Information processing method, information processing device, program, and recording medium

Abstract

To provide an information processing method and an information processing device capable of designing and analyzing, for example, three-dimensional components, using physically meaningful shape representation and control method, and a program, as well as a recording medium.SOLUTION: Using a quadratic equation of Zi,j, the Gaussian curvature k1(xi,yi) k2(xi,yi) for all grid points i except a boundary is specified, and a grid spacing is reduced and converged, thereby generating a curve surface with the specified Gaussian curvature. According to the present technology, it is possible to perform partial correction and smoothing by controlling the curvature, which is a representation of a physical shape.SELECTED DRAWING: Figure 8

Description

[Technical field]

[0001] The present technology relates to an information processing method, an information processing device, a program, and a recording medium for designing and analyzing, for example, three-dimensional parts. [Background technology]

[0002] Conventionally, three-dimensional parts and the like are designed and analyzed using the calculation capabilities, memory capabilities, analytical capabilities, and the like of computers. Design data for three-dimensional parts is created using software called CAD (Computer Aided Design), for example, and NC (Numerical Control) data based on the design data is created using software called CAM (Computer Aided Manufacturing) (see, for example, Patent Document 1). NC data is numerical information necessary to control the relative position between the part or die to be machined and the tool that cuts the part or die, the tool path, the tool rotation speed, and the like, in order to automatically machine three-dimensional parts with an NC (Numerical Control) machine tool (including a machining center). [Prior art documents] [Patent documents]

[0003] [Patent Document 1] JP 2004-021474 A Summary of the Invention [Problem to be solved by the invention]

[0004] A B-spline curve is an example of a method for drawing a curve in the above-mentioned CAD, etc. However, since a B-spline curve is expressed by a plurality of control points and a B-spline basis function, it is difficult to modify the curve using a physically meaningful shape expression or control method.

[0005] The present technology has been proposed in light of the above-mentioned conventional situation, and provides an information processing method, information processing device, program, and recording medium that can design and analyze, for example, three-dimensional parts using physically meaningful shape representations and control methods. [Means for solving the problem]

[0006] As a result of intensive research, the inventors of the present application have developed a method for discretizing differential equations having physical meanings and using functions introduced to express local relationships in physical phenomena. By the following technologies It has been found that the above problems can be solved.

[0007] [1] The computer divides the parabola into i equal parts in the x direction. i The function f(x i ), and f(x i ) second derivative f''(x i ), curvature k(x i ), and the second derivative of the specified point P i '', to define a curve with both ends of the parabola fixed, as shown in the following formulas (B-1), (B-2), and (B-3), x i The function g(x i The differential equations that the curve should satisfy are defined as the following equations (C-1), (C-2), and (C-3), and a specified point P k is moved by Δy in the y direction, and α and β shown in the following equations (E-1) and (E-2) are calculated so as to satisfy the following equations (D-1) and (D-2), An information processing method in which a parabola with fixed ends is modified by moving a specified point. [2] The computer divides the parabola into i equal parts in the x direction. i The function f(x i ), and f(x i ) second derivative f''(x i ), the curvature weighting function k(i), and the second derivative at the specified point P i '', a curve with both ends of the parabola fixed is defined as shown in the following formulas (F-1), (F-2), and (F-3), x i i≦k 1 、(k 1 +1)≦i≦k 2 , and (k 2 +1)≦i≦k 3 and define curvature weighting functions k(i) having function values ​​shown in the following equations (G-1), (G-2), and (G-3), respectively: A specified point P k1,P k2 ,P k3 Calculate α shown in the following formula (H) so that it passes through An information processing method for correcting a parabola into a straight line from a specified point to its end. [3] The computer determines the starting point P 0 From the specified point P k A parabola up to and a specified point P k From end point P n For a straight line up to x, divide the line into i equal parts in the x direction. i The function f(x i ), and f(x i ) second derivative f''(x i ) is defined by the following formula (I): x i The function g(x i The differential equations to be satisfied by the following equations (J-1) and (J-2) are defined, and α shown in the following equation (L) is calculated so as to satisfy the following equation (K): An information processing method for correcting a straight line from a specified point to the end of a parabola into a curve. [4] A parabola is divided into i equal parts in the x direction. i The function f(x i ), and f(x i ) second derivative f''(x i ), curvature k(x i ), and the second derivative of the specified point P i '', to define a curve with both ends of the parabola fixed, as shown in the following formulas (B-1), (B-2), and (B-3), x i The function g(x i The differential equations that the curve should satisfy are defined as the following equations (C-1), (C-2), and (C-3), and a specified point P k is moved by Δy in the y direction, and α and β shown in the following equations (E-1) and (E-2) are calculated so as to satisfy the following equations (D-1) and (D-2), An information processing device that corrects a parabola with fixed ends by moving a specified point. [5] A parabola is divided into i equal parts in the x direction. i The function f(x i ), and f(x i ) second derivative f''(x i ), the curvature weighting function k(i), and the second derivative at the specified point P i '', a curve with both ends of the parabola fixed is defined as shown in the following formulas (F-1), (F-2), and (F-3), x i i≦k 1 、(k 1 +1)≦i≦k 2 , and (k 2 +1)≦i≦k 3 and define curvature weighting functions k(i) having function values ​​shown in the following equations (G-1), (G-2), and (G-3), respectively: A specified point P k1 ,P k2 ,P k3 Calculate α shown in the following formula (H) so that it passes through An information processing device that corrects a parabola into a straight line from a specified point to its end. [6] The computer determines the starting point P 0 From the specified point P k A parabola up to and a specified point P k From end point P n For a straight line up to x, divide the line into i equal parts in the x direction. i The function f(x i ), and f(x i ) second derivative f''(x i ) is defined by the following formula (I): x i The function g(x i The differential equations to be satisfied by the following equations (J-1) and (J-2) are defined, and α shown in the following equation (L) is calculated so as to satisfy the following equation (K): An information processing method for correcting a straight line from a specified point to the end of a parabola into a curve. [7] A program for causing a computer to execute the information processing method according to any one of claims [1] to [3]. [8] A recording medium having the program according to claim [7] recorded thereon.

[0010]

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[0011] According to this technology, Because functions are introduced to express local relationships in physical phenomena, they can be modified using physically meaningful shape expressions and control methods. . [Brief description of the drawings]

[0012] [Figure 1] FIG. 1 is a block diagram showing an outline of the configuration of an information processing device. [Diagram 2] FIG. 2 is a flowchart showing a method for designing an aspheric lens. [Diagram 3] FIG. 3 is a diagram for explaining an example of calculation of an algorithm function in the first curved surface generating step. [Figure 4] FIG. 4 is a diagram for explaining an example of calculation of the algorithm function in the second curved surface generating step. [Diagram 5] FIG. 5A is a diagram showing the cross-sectional shape of an aspherical lens that achieves a focal length of 10, and FIG. 5B is a diagram showing the results of tracing rays through the aspherical lens. [Figure 6] FIG. 6A is a diagram showing the cross-sectional shape of an aspheric lens that achieves a focal length of 5, and FIG. 6B is a diagram showing the results of tracing rays through the lens. [Figure 7] FIG. 7(A) is a diagram showing the XY plane of the sphere and the rectangular prism, and FIG. 7(B) is a diagram showing the XZ plane of the sphere and the rectangular prism. [Figure 8] FIG. 8 is a diagram showing an example of a program for a surface having a Gaussian curvature. [Figure 9] FIG. 9 is a diagram showing a lattice specifying a Gaussian curvature and a lattice specifying a mean curvature. [Figure 10] FIG. 10(A) is a graph showing the minimum curvature and maximum curvature when the boundary line is a straight line, and FIG. 10(B) is a graph showing the Gaussian curvature and mean curvature when the boundary line is a straight line. [Figure 11] FIG. 11(A) is a graph showing the minimum curvature and maximum curvature when the boundary line is an arc, and FIG. 11(B) is a graph showing the Gaussian curvature and mean curvature when the boundary line is an arc. [Figure 12] FIG. 12 is a diagram for explaining a correction method for moving the end point Pn of the broken line by Δy in the y direction. [Figure 13]FIG. 13A is a graph showing the calculation results of the curve, and FIG. 13B is a graph showing the calculation results of the radius of curvature of the curve. [Figure 14] FIG. 14 is a diagram for explaining a method of moving a designated point Pk by Δy in the y direction on a parabola with both ends fixed. [Figure 15] FIG. 15(A) is a graph showing the calculation results of a parabola, and FIG. 15(B) is a graph showing the calculation results of the radius of curvature of the parabola. [Figure 16] FIG. 16 is a diagram for explaining a method of correcting a curve to make a straight line from a specified point to an end point. [Figure 17] FIG. 17(A) is a graph showing the calculation results of the second-order differential in the X direction using the grid points before smoothing, and FIG. 17(B) is a graph showing the calculation results of the second-order differential in the Y direction using the grid points before smoothing. [Figure 18] FIG. 18(A) is a graph showing the calculation results of the second-order differential in the X direction using the grid points after smoothing, and FIG. 18(B) is a graph showing the calculation results of the second-order differential in the Y direction using the grid points after smoothing. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0013] Hereinafter, embodiments of the present technology will be described in detail in the following order. 1. Overview of Technology 2. Information processing equipment 3. Specific examples

[0014] <1. Overview of Technology> This technology discretizes differential equations (hereinafter also referred to as "equations with physical meaning") that are expressed in physically meaningful shape expressions and control methods, and performs information processing using functions (hereinafter also referred to as "algorithm functions") that are introduced to express local relationships in physical phenomena. This can be done by defining differentiation on discrete geometric structures (polygons and polyhedra) in a manner similar to the construction of calculus.

[0015] First, let us organize the symbol system. We define the difference operator, derivative, partial derivative, etc. as shown in the formula below. Since there are no limit operations, no complicated discussion is necessary.

[0016]

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[0017] Next, we will derive the formulas. As shown in the formulas below, we can derive the discrete integral formula, the discrete partial integral formula, and the discrete Stokes' theorem. These are not directly useful for calculations, but are thought to be necessary for theoretical construction.

[0018]

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[0019] Next, we derive the algorithm function. The algorithm function is a function introduced to express local relationships in physical phenomena, and in its general definition, it specifies multiple relational expressions that should be satisfied by data associated with each vertex and adjacent vertices for each vertex of a broken line or polyhedron. This is based on the knowledge that many physical phenomena can be recognized when local relationships are in harmony as a whole.

[0020] Finally, the algorithm function is calculated. This is a sequential process that solves the specified relational equation at each point and updates the data. When it converges, the value of the algorithm function is determined, and the result is considered to give the correct solution.

[0021] In this way, by formulating calculus for the discrete structure of equations that have physical meaning, constructing a finite difference method as an independent system, turning the solution of the discretized partial differential equation into a function, and calculating the algorithm function, it is possible to design and analyze, for example, three-dimensional parts using shape representations and control methods that have physical meaning.

[0022] Examples of equations having physical meaning include the differential equation of curvature, the Poisson equation (Laplace equation), the heat conduction equation, the Navier-Stokes equation, etc. For discretization, the difference method, the finite element method, the finite volume method, the boundary element method, etc. can be used, and among these, the difference method is preferably used from the viewpoint of ease of calculation processing.

[0023] Below, some examples of equations having physical meanings that are replaced with algorithmic functions will be described.

[0024] [Curvature differential equation] When the curvature κ is expressed using a curve expressed by the explicit function y=f(x), it is expressed as a differential equation of curvature as shown in equation (1). When equation (1) is discretized and expressed using the central difference of equations (2-1) and (2-2), it is expressed as equation (3), and when the curvature κ=C is specified at lattice point i, it becomes an algorithm function as shown in equation (4).

[0025]

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[0026] Then, using equation (4), a curve can be obtained by specifying the curvature C for all lattice points i except for the boundaries and repeatedly updating the value of yi. Since curvature is a quantity that represents the degree of curvature of a curve or curved surface, partial correction and smoothing can be performed by controlling the curvature, which is a physical shape expression.

[0027] [Poisson Equation] In the Poisson equation expressed as equation (5), if the x direction is divided into i equal parts and the y direction is divided into j equal parts, it can be expressed as equation (6). If equation (6) is discretized and expressed using the central difference of equations (7-1) and (7-2), it becomes an algorithm function as shown in equation (8).

[0028]

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[0029] The usual way is to write it as a linear equation: i,j In the calculation of the algorithm function, z is calculated from (8). i,j The process of calculating is repeated, and the converged one is regarded as the solution. The Poisson equation can describe, for example, the electrostatic potential when a charge distribution is given, the gravitational potential when a mass distribution is given, the temperature distribution when a heat source exists, and the substance concentration distribution when a source of substance generation or destruction exists.

[0030] [Heat Equation] In the heat conduction equation expressed as equation (9), if the x direction is divided into i equal parts and the y direction is divided into j equal parts, it can be expressed as equation (10). If equation (10) is discretized and expressed using the central difference of equations (11-1) and (11-2), it becomes an algorithm function as shown in equation (12).

[0031]

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[0032] This algorithm function repeats a process of calculating using past data with respect to time, and the converged solution is the solution. The heat conduction equation can be applied not only to heat conduction but also to diffusion phenomena in general.

[0033] [Navier-Stokes equations] In the Navier-Stokes equations (two-dimensional, incompressible, steady flow) expressed as equations (13), (14), and (15), if the x direction is divided into i equal parts and the y direction is divided into j equal parts, and equations (13), (14), and (15) are discretized and displayed using central differences, the algorithm functions shown in equations (16), (17), and (18) are obtained.

[0034]

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[0035] This algorithm function only needs to satisfy equations (16), (17), and (18) at each point. Since equation (18) does not include information about the point of interest, the relational equations for adjacent points are used. Specifically, the velocity and pressure for the point of interest are determined by the relational equations for the velocity and pressure at 12 points around the point of interest. Therefore, the solution of the algorithm function is found as the minimum value of a total of 14 relational equations.

[0036] [Effect of algorithm function] As described above, by using an algorithm function, information processing can be performed using a physically meaningful shape expression and control method.

[0037] In addition, since the algorithmic function is easy to handle, it can be calculated by a computer in a simple manner. In particular, since the calculation process of the algorithmic function depends only on local information, there is no need to reserve a large-capacity memory or virtual memory, and no specialized library for solving large-scale linear equations is required.

[0038] The algorithm function can also be applied to inverse analysis problems, where the coefficients and boundary conditions of differential equations are changed to find the optimal solution to the problem.Also, since the algorithm function is defined by the adjacent relationship of information at each point, it has the ability to be expanded, such as by reducing the lattice spacing and, when convergence occurs, forming a curved surface with the specified curvature.

[0039] <2. Information processing device> Next, an information processing device that calculates the above-mentioned algorithm function will be described. Fig. 1 is a block diagram showing an outline of the configuration of the information processing device. As shown in Fig. 1, the information processing device 10 includes a CPU (Central Processing Unit) 11, a storage device 12, an input unit 13, and an output device 14.

[0040] The CPU 11 performs calculation processing of the algorithm function based on the data stored in the storage device 12 and the numerical values ​​and instructions input from the input unit 13 according to the program. As will be described later, the CPU 11 also functions as a first curved surface generating unit that uses the algorithm function to specify the curvature of the first curved surface of the aspheric lens and generate the first curved surface. As will be described later, the CPU 11 also functions as a second curved surface generating unit that uses the algorithm function to calculate the curvature of the second curved surface so that the intersection of the optical axis of the aspheric lens and a light ray is the same as a previously specified focal length, and generates the second curved surface.

[0041] The storage device 12 is composed of a primary storage device such as a RAM (Random Access Memory) and a secondary storage device such as a hard disk drive (HDD) and an SSD (Solid State drive), and stores various data. The storage device 12 also stores a program that causes the storage device 12 to function as a first curved surface generator and a second curved surface generator. The storage device 12 also stores various data related to the refractive index of the lens material, etc.

[0042] The input unit 13 is a user interface for inputting various data, such as a touch panel, a keyboard, etc. The input unit 13 is used to input values ​​that can be taken by control factors such as the curvature of the lens surface and the refractive index of the lens material.

[0043] The output device 14 is, for example, a display, a printer, etc., and is capable of processing such as displaying, printing, and outputting the results of the calculation processing of the algorithm function.

[0044] The function of calculating the algorithm function is not limited to the configuration of the information processing device described above, and may be configured by multiple hardware devices such as terminal devices and server devices, and may be configured as cloud computing in which one function is shared and processed jointly by multiple devices via a network. In addition, the function of calculating the algorithm function can be realized by creating a computer program and implementing it in a computer. It is also possible to provide a computer-readable recording medium in which such a computer program is stored. Examples of the recording medium include a magnetic disk, an optical disk, a magneto-optical disk, and a flash memory. In addition, the computer program can be distributed, for example, via a network, without using a recording medium.

[0045] <Example> Next, a specific example of information processing using the above-mentioned algorithm function will be described.

[0046] [Aspheric lens design method] Fig. 2 is a flow chart showing a method for designing an aspheric lens. As shown in Fig. 2, the method for designing an aspheric lens according to this embodiment includes a first curved surface generating step (S1) for generating a first curved surface using a function obtained by discretizing a differential equation of curvature, and a second curved surface generating step (S2) for generating a second curved surface using the function. Note that a description of various data related to the refractive index of the lens material and the like is omitted here.

[0047] The function obtained by discretizing the curvature differential equation is the algorithm function expressed by the following equation (4) when the curvature κ=C is specified at the lattice point i.

[0048]

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[0049] In the first curved surface generating step (S1), the curvature of the first curved surface of the aspheric lens is specified to generate the first curved surface using an algorithm function expressed by formula (4). When the aspheric lens is a symmetrical optical system in which the lens shape is symmetrical on a cross section perpendicular to the lens including the optical axis, it is preferable to specify the curvature C of the first curved surface in the outer circumferential direction from the optical axis of the aspheric lens and generate the first curved surface by rotating it about the optical axis. This makes it possible to reduce the amount of calculation of the algorithm function that generates the first curved surface.

[0050] Fig. 3 is a diagram for explaining an example of calculation of an algorithm function in the first curved surface generation step. In Fig. 3, in the first curved surface generation step (S1), a curve is obtained by specifying the curvature C of each lattice point i from x=0 to x=a and repeatedly updating the value of yi. Then, by inverting it about the optical axis (y-axis), a cross section of the first curved surface of the aspheric lens can be generated. Also, by rotating it about the optical axis (y-axis), the first curved surface of the aspheric lens can be generated.

[0051] In the second curved surface generating step (S2), the curvature of the second curved surface is calculated using an algorithm function expressed by formula (4) so ​​that the intersection point between the optical axis of the aspheric lens and the light ray is the same as a pre-specified focal length, and the second curved surface is generated. As in the first curved surface generating step (S1), when the aspheric lens is a symmetrical optical system in which the lens shape is symmetrical on a cross section perpendicular to the lens including the optical axis, it is preferable to calculate the curvature C of the second curved surface in the outer circumferential direction from the optical axis of the aspheric lens, and generate the second curved surface by rotating it about the optical axis. This makes it possible to reduce the amount of calculation of the algorithm function for generating the second curved surface.

[0052] Fig. 4 is a diagram for explaining an example of calculation of the algorithm function in the second curved surface generation step. In Fig. 4, in the second curved surface generation step (S2), first, the curvature of the second curved surface 22 is made the same as that of the first curved surface 21, and then the curvature of the second curved surface 22 is changed, and the solution of the algorithm function is converged so that the intersection of the optical axis of the aspheric lens and the light ray coincides with a pre-specified focal length, thereby generating the second curved surface 22. In addition, the tracing of the light ray can be obtained by inputting the refractive index of the lens material, etc.

[0053] In the second curved surface generating step (S2), an evaluation function represented by the sum of the intersection point between the optical axis of the aspheric lens and a light ray and a pre-specified focal length may be used to make the curvature of the second curved surface 22 the same as that of the first curved surface 21, and then the curvature of the second curved surface 22 may be changed to generate the second curved surface 22 so that the evaluation function is minimized.

[0054] Fig. 5(A) is a diagram showing the cross-sectional shape of an aspherical lens that achieves a focal length of 10, and Fig. 5(B) is a diagram showing the results of tracing rays through the aspherical lens. Also, Fig. 6(A) is a diagram showing the cross-sectional shape of an aspherical lens that achieves a focal length of 5, and Fig. 6(B) is a diagram showing the results of tracing rays through the lens.

[0055] According to such a design method for an aspheric lens, since an algorithm function is used, calculations can be easily performed. In addition, partial correction and smoothing can be performed by controlling the curvature, which is a physical shape expression. In addition, the above-mentioned design method for an aspheric lens can also be applied to the design of a Paulel lens, which generates a line light with a uniform distribution by diverging collimated light in a fan shape in one direction. In addition, by using the design data obtained by such a design method, a lens can be manufactured by controlling the curvature by processing the lens by a pressing method using a mold or a direct processing method by grinding and polishing.

[0056] [Specified Gaussian surface generation method] The Gaussian curvature and the mean curvature are expressed by equations (19-1) and (19-2), respectively. E, F, and G are z i,jSince it does not include z, it can be expressed by equation (20). If α, β, and γ are expressed by equations (21-1), (21-2), and (21-3), respectively, we get equation (22), and z i,j This can be solved using a quadratic equation.

[0057] In equation (22), к1(x i ,y i )к2(x i ,y i ) is the specified Gaussian curvature value, and ∂ 2 z i,j / ∂x i ∂y i is (z i+1,j+1 -z i+1,j-1 -z i-1,j+1 -z i-1,j-1 ) / 4ΔxΔy, and z i,j Since there is no term, it can be treated as a constant.

[0058]

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[0059] Next, we will consider a method of using an algorithm function to find a curved surface obtained by cutting a sphere of radius R with a rectangular prism of 2L. Fig. 7(A) is a diagram showing the XY plane of the sphere and rectangular prism, and Fig. 7(B) is a diagram showing the XZ plane of the sphere and rectangular prism. The calculation results of the algorithm function when R=10 and L=5 showed that in the case of 20x20 division, the maximum error was 0.0 and the minimum error was -0.0014. Also, in the case of 200x200 division, the maximum error was 0.0 and the minimum error was -0.00042.

[0060] Figure 8 shows an example of a program for a surface with Gaussian curvature. This program can perform one calculation for all lattice points. In this way, a surface with Gaussian curvature can be calculated with a simple program.

[0061] Next, we consider the creation of a surface with specified Gaussian curvature and mean curvature. Figure 9 shows a lattice that specifies the Gaussian curvature and a lattice that specifies the mean curvature. As shown in Figure 9, the Gaussian curvature is specified in the black circle a, and the mean curvature is specified in the white circle b to obtain the algorithm function.

[0062] Fig. 10(A) is a graph showing the minimum and maximum curvatures when the boundary line is a straight line, and Fig. 10(B) is a graph showing the Gaussian curvature and the mean curvature when the boundary line is a straight line. In Fig. 10(A), a indicates the minimum curvature, and b indicates the maximum curvature, and in Fig. 10(B), c indicates the Gaussian curvature, and b indicates the mean curvature. Fig. 11(A) is a graph showing the minimum and maximum curvatures when the boundary line is a circular arc, and Fig. 11(B) is a graph showing the Gaussian curvature and the mean curvature when the boundary line is a circular arc. In Fig. 11(A), a indicates the minimum curvature, and b indicates the maximum curvature, and in Fig. 11(B), c indicates the Gaussian curvature, and b indicates the mean curvature.

[0063] From the graphs shown in Figures 10 and 11, we can see that the conditions specified at each point are satisfied, and by decreasing the grid spacing, when convergence occurs, a surface with the specified curvature is obtained.

[0064] [How to correct broken lines] Next, we consider the correction of two-dimensional broken lines. n 4 is a diagram for explaining a correction method for moving the image in the y direction by Δy.

[0065] Let P be the vertex of the two-dimensional broken line. i (x i ,y i ), 0≦i≦n. Also, x i The intervals are equal and are Δx. n Consider the problem of finding a smooth curve b by moving Δy in the y direction. k It will not change until

[0066] Linear x i x iThe function f(x i ), then x i In the discrete second derivative f''(x i ) can be defined. If Δx is equally spaced, it becomes equation (23). New x i The function g(x i The algorithm function that x must satisfy is defined as (24-1) and (24-2), and α is calculated to obtain (25). Using the formulas of discrete differentiation and geometry, α can be calculated as (26). This operation is k Weight 1 up to x k From (1+α(xx k )), this ensures the continuity of the second derivative.

[0067]

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[0068] Fig. 13(A) is a graph showing the calculation results of the curve, and Fig. 13(B) is a graph showing the calculation results of the radius of curvature of the curve. As shown in Fig. 13(A), the corrected curve b is scaled approximately twice as large in the Y direction as the curve a before correction, and as shown in Fig. 13(B), the radius of curvature of the corrected curve b has changed significantly compared to the radius of curvature of the curve a before correction. There is no point where the curvature vector is 0, so there is no inflection point. For this reason, it can be seen that the correction has been made while maintaining the convexity.

[0069] [How to modify a parabola by moving a specified point with both ends fixed] Next, we consider the movement of a specified point of a parabola with both ends fixed. Figure 14 shows the movement of a specified point P k 14 is a diagram for explaining a method of moving the parabola by Δy in the y direction. In Fig. 14, a is the parabola before the movement, and b is the parabola after the movement.

[0070] First, find the curve that satisfies equations (27-1), (27-2), and (27-3). i The function g(x iThe algorithm functions that should be satisfied by (28-1), (28-2), and (28-2) are defined, and α and β are determined so as to obtain (29-1) and (29-2). α and β can be calculated specifically, and are given by (30-1) and (30-2).

[0071]

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[0072] Fig. 15(A) is a graph showing the calculation results of a parabola, and Fig. 15(B) is a graph showing the calculation results of the radius of curvature of the parabola. In Fig. 15(A), a is a parabola that is 100 between P0 and P1 and -10 at position 50, and b is a parabola that is moved -2 ​​in the y direction at position 60 with respect to curve a. In Fig. 15(B), a indicates the radius of curvature of parabola a shown in Fig. 15(A), and b indicates the radius of curvature of parabola b shown in Fig. 15(B). It can be seen from Fig. 15(B) that the radius of curvature changes significantly at the specified point position 60.

[0073] [How to modify a curve to make it straight from a specified point to the end] FIG. 16 is a diagram for explaining a method of modifying a curve to make it linear from a specified point to the end point. First, a curve that satisfies equations (31-1), (31-2), and (31-3) is obtained. The curve is modified to be linear from the specified point k1. In order for the curvature to be continuous, a portion that connects the curvature is necessary. For this reason, the portion to be modified is divided into three, and the second-order differential of the original point sequence is multiplied by a curvature weighting function K(x) having a function value such as equations (32-1), (32-2), and (32-3). The K(x) function is defined by equation (34) using α, and this modification is possible by adjusting α so that it passes through the specified point at the end point.

[0074]

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[0075] [B-spline surface smoothing method] The method for smoothing a B-spline surface shown as a specific example involves considering the B-spline surface as a manifold, approximating it with a quadratic polynomial at each point, and calculating the second derivative. The surface with these second derivatives is then reproduced with an algorithm function.

[0076] First, the B-spline surface is polygonized (STL converted) with an error of, for example, 0.001 mm, and this is covered with, for example, 121 x 121 grid points. Then, quadratic polynomial approximation is performed at the grid points.

[0077] Specifically, a quadratic polynomial approximation is performed using STL vertices with a radius of R or less, centered on the grid points. Since the STL vertices are on the original surface, a relatively accurate approximation can be obtained. The B-spline surface is then considered as a manifold and is covered with quadratic polynomials at the grid points.

[0078] Next, multiply the second derivatives in the X and Y directions to get G ij Calculate (i,j) and calculate the algorithm function using the relationship in the lattice as equation (35).

[0079]

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[0080] Fig. 17(A) is a graph showing the calculation result of the second order differential in the X direction using the grid points before smoothing, and Fig. 17(B) is a graph showing the calculation result of the second order differential in the Y direction using the grid points before smoothing. Fig. 18(A) is a graph showing the calculation result of the second order differential in the X direction using the grid points after smoothing, and Fig. 18(B) is a graph showing the calculation result of the second order differential in the Y direction using the grid points after smoothing. A body part with a size of 1400 mm in the horizontal (X) direction and 1000 mm in the vertical (Y) direction was generated with a B-spline surface and slightly modified using an algorithm function to smooth the second order differential.

[0081] As can be seen from the graphs in Figures 17 and 18, there is noise before the correction, but it has become smooth after smoothing. Also, the amount by which the vertex was moved is very small, at 0.402 mm, and it can be seen that the second derivative can be smoothed by making small corrections.

[0082] Moreover, the present technology can be expressed as follows (1) to (29). (1) a first curved surface generating step of specifying a curvature C for all lattice points i excluding boundaries using a function obtained by discretizing a differential equation for curvature expressed using a curve expressed as an explicit function, updating the value of the curve at the lattice point i, and repeating this process to obtain a curve, and rotating the curve about the optical axis to generate a first curved surface of the aspheric lens; and a second curved surface generation step of using the function to specify a curvature C for all lattice points i excluding boundaries so that the intersection of the optical axis of the aspherical lens and the light ray is the same as a pre-specified focal length, updating the value of the curve at the lattice point i, and repeating this process to obtain a curve, and rotating the curve about the optical axis to generate a second curved surface of the aspherical lens. (2) The information processing method according to (1), wherein the function is obtained by discretizing a differential equation of the curvature using a finite difference method. (3) The information processing method according to (2), wherein the difference method is a central difference method. (4) The information processing method according to any one of (1) to (3), wherein in the second curved surface generation step, after the curvature C of the second curved surface is specified to be the same as the curvature C of the first curved surface, the curvature C of the second curved surface is changed, and the solution of the function is converged so that the intersection of the optical axis of the aspheric lens and a light ray is the same as a pre-specified focal length, thereby generating the second curved surface. (5) Using an evaluation function expressed by the sum of the intersection point between the optical axis of the aspherical lens and the light ray and a pre-specified focal length, The information processing method according to any one of (1) to (3), wherein in the second surface generation process, the curvature C of the second surface is specified to be the same as the curvature C of the first surface, and then the curvature C of the second surface is changed to generate the second surface so that the evaluation function is minimized. (6) The information processing method according to any one of (1) to (5), wherein the function is expressed by the following formula (1) when a curvature κ=C is specified at a lattice point i.

number

[0083] 10 information processing device, 11 CPU, 12 storage device, 13 input unit, 14 output device, 21 first curved surface, 22 second curved surface

Claims

1. The computer divides the parabola into i equal parts in the x direction. ｉ The function f(x ｉ ) and f(x ｉ ) second derivative f''(x ｉ ), curvature k(x ｉ ), and the second derivative of the specified point P ｉ '', a curve with both ends of the parabola fixed is defined as shown in the following formulas (B-1), (B-2), and (B-3), x ｉ The function g(x ｉ The differential equations to be satisfied by the curve are defined as the following formulas (C-1), (C-2), and (C-3), and a specified point P ｋ is moved by Δy in the y direction, and α and β shown in the following formulas (E-1) and (E-2) are calculated so as to satisfy the following formulas (D-1) and (D-2), An information processing method in which a parabola with fixed ends is modified by moving a specified point. [0010]

2. A parabola is divided into i equal parts in the x direction. ｉ The function f(x ｉ ) and f(x ｉ ) second derivative f''(x ｉ ), curvature k(x ｉ ), and the second derivative of the specified point P ｉ '', a curve with both ends of the parabola fixed is defined as shown in the following formulas (B-1), (B-2), and (B-3), x ｉ The function g(x ｉ The differential equations to be satisfied by the curve are defined as the following formulas (C-1), (C-2), and (C-3), and a specified point P ｋ is moved by Δy in the y direction, and α and β shown in the following formulas (E-1) and (E-2) are calculated so as to satisfy the following formulas (D-1) and (D-2), An information processing device that corrects a parabola with fixed ends by moving a specified point. [0025]

3. A program for causing a computer to execute the information processing method described in claim 1.

4. A recording medium having the program described in claim 3 recorded thereon.

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