A three-coordinate measurement and parameter fitting method for hyperbolic aspheric surfaces

Abstract

The present invention relates to the field of optical processing and detection technology, and specifically to a three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface. A three-coordinate measuring machine is used to measure a hyperbolic surface to obtain a set of three-coordinate values ​​of the hyperbolic surface, wherein x is a non-plane direction coordinate value, y is a plane direction coordinate value, and z is a vector height direction coordinate value; a spatial conic curve equation and a linear fitting method are used to obtain correlation coefficient matrices S and L; the eigenvalue λ and the eigenvector Q of S are solved, and the eigenvector Q is converted into a rotation matrix; the cross term of the conic curve equation is eliminated using a rotation axis formula; the cone coefficient and the curvature radius of the hyperbolic surface are solved and the surface shape residual σ is calculated. The present invention uses the spatial conic curve equation and the linear fitting method to numerically calculate the cone coefficient and the curvature radius of the hyperbolic surface, thereby improving the calculation speed and the accuracy of fitting the hyperbolic surface shape, and can be applied to guiding the fields of optical processing and detection of hyperbolic mirrors.

Description

Technical Field

[0001] The invention relates to the technical field of optical processing and detection, and can be used for processing and detecting aspheric hyperbolic aspheric mirrors, and in particular to a three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface. Background Art

[0002] The continuous development of science and technology has put forward new requirements for optical systems, such as simplifying the system, improving imaging quality, improving illumination uniformity, etc. It is difficult to meet these requirements by relying solely on traditional plane and spherical aspherical optical elements. Aspherical elements have characteristics that traditional planes and spherical surfaces do not have, which is conducive to correcting aberrations, improving imaging quality, and increasing the freedom of optical design. At present, the application of aspherical optical elements has penetrated into our lives, work and scientific research. Among them, the hyperbolic mirror has better design freedom due to its two different curvatures in two directions and aspherical coefficients. It is widely used in astronomical telescopes, space remote sensing mapping, automobiles and other fields. However, the hyperbolic mirror does not have rotational symmetry, there is no rotational symmetry axis, the radius of curvature of each part of the surface is different, the processing technology is relatively complicated, and the detection difficulty is also relatively large. The detection of the hyperbolic mirror mainly includes the profilometer method and computer holography method.

[0003] In the early processing stage of high-precision hyperbolic mirrors, namely the grinding and rough polishing stage, the mirror surface is not polished and cannot be inspected by computer holography. The three-coordinate profile measurement technology becomes the main detection method for surface shape errors in the early stage of processing. The cone coefficient and curvature radius of the hyperbolic surface determine the surface shape results of the hyperbolic surface. Therefore, it is very important to obtain effective and accurate cone coefficient and curvature radius during the detection process. Summary of the invention

[0004] In view of the above problems, the present invention provides a three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface, which solves the problem that when the three-coordinate measurement method is used to detect a standard hyperbolic surface, it is difficult to directly use the measurement data to obtain the aspheric coefficients and curvature radius of the hyperbolic surface because the mathematical model of the standard hyperbolic surface is relatively complex.

[0005] The technical solution adopted by the present invention to solve the technical problem is: a three-coordinate measurement and parameter fitting method of a hyperbolic aspheric surface, comprising the following steps:

[0006] Step S1: Use a three-coordinate measuring machine to measure the hyperbolic surface to obtain a set of three-coordinate values ​​of the hyperbolic surface, where x is the coordinate value in the non-plane direction, y is the coordinate value in the plane direction, and z is the coordinate value in the sagittal height direction;

[0007] Step S2: Using the spatial conic section equation and the linear fitting method, the correlation coefficient matrices S and L are obtained;

[0008] Step S3: Solve the eigenvalue λ and eigenvector Q of the matrix S, and transform the eigenvector Q into an orthogonal unit matrix, i.e., a rotation matrix;

[0009] Step S4: Eliminate the cross terms of the conic section equation using the rotation matrix and the rotation axis formula;

[0010] Step S5: using the conic section equation with the cross terms eliminated, solve the cone coefficient and curvature radius of the hyperbolic surface and calculate the surface shape residual σ.

[0011] The beneficial effects of the present invention are as follows: the present invention provides a three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface, which utilizes a spatial conic curve equation and a linear fitting algorithm to fit the cone coefficient and curvature radius of the hyperbolic surface, thereby realizing rapid numerical calculation; utilizing a rotation axis formula to eliminate the cross terms of the conic curve equation, thereby reducing the influence of the original data measurement error, and can be applied to guide the fields of optical processing and detection of hyperbolic mirrors. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 It is a schematic flow chart of a three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface according to the present invention;

[0013] Figure 2 This is a set of three-coordinate measurement example diagrams of hyperbolic surfaces;

[0014] Figure 3 This is an example diagram of the surface shape residual after fitting the hyperbolic surface data in three-coordinate detection. DETAILED DESCRIPTION

[0015] It should be noted that the following detailed descriptions are illustrative and are intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used herein have the same meanings as those commonly understood by those skilled in the art to which the present application belongs.

[0016] It should be noted that the terms used herein are only for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular form is also intended to include the plural form. In addition, it should be understood that when the terms "comprise" and / or "include" are used in this specification, it indicates the presence of features, steps, operations, devices, components and / or combinations thereof.

[0017] For the convenience of description, if the words "up", "down", "left" and "right" appear in the present invention, they only indicate that they are consistent with the up, down, left and right directions of the drawings themselves, and do not limit the structure. They are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the referred device or element must have a specific orientation, be constructed and operated in a specific orientation. Therefore, it cannot be understood as a limitation on the present invention.

[0018] The present invention is a flow chart of a three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface, as shown in FIG. Figure 1 As shown, it includes the following steps:

[0019] Step S1: Use a three-coordinate measuring machine to measure the hyperbolic surface, and obtain a set of three-coordinate values ​​of the hyperbolic surface as [x, y, z], where x is the coordinate value in the non-planar direction, y is the coordinate value in the planar direction, and z is the coordinate value in the sagittal direction. For example, Figure 2 As shown (c x =1 / 1994.89, k x =0, c y =1 / 2005.12, k y =0.00513);

[0020] The step S1 specifically includes: measuring the hyperbolic surface using a three-coordinate measuring machine to obtain a set of three-coordinate values ​​of the hyperbolic surface as [x, y, z], and the standard hyperbolic surface equation is:

[0021]

[0022] where c x is the x curvature, k x is the x-cone coefficient, c y is the y curvature, k y is the y cone coefficient.

[0023] Step S2: Using the spatial conic section equation and the linear fitting method, the correlation coefficient matrices S and L are obtained;

[0024] The step S2 is specifically as follows: the conic section equation expresses the spatial surface equation as:

[0025] F(x,y,z)=ax 2 +by 2 +cz 2 +dxy+exz+fyz+gx+hy+kz

[0026] Let z = ax 2 +by 2 +cz 2 +dxy+exz+fyz+gx+hy+k

[0027] Let A = x2 y 2 z 2 x*yx*zy*zxym,

[0028] Where E = (A T A) -1 A T z.

[0029] After finding the correlation coefficient, the original conic section equation is rearranged as follows:

[0030]

[0031] make

[0032] Step S3: solve the eigenvalue λ and eigenvector Q of S, and convert the eigenvector Q into a rotation matrix;

[0033] The step S3 specifically comprises: solving the eigenvalue λ and the eigenvector Q of S:

[0034]

[0035] If the matrix composed of eigenvectors is not a standard orthogonal matrix, it must be converted into a standard orthogonal matrix. x , y-axis θ y and z-axis θ z The rotation angle will be relatively small, that is, cosθ≈1, sinθ≈0, and the rotation matrix can be simplified to:

[0036]

[0037] From the geometric properties of the conic section equation, we know that the matrix Q is the rotation information matrix of the space surface. According to the right-hand rule, the matrix Q is adjusted to And adjust the order of λ1,λ2,λ3 accordingly. After the adjustment, |Q|=1, and Q is converted into a rotation matrix.

[0038] Step S4: Eliminate the cross terms of the conic section equation using the axis rotation formula;

[0039] The step S4 is specifically as follows: the rotation axis formula is: [x, y, z] = [x', y', z'] * Q T , the coefficient of the first-order term is At this time, the surface equation is: F = λ1x' 2 +λ2y' 2 +λ3z' 2 +g'x'+h'y'+lz'+k.

[0040] Step S5: Solve the cone coefficient, curvature radius and surface shape residual σ of the hyperbolic surface, such as Figure 3 shown.

[0041] The step S5 specifically includes: solving the cone coefficient and curvature radius of the hyperbolic surface, and then obtaining the vector height z according to the standard hyperbolic surface equation. σ , and then calculate the surface residual σ.

[0042] When all eigenvalues ​​are equal, the surface is a sphere, which is not in the required range and will not be discussed. When two eigenvalues ​​are equal, if λ1=λ2, the rotation axis is on the X-axis or Y-axis, which is a general quadratic surface, which is not in the required range and will not be discussed. When λ1=λ3=0, the surface is two parallel planes and will not be discussed. Therefore, only the case where λ1=λ3~=0 or the eigenvalues ​​are not equal is considered.

[0043] (1) When λ1λ2λ3≠0

[0044]

[0045] make

[0046]

[0047] You can continue to discuss:

[0048] (a) When λ1'λ2'λ3'>0, the surface is an ellipsoid, let

[0049] The translation terms are:

[0050] (b) When λ3'>0, λ1'<0, λ2'<0, the surface is a bilobed hyperboloid, let

[0051] The translation terms are:

[0052] (c) When λ3'<0, λ1'>0, λ2'>0, the surface is a single-leaf hyperboloid. Let

[0053] The translation terms are:

[0054] (2) When λ1λ2λ3=0 and λ3=0

[0055] (a) When l = 0, the surface is a non-spherical cylinder with the bottom surface perpendicular to the Z axis (when λ1λ2<0, it is a hyperbolic cylinder, and when λ1λ2>0, it is an ellipsoidal cylinder).

[0056]

[0057]

[0058] make

[0059] The translation terms are:

[0060] (b) When l≠0, the surface is a non-spherical parabolic surface with the bottom surface perpendicular to the Z axis (when λ1λ2<0, it is a saddle surface, and when λ1λ2>0, it is an ellipsoidal parabolic surface).

[0061]

[0062]

[0063] make

[0064]

[0065] You can continue to discuss:

[0066] (i) When λ1λ2>0, the surface is an ellipsoidal parabolic surface.

[0067] The translation terms are:

[0068] (ii) When λ1λ2<0, the surface is a saddle surface. Let λ1>0,λ2<0,

[0069] The translation terms are:

[0070] (3) Similarly, we can deduce the cone coefficient and curvature radius of the surface equation when λ1λ2λ3=0, λ1=0 or λ2=0.

[0071] (4) When λ1=λ3≠0

[0072] (a) If λ2=0,

[0073]

[0074] (i) h'=0, the surface is a cylinder with the bottom perpendicular to the Y axis,

[0075]

[0076] make The translation terms are:

[0077] (ii) h'≠0, the surface is a parabolic surface with the bottom surface perpendicular to the Z axis,

[0078]

[0079] make The translation terms are:

[0080] (b) If λ2≠0,

[0081] (i)λ2>0, the surface is a rotating ellipsoid with the bottom perpendicular to the Z axis,

[0082]

[0083] make

[0084] The translation terms are:

[0085] (ii) λ1>0, λ2<0, the surface is a double-leaf rotation hyperboloid with the bottom surface perpendicular to the Z axis,

[0086]

[0087] make

[0088] The translation terms are:

[0089] (iii)λ1<0,λ2>0, the surface is a single-leaf rotation hyperboloid with the bottom surface perpendicular to the Z axis,

[0090]

[0091] make

[0092] The translation terms are:

[0093] After solving the cone coefficient and curvature radius of the hyperbolic surface, the vector height z is obtained according to the standard hyperbolic surface equation. σ ;

[0094]

[0095] Then the surface residual σ is calculated.

[0096] σ=z'-zσ

[0097] Examples of applications of the present invention are listed below:

[0098] Application example 1: Application of a three-coordinate measurement and parameter fitting method flow chart of a hyperbolic aspheric surface in the field of optical processing of hyperbolic mirrors.

[0099] The present invention can be applied to guide the optical processing of a hyperbolic mirror. Due to the existence of original measurement errors, the three-coordinate measurement and parameter fitting method of a hyperbolic aspheric surface proposed by the present invention can accurately numerically calculate the cone coefficient and the curvature radius of the hyperbolic surface, thereby improving the calculation speed and the accuracy of fitting the hyperbolic surface shape.

[0100] The above-mentioned specific implementations are only specific cases of the present invention. The patent protection scope of the present invention includes but is not limited to the product form and style of the above-mentioned specific implementations. Any appropriate changes or modifications made to them by ordinary technicians in the technical field that conform to the claims of the present invention shall fall within the patent protection scope of the present invention.

Claims

1. A three-coordinate measurement and parameter fitting method for a hyperbolic aspheric surface, characterized in that: The following steps are involved: Step S1: Use a three-coordinate measuring machine to measure the hyperbolic surface and obtain a set of three-coordinate values ​​of the hyperbolic surface: , where x is the non-plane direction coordinate value, y is the plane direction coordinate value, and z is the vector height direction coordinate value; Step S2: Use the spatial conic section equation and linear fitting method to find Correlation coefficient matrices S and L; Step S3: Solve the eigenvalue λ and eigenvector Q of the matrix S, and transform the eigenvector Q into an orthogonal unit matrix, i.e., a rotation matrix; Step S4: Eliminate the cross terms of the conic section equation using the rotation matrix and the rotation axis formula; Step S5: using the conic section equation with the cross terms eliminated, solve the cone coefficient and curvature radius of the hyperbolic surface and calculate the surface shape residual σ.

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