud and the cone. In the calculation process, the ideal model does not need to be discretized into point cloud, the feature distance of the measured point cloud to the initial cone and the corresponding points in the feature matching, the distance of the point to the cone in the bidirectional matching, and the vertex angle of the cone are all analytical solutions, which have high calculation efficiency and precision and low memory consumption. BRIEF DESCRIPTION OF DRAWINGS

[0062] In order to more clearly illustrate the technical scheme in the exemplary embodiments of the present application, the following will briefly introduce the drawings needed to be used in the embodiments, and it should be understood that the following drawings only show some embodiments of the present application, and therefore should not be regarded as a limitation on the scope, and for those skilled in the art, other related drawings can also be obtained without paying creative labor. In the drawings:

[0063] Figure 1 It is a total flow chart of the cone topography error evaluation method based on bidirectional matching;

[0064] Figure 2 It is a feature matching algorithm flow chart of the measured point cloud and the initial cone based on hybrid distance;

[0065] Figure 3 Flow chart of bidirectional matching algorithm based on measured point cloud and cone model;

[0066] Figure 4 Conical surface topography error result graph after feature matching based on hybrid distance;

[0067] Figure 5 Conical surface topography error result graph after bidirectional matching based on point-to-cone projection distance. DETAILED DESCRIPTION

[0068] In order to make the objects, technical solutions and advantages of the present application clearer, further detailed description will be given to the present application in combination with embodiments and drawings, and the illustrative embodiments of the present application and their descriptions are only used to explain the present application, and do not limit the present application.

[0069] Embodiment 1

[0070] A conical surface topography error evaluation method, Figures 1 to 4 The method comprises the following steps:

[0071] Step 1, obtaining all measurement point coordinate data of the conical surface by using a coordinate measuring system and a unit normal .

[0072] Step 2, solving the average curvature of the measured point cloud according to the measurement point coordinates and the unit normal, and the specific steps are as follows:

[0073] Step 2-1, establishing a local coordinate system with the data point as the origin of the coordinate system , the unit normal at the data point as the coordinate axis , and the coordinate axis arbitrarily selected under the premise of conforming to the right-hand orthogonal rule;

[0074] Step 2-2, performing local coordinate transformation to convert the neighborhood data points of from the original coordinate system to the local coordinate system, and performing local quadratic surface fitting on the neighborhood data points of after coordinate conversion by using the least square method, and obtaining the quadratic surface equation ;

[0075] Step 2-3, calculating the average curvature at the point , and obtaining the average curvature set of the measured point cloud.

[0076] Step 3, optimizing the hybrid distance weight factor, and matching the measured point cloud with the initial conical surface performing feature matching to determine the feature-matched point cloud; wherein for the vertex coordinates , the axial direction of the conical surface , the initial vertex angle of the conical surface; the specific steps are as follows:

[0077] Step 3-1, defining the mixed distance of the measurement point and the target point as , wherein , ; for , the algorithm is used to calculate the distance set of the measurement point cloud position , to the conical surface under the distance weight factor ;

[0078] Step 3-2, selecting the measurement point cloud corresponding to the minimum error peak value as the initial measurement point cloud for subsequent precise matching .

[0079] Further optimization, in the step 3-1, the algorithm includes the following specific steps:

[0080] Step 3-1a, for each , the algorithm is used to calculate the point on the conical surface with the minimum mixed distance and the average curvature ; including the following specific steps:

[0081] (a) let , , the is calculated as follows:

[0082] ;

[0083] (b) let , , , wherein , , ;

[0084] According to the range of , the conical surface point and the average curvature corresponding to are calculated, and the specific steps are as follows:​​

[0085] (b.1) If , , calculate the corresponding conic points and hybrid distance, take the corresponding point of the minimum hybrid distance as ;

[0086] (b.2) If , calculate all positive roots of the cubic equation , calculate the corresponding conic points and hybrid distance, take the corresponding point of the minimum hybrid distance as , record the corresponding ; (b.3) If , calculate all positive roots of the cubic equation

[0087] , and calculate , calculate the corresponding conic points and hybrid distance, take the corresponding point of the minimum hybrid distance as , record the corresponding ; (b.4) Calculate its average curvature . Step 3-1b, solve the measured point After the rigid body changes, the hybrid distance of the corresponding target point

[0088] The minimum optimization problem is solved as follows:

[0089] ; Where is the total number of measured point cloud points,

[0090] and are the rotation matrix and translation vector respectively; let

[0091] ; ;

[0092] ;

[0093] SVD decomposition is performed on , where is a diagonal matrix with all singular values being non-negative, and the solution is , :

[0094] ;

[0095] ;

[0096] ; ​​​​

[0097] Step 3-1c, check if the iteration converges; if the absolute value of the difference between the root mean square value of the current measured point cloud position to the target point cloud hybrid distance and the calculated value of the last iteration is less than a given threshold, terminate the iteration and output the position of the measured point cloud after feature matching ; otherwise, update the position of the measured point cloud , continue to execute step 3-1a;

[0098] Step 3-1d, use the algorithm to calculate the measured point cloud to the apex angle conical surface projection and distance.

[0099] Step 4, use the bidirectional matching algorithm based on the measured point cloud and the reconstructed conical surface to perform fine matching on the measured point cloud after feature matching , output the distribution, root mean square value and peak-to-valley value of the measured point cloud to the conical surface distance, and the flowchart is shown in Figure 3 . Among them, the bidirectional matching algorithm based on the measured point cloud and the reconstructed conical surface includes the following specific steps:

[0100] Step 4-1, calculate the best approximation conical surface of the measured point cloud under the current state, the conical surface apex coordinates , the conical surface axis , and the apex angle are obtained according to the least square fitting method of the conical surface distance:

[0101] ;

[0102] wherein ;

[0103] Step 4-2, use to calculate the point cloud to the conical surface projection point set , and use the SVD method to solve the optimization problem of the measured point cloud rigid body change and the corresponding target point cloud distance minimum,

[0104] ;

[0105] update the measured point cloud position , continue to find the projection point of to the conical surface , until the convergence condition of the problem is met, at this time record the measured point cloud position , point cloud​ To the cone surface distance set ,calculate The root mean square value and peak-to-valley value;

[0106] If the root mean square value of the distance from the current measurement point to the target cone is less than the calculated value in the previous iteration, the iteration is terminated and the position of the measurement point after fine matching is output; otherwise, return to step 4-1.

[0107] In the above scheme, the algorithm The specific steps include the following:

[0108] (a) Assume the axial direction of the conical surface Then point To the cone surface distance :

[0109] ;

[0110] (b) Let Solve the following equations to obtain the projection point. :

[0111] ;

[0112] We can obtain:

[0113]

[0114]

[0115] (c) Traversal At each point in the cone, the measured point cloud is obtained. The projection is denoted as Measure point cloud to cone surface distance set .

[0116] The principle of this invention: In order to more accurately and comprehensively evaluate the topographic error of a conical surface, this technical solution uses a coordinate measurement system to acquire the measurement point cloud data of the conical surface and proposes a topographic error evaluation method based on coarse and fine matching of the measurement point cloud and the conical surface model. Compared with traditional methods, the rating method designed in this invention is divided into feature matching and fine matching. Feature matching uses the mixed distance from the measurement point cloud to the conical surface point cloud model as the objective function for matching, and selects different weights to optimize the feature matching results, thereby improving the initial state of the measurement point cloud before fine matching. Fine matching uses the projection distance from the measurement point cloud to the conical surface model as the objective function for matching. The final topographic error is calculated through the bidirectional matching between the measurement point cloud and the conical surface.

[0117] Example 2

[0118] This embodiment 2 is a further optimization based on embodiment 1, providing an implementation method for evaluating the morphological error of a conical surface. The specific implementation method includes the following steps:

[0119] Step 1: Use the REVO five-axis coordinate measuring system to obtain the position and normal information of the conical surface. A total of 1212 measurement points are used.

[0120] Step 2: Use the local quadratic surface fitting method to calculate the average curvature of all current measurement points.

[0121] Step 3: Set the initial vertex angle of the cone surface. °, perform feature matching between the measured point cloud and the initial cone surface. Let... Through screening, it was found that when At that time, the point cloud after feature matching is compared with the initial cone model. The PV value of the error is minimized. The topographic error result of the conical surface after feature matching is shown in the figure below. Figure 4 As shown, the RMS value of the topography error is 9.342. m, PV value is 36.109 m.

[0122] Step 4: Perform least-squares fitting on the measured point cloud after feature matching to obtain an initial vertex angle of 159.9505° for the cone model before fine matching. Then, perform fine matching between the measured point cloud and the cone model using a bidirectional approximation method. After fine matching, the cone angle is 159.9852°, and the RMS value of the surface topography error is 1.125. m, PV value is 4.013 m.

[0123] This algorithm outputs the topographic error distribution, RMS value, and PV value of the conical surface, such as... Figure 5 As shown in the figure, the specific calculation results are shown in Table 1.

[0124]

[0125] Table 1. Topographic error RMS value, PV value, cone apex angle, and number of iterations during the matching process.

[0126] The specific embodiments described above further illustrate the purpose, technical solution, and beneficial effects of the present invention. It should be understood that the above description is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for evaluating the surface topography error of a conical surface, characterized in that, Includes the following steps: Step 1: Use a coordinate measuring system to obtain the coordinate data of all measurement points on the surface of the cone. and unit legal direction ; Step 2: Solve for the average curvature of the measured point cloud based on the coordinates of the measured points and their unit normals; Step 3: Optimize the hybrid distance weighting factor for the measured point cloud. With the initial cone surface Perform feature matching to determine the measured point cloud after feature matching, where Vertex coordinates axial direction of the conical surface Initial vertex angle The conical surface; Step 4: Employ a bidirectional matching algorithm based on the measured point cloud and the reconstructed cone surface. Measured point cloud after feature matching Perform fine matching and output the distribution, root mean square value, and peak-valley value of the distance from the measured point cloud to the cone surface; Step 3 includes the following specific steps: Step 3-1, Define measurement points and target point The mixing distance is ,in , ;against Algorithm Calculate distance weight factors The measurement point cloud position below , To the cone surface distance set ;in, for The average curvature at that point, For cone surface points The average curvature at that point; Step 3-2, Select error peak and valley values The minimum corresponding measurement point cloud is used as the initial measurement point cloud for subsequent fine matching. ; In step 4, a bidirectional matching algorithm based on the measured point cloud and the reconstructed cone surface is used. The specific steps include the following: Step 4-1: Calculate the measured point cloud under the current state. The best approximation cone The coordinates of the vertex of the cone axial direction of the conical surface The apex according to The distance to the cone surface is obtained using the least squares fitting method: ； in, ; Step 4-2, using Computational point cloud To the cone surface Set of projection points The SVD method is used to solve the measured point cloud. Rigid body transformation and corresponding target point cloud The optimization problem of minimizing distance. ； In the formula, and These are the rotation matrix and the translation vector, respectively. Update measurement point cloud location Continue searching To the cone surface The projection points are calculated until the convergence condition of the problem is met, at which point the measured point cloud position is recorded. Point clouds To the cone surface distance set ,calculate The root mean square value and peak-to-valley value; In calculation When calculating the root mean square value and peak-valley value, if the ratio of the root mean square value of the distance from the current measurement point to the target cone surface to the calculated value of the previous iteration is less than a given threshold, the iteration is terminated and the position of the measurement point after fine matching is output; otherwise, return to step 4-1.

2. The method for evaluating the surface morphology error of a conical surface according to claim 1, characterized in that, Step 2 includes the following specific steps: Step 2-1, establish a data point Local coordinate system as the origin of the coordinate system Unit normal at data point coordinate axes coordinate axes Choose any option provided it conforms to the right-hand orthogonal rule; Step 2-2, perform local coordinate transformation, and... The neighborhood data points are transformed from the original coordinate system to the local coordinate system. After the coordinate transformation... The neighborhood data points are fitted with a local quadratic surface using the least squares method. The equation of the fitted quadratic surface is as follows: ; Steps 2-3, calculate points mean curvature at The average curvature set of the measured point cloud is obtained. .

3. The method for evaluating the surface topography error of a conical surface according to claim 1, characterized in that, In step 3-1, the algorithm The specific steps include the following: Step 3-1a, for each Algorithm Calculate the cone surface Above and The point with the smallest mixing distance and mean curvature ; Step 3-1b, Solve for the measured points After rigid body transformation and corresponding target point The optimization problem of minimizing the mixing distance is solved as follows: ； in This represents the total number of points in the measured point cloud. and Let them be the rotation matrix and the translation vector, respectively; denoted as... ； right Perform SVD decomposition ,in Given a diagonal matrix with all non-negative singular values, the solution is: , for: ； ； ； Step 3-1c: Check if the iteration convergence condition is met; if the absolute value of the difference between the root mean square value of the mixed distance from the current measurement point position to the target point cloud and the calculated value of the previous iteration is less than a given threshold, then terminate the iteration and output the position of the measured point cloud after feature matching. Otherwise, update the position of the measured point cloud. Continue with step 3-1a; Step 3-1d: Calculate the measured point cloud. To the top corner Conical projection and distance.

4. The method for evaluating the surface topography error of a conical surface according to claim 3, characterized in that, In step 3-1d, an algorithm is used. Calculate the measurement point cloud To the top corner Conical projection and distance.

5. The method for evaluating the surface morphology error of a conical surface according to claim 3, characterized in that, In step 3-1a, the algorithm is adopted. Calculate the relationship between the cone surface and The point with the smallest mixing distance and mean curvature The specific steps include the following: (a) Let , The following formula is used for calculation. : ； (b) Let , , ,in , , ;according to The range, calculation Corresponding cone point and mean curvature .

6. The method for evaluating the surface morphology error of a conical surface according to claim 5, characterized in that, In step (b), the method based on The range, calculation Corresponding cone point and mean curvature The specific steps include the following: (b.1) If , Calculate directly based on the equation of the cone surface ; (b.2) If Calculate the cubic equation All positive roots ,calculate The corresponding cone point and the mixing distance are used to select the point with the smallest mixing distance. Record the corresponding ; (b.3) If Calculate the cubic equation All positive roots and calculate , calculate and The corresponding cone point and the mixing distance are used to select the point with the smallest mixing distance. Record the corresponding ; (b.4) Calculate its mean curvature .

7. The method for evaluating the surface morphology error of a conical surface according to claim 6, characterized in that, The algorithm The specific steps include the following: (a) Assume the axial direction of the conical surface Then point To the cone surface distance : ； (b) Let Solve the following equations to obtain the projection point. : ； We can obtain: (c) Traversal At each point in the cone, the measured point cloud is obtained. The projection is denoted as Measure point cloud to cone surface distance set .

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