JP7291356B2 - Information processing method, information processing device, program, and recording medium - Google Patents

Description

Article 30, Paragraph 2 of the Patent Act is applied Published at the 45th VCAD System Study Group on February 24, 2017 (Method of generating a physical curved surface using an algorithm function)

The present technology relates to an information processing method, an information processing apparatus, a program, and a recording medium for designing and analyzing a three-dimensional part, for example.

2. Description of the Related Art Conventionally, three-dimensional parts and the like are designed and analyzed by using computing power, storage power, and analysis power of computers. Design data for a three-dimensional part is created using, for example, software called CAD (Computer Aided Design), 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 used to automatically process three-dimensional parts with NC (numerical control) machine tools (including machining centers). It is the numerical information necessary to control the relative position between the tools, the tool path (tool path), the tool rotation speed, etc.

JP 2004-021474 A

A B-spline curve is an example of a method of drawing a curve in the above-described CAD or the like. However, since B-spline curves are represented by multiple control points and B-spline basis functions, it is difficult to modify them using physically meaningful shape representations and control methods. is.

This technology has been proposed in view of such conventional circumstances, and uses physically meaningful shape representation and control methods to enable the design and analysis of, for example, three-dimensional parts. A method, an information processing device, a program, and a recording medium are provided.

As a result of intensive studies, the inventors of the present application have found that the above problem can be solved by discretizing differential equations having physical meaning and using functions introduced to express local relationships of physical phenomena.

That is, in the information processing method according to the present technology, the computer divides the x direction into i equal parts and the y direction into j equal parts on a curved surface z=f(x, y) in a three-dimensional space. Gaussian _{curvature κ 1} (xi , _yi ) _{κ 2} ( _{xi ,} y _i ) to generate a curved surface.

Further, the information processing apparatus according to the present technology is configured such that, on a curved surface z=f(x, y) in a three-dimensional space, Z i _{, Gaussian} curvature κ ₁ ( _{xi , yi )} κ ₂ (xi , yi ) specified for all grid points i ( _xi , _{yi )} excluding the boundary of the surface using the quadratic equation in _j Generate a surface based on .

Further, the program according to _the present technology is a curved surface z=f(x, y) in a three-dimensional space. Based on _the Gaussian curvature κ ₁ ( _xi , _{yi )} κ ₂ ( _xi , yi ₎ specified for all grid points i (xi , yi ) excluding the boundary of the surface using a quadratic equation A computer is caused to execute a process of generating a curved surface by

According to this technology, since a function obtained by discretizing a differential equation of curvature is used, partial correction and smoothing can be performed by controlling the curvature, which is a physical shape expression.

FIG. 1 is a block diagram showing a schematic configuration of an information processing apparatus. FIG. 2 is a flow chart showing a method of designing an aspherical lens. FIG. 3 is a diagram for explaining a calculation example of the algorithm function in the first curved surface generation step. FIG. 4 is a diagram for explaining a calculation example of the algorithm function in the second curved surface generation step. 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 result of tracing light rays with respect to the aspherical lens. FIG. 6A is a diagram showing the cross-sectional shape of an aspherical lens that achieves a focal length of 5, and FIG. 6B is a diagram showing the result of tracing light rays with respect to the lens. FIG. 7A is a diagram showing the XY plane of the sphere and prism, and FIG. 7B is a diagram showing the XZ plane of the sphere and prism. FIG. 8 is a diagram showing a program example of a curved surface with Gaussian curvature. FIG. 9 shows a grid specifying Gaussian curvature and a grid specifying mean curvature. FIG. 10(A) is a graph showing the minimum curvature and maximum curvature when the boundary line is straight, and FIG. 10(B) is a graph showing the Gaussian curvature and average curvature when the boundary line is straight. be. FIG. 11A is a graph showing the minimum curvature and maximum curvature when the boundary line is an arc, and FIG. 11B is a graph showing the Gaussian curvature and average curvature when the boundary line is an arc. be. FIG. 12 is a diagram for explaining a correction method for moving the end point _Pn of the polygonal line by Δy in the y direction. FIG. 13A is a graph showing the calculation result of the curve, and FIG. 13B is a graph showing the calculation result of the radius of curvature of the curve. FIG. 14 is a diagram for explaining a method of moving the designated point _Pk by Δy in the y direction with respect to a parabola whose both ends are fixed. FIG. 15A is a graph showing the calculation result of the parabola, and FIG. 15B is a graph showing the calculation result of the radius of curvature of the parabola. 16A and 16B are diagrams for explaining a method of correcting a curve to form a straight line from a specified point to an end. FIG. 17A is a graph showing the calculation result of the second derivative in the X direction using the grid points before smoothing, and FIG. It is a graph which shows the calculation result of differentiation. FIG. 18A is a graph showing the calculation result of the second derivative in the X direction using the grid points after smoothing, and FIG. It is a graph which shows the calculation result of differentiation.

Hereinafter, embodiments of the present technology will be described in detail in the following order.
1. Outline of technology2. Information processing device 3. Concrete example

<1. Overview of technology>
This technology discretizes differential equations expressed by physically meaningful shape representations and control methods (hereinafter also referred to as “physically meaningful equations”), and expresses local relationships in physical phenomena. Information processing is performed using a function (hereinafter also referred to as an “algorithm function”) introduced for the purpose. This can be done in a similar way to the construction of calculus, defining the derivative over a discrete geometric structure (polyline or polyhedron).

First, the symbol system is prepared. Define the difference operator, differentiation, partial differentiation, etc. as shown in the following equations. Since there is no limit operation, complicated arguments are not necessary.

Next, we derive the formula. As shown in the following equations, the discrete integral formula, discrete partial integral formula, discrete Stokes' theorem, etc. can be derived. Although these are not directly useful for calculation, they are considered to be necessary for theory construction.

Next, we derive the algorithm function. Algorithmic functions are functions introduced to express local relationships about physical phenomena. In general definition, for each vertex of a polygonal line or polyhedron, the data attached to each vertex and its adjacent vertices satisfy It specifies multiple relational expressions to powers. This is based on the knowledge that many physical phenomena are often perceived when local relationships are harmonized as a whole.

Finally, we compute the algorithm function. It is calculated by a sequential process that solves the relational expression specified at each point and updates the data. When it converges, the value of the algorithmic function is fixed and the result seems to give the correct solution.

In this way, by formulating calculus for the discrete structure of equations with physical meaning, constructing the finite difference method as an independent system, converting the solution of the discretized partial differential equation into a function, and calculating the algorithmic function. For example, three-dimensional parts can be designed and analyzed using physically meaningful shape representations and control methods.

Equations with physical meaning include the differential equation of curvature, the Poisson equation (Laplace equation), the heat conduction equation, the Navier-Stokes equation, and the like. For discretization, the finite difference method, the finite element method, the finite volume method, the boundary element method, etc. can be used. Among these, the finite difference method is preferably used from the viewpoint of ease of calculation.

Examples of replacing several equations with physical meanings with algorithmic functions will be described below.

[Differential equation of curvature]
When the curvature κ is expressed using a curve represented by an explicit function y=f(x), it is expressed as a differential equation of curvature as shown in Equation (1). By discretizing the equation (1) and displaying it using the central difference of the equations (2-1) and (2-2), it is expressed by the equation (3). (4) It becomes an algorithm function like the formula.

Then, using equation (4), a curve can be obtained by specifying the curvature C for all grid points i excluding the boundary and repeating the update of the value of yi. Curvature is a quantity that represents the degree of curvature of a curve or curved surface, so it is possible to perform partial correction or smoothing by controlling the curvature, which is a representation of a physical shape.

[Poisson equation]
In the Poisson's equation represented by the formula (5), when the x direction is equally divided into i and the y direction is equally divided into j, it is represented by the formula (6). When formula (6) is discretized and displayed using the central difference of formulas (7-1) and (7-2), it becomes an algorithm function as shown in formula (8).

In the usual method, z _i,j is calculated as a linear equation, but in the calculation of the algorithm function, the process of calculating z _i,j from equation (8) is repeated, and the converged result is taken as the solution. Poisson's equation gives, for example, the electrostatic potential when the charge distribution is given, the gravitational potential when the mass distribution is given, the temperature distribution when there is a source of heat generation, It can describe substance concentration distribution and the like.

[Heat conduction equation]
In the heat conduction equation represented by the formula (9), when the x direction is divided into i equal parts and the y direction is divided into j equal parts, it is expressed by the formula (10). By discretizing the equation (10) and displaying it using the central difference of the equations (11-1) and (11-2), an algorithm function is obtained as in the equation (12).

This algorithm function repeats the process of calculating using the past time, and takes the converged one as the solution. The heat conduction equation can be applied not only to heat conduction but also to diffusion phenomena in general.

[Navier-Stokes equation]
In the Navier-Stokes equations (two-dimensional, incompressible, steady flow) represented by equations (13), (14), and (15), the x direction is equally divided into i and the y direction is equally divided into j Then, by discretizing equations (13), (14), and (15) and displaying them using the central difference, algorithm functions are obtained as equations (16), (17), and (18).

This algorithm function should satisfy the equations (16), (17) and (18) at each point. Here, equation (18) does not include information on the point of interest, so a relational expression regarding adjacent points is used. Specifically, the velocity and pressure for the point of interest are determined by a relational expression regarding velocity and pressure at 12 points around the point of interest. Therefore, the solution of the algorithm function is obtained as a local minimum of a total of 14 relational expressions.

[Effect of algorithm function]
As described above, by using algorithmic functions, information processing can be performed using physically meaningful shape representations and control methods.

Algorithmic functions are also easy to work with and can be computed in a simple manner using a computer. In particular, computational processing of algorithm functions depends only on local information, so there is no need to allocate large-capacity memory or virtual memory, and there is also a specialized library for solving large-scale linear equations. do not need.

Algorithm functions can also be applied to methods of inverse analysis problems in which the coefficients and boundary conditions of differential equations are changed to find the optimum solution to the problem. In addition, since the algorithm function is defined by the adjacency relationship of information at each point, it has expandability such that the interval of the lattice is reduced and, when converging, a curved surface having the indicated curvature is obtained.

<2. Information processing device>
Next, an information processing device for calculating the algorithmic functions described above will be described. FIG. 1 is a block diagram showing a schematic configuration of an information processing apparatus. As shown in FIG. 1 , the information processing device 10 includes a CPU (Central Processing Unit) 11 , a storage device 12 , an input section 13 and an output device 14 .

The CPU 11 performs algorithm function calculation processing according to a program, based on data stored in the storage device 12 and numerical values and instructions input from the input unit 13 . In addition, as will be described later, the CPU 11 functions as a first curved surface generation unit that uses an algorithm function to specify the curvature of the first curved surface of the aspherical lens to generate the first curved surface. In addition, as will be described later, the CPU 11 uses an algorithm function to calculate the curvature of the second curved surface so that the intersection of the optical axis of the aspherical lens and the light ray is the same as the focal length specified in advance. functions as a second curved surface generation unit that generates a second curved surface by

The storage device 12 includes, for example, a primary storage device such as a RAM (Random Access Memory), a secondary storage device such as a hard disk drive (HDD), an SSD (Solid State drive), and the like, and stores various data. The storage device 12 also stores a program that functions as the first curved surface generator and the second curved surface generator. The storage device 12 also stores various data regarding the refractive index of the lens material.

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

The output device 14 is, for example, a display, a printer, or the like, and is capable of processing such as display, printing, and output of calculation processing results of algorithm functions.

Note that the function of calculating the algorithm function is not limited to the configuration of the information processing device described above, and can be configured by, for example, a plurality of hardware such as a terminal device and a server device. A configuration of cloud computing in which a plurality of apparatuses share and jointly process may be used. Also, the function of calculating the algorithm function can be implemented in a computer by creating a computer program for realizing this. It is also possible to provide a computer-readable recording medium storing such a computer program. Recording media include, for example, magnetic disks, optical disks, magneto-optical disks, and flash memories. Also, the computer program can be distributed, for example, via a network without using a recording medium.

<Specific example>
Next, a specific example of information processing using the algorithm functions described above will be described.

[Method for designing an aspherical lens]
FIG. 2 is a flow chart showing a method of designing an aspherical lens. As shown in FIG. 2, the method for designing an aspherical lens according to the present embodiment comprises a first curved surface generating step (S1) of generating a first curved surface using a function obtained by discretizing a differential equation of curvature; and a second curved surface generation step (S2) of generating a second curved surface using the function. Here, explanation of various data relating to the refractive index of the lens material and the like is omitted.

A function obtained by discretizing the differential equation of curvature is an algorithm function represented by the following equation (4) when curvature κ=C is specified at grid point i.

In the first curved surface generation step (S1), the algorithm function represented by the formula (4) is used to designate the curvature of the first curved surface of the aspherical lens and generate the first curved surface. If the aspherical 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, specify the curvature C of the first curved surface in the outer peripheral direction from the optical axis of the aspherical lens, Preferably, the first curved surface is generated by rotating about the optical axis. This can reduce the computational complexity of the algorithm function for generating the first curved surface.

FIG. 3 is a diagram for explaining a calculation example of the 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 designating the curvature C of each grid point i from x=0 to x=a and repeating updating of the value of yi. Then, by inverting with respect to the optical axis (y-axis), a cross section of the first curved surface of the aspherical lens can be generated. Also, the first curved surface of the aspherical lens can be generated by rotating it about the optical axis (y-axis).

In the second curved surface generation step (S2), the algorithm function represented by the formula (4) is used so that the intersection point of the optical axis of the aspherical lens and the light ray is the same as the focal length specified in advance. A second curved surface is generated by calculating the curvature of the second curved surface. As in the first curved surface generation step (S1), when the aspherical lens is a symmetrical optical system in which the lens shape is bilaterally symmetrical on a cross section perpendicular to the lens including the optical axis, Preferably, the curvature C of the second curved surface is calculated in the direction and rotated about the optical axis to generate the second curved surface. This can reduce the computational complexity of the algorithm function for generating the second curved surface.

FIG. 4 is a diagram for explaining a calculation example of the algorithm function in the second curved surface generation step. In FIG. 4, in the second curved surface generating step (S2), first, the curvature of the second curved surface 22 is set to be the same as the curvature of the first curved surface 21, and then the curvature of the second curved surface 22 is changed. The solution of the algorithm function is converged so that the point of intersection between the ray and the light ray is the same as the focal length specified in advance, and the second curved surface 22 is generated. Also, ray tracing can be obtained by inputting the refractive index of the lens material.

Further, in the second curved surface generation step (S2), the curvature of the second curved surface 22 is calculated using an evaluation function represented by the sum of the intersection of the optical axis of the aspherical lens and the light ray and the focal length specified in advance. After making the curvature the same as that of the first curved surface 21, the curvature of the second curved surface 22 may be changed to generate the second curved surface 22 so as to minimize the evaluation function.

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 result of tracing light rays with respect to the aspherical lens. FIG. 6A is a diagram showing the cross-sectional shape of an aspherical lens that achieves a focal length of 5, and FIG. 6B is a diagram showing the result of tracing light rays with respect to the lens.

According to such an aspherical lens design method, since an algorithm function is used, calculation can be easily performed. In addition, partial correction and smoothing can be performed by controlling the curvature, which is a physical representation of the shape. In addition, the design method of the aspherical lens described above can also be applied to the design of a Pourell lens that generates uniformly distributed line light by diverging collimated light in a fan shape in one direction. In addition, by using the design data obtained by such a design method, the lens can be manufactured by controlling the curvature by processing the lens by a press method using a mold or the like or a direct processing method by grinding and polishing. can be done.

[Generation method of specified Gaussian surface]
Gaussian curvature and mean curvature are represented by equations (19-1) and (19-2), respectively, and E, F, and G are equations that do not include z _{i, j,} so they are represented by equation (20) be able to. (21-1), (21-2), and (21-3) for α, β, and γ, respectively, the equation (22) is obtained, which can be solved by a quadratic equation of z _i,j .

(22) where κ ₁ (x _i , y _i ) κ ₂ (x _i , y _i ) is the value of the specified Gaussian curvature 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. can handle.

Next, a method of obtaining a curved surface obtained by cutting a spherical surface of radius R with a prism of 2L using an algorithm function will be examined. FIG. 7A is a diagram showing the XY plane of the sphere and prism, and FIG. 7B is a diagram showing the XZ plane of the sphere and prism. The calculation result of the algorithm function when R=10 and L=5 had a maximum error of 0.0 and a minimum error of −0.0014 for 20×20 divisions. In the case of 200×200 division, the maximum error was 0.0 and the minimum error was −0.00042.

FIG. 8 is a diagram showing a program example of a curved surface with Gaussian curvature. This program can perform one calculation at all grid points. A surface with Gaussian curvature like this can be calculated with a simple program.

Next, consider creating a surface with a specified Gaussian and mean curvature. FIG. 9 shows a grid specifying Gaussian curvature and a grid specifying mean curvature. As shown in FIG. 9, the black circle a designates the Gaussian curvature and the white circle b designates the average curvature to obtain the algorithm function.

FIG. 10(A) is a graph showing the minimum curvature and maximum curvature when the boundary line is straight, and FIG. 10(B) is a graph showing the Gaussian curvature and average curvature when the boundary line is straight. be. In FIG. 10A, a indicates the minimum curvature and b indicates the maximum curvature, and in FIG. 10B, c indicates the Gaussian curvature and b indicates the average curvature. Further, FIG. 11A is a graph showing the minimum curvature and maximum curvature when the boundary line is an arc, and FIG. 11B shows the Gaussian curvature and average curvature when the boundary line is an arc. graph. In FIG. 11A, a indicates the minimum curvature and b indicates the maximum curvature, and in FIG. 11B, c indicates the Gaussian curvature and b indicates the average curvature.

From the graphs shown in FIGS. 10 and 11, it can be seen that the conditions specified at each point are satisfied, and when the grid spacing is reduced and converged, the curved surface has the specified curvature. .

[How to correct polygonal lines]
Next, the correction of a two-dimensional polygonal line will be considered. FIG. 12 is a diagram for explaining a correction method for moving the end point _Pn of the polygonal line by Δy in the y direction.

Let P _i (x _i , y _i ), 0≦i≦n be the vertex of the two-dimensional polygonal line. Also, the intervals of x _i are equal intervals, and are assumed to be Δx. Consider the problem of obtaining a smooth curve b by moving the end point _Pn in the y direction by Δy with respect to the polygonal line a. It is also assumed that P ₀ to P _k do not change.

Assuming that the polygonal line x _i is a function f(x _i ) on x _i , a discrete second-order differential f''(x _i ) can be defined at x _i . If Δx is evenly spaced, the equation (23) is obtained. A new algorithm function to be satisfied by the function g(x _i ) on x _i is defined by the equations (24-1) and (24-2), and α is found so as to satisfy the equation (25). α can be calculated by using the formula of discrete differentiation and geometry, resulting in the formula (26). Since this operation is weighted by 1 up to x _k and multiplied by (1+α(x−x _k )) from x _k , the continuity of the second derivative is ensured.

FIG. 13A is a graph showing the calculation result of the curve, and FIG. 13B is a graph showing the calculation result of the radius of curvature of the curve. As shown in FIG. 13(A), the corrected curve b is scaled about twice as much as the uncorrected curve a in the Y direction, and as shown in FIG. The radius of curvature of the corrected curve b changes greatly with respect to the radius of curvature of a. Since there is no point where the curvature vector is 0, there is no point of inflection. For this reason. It can be seen that the convexity can be maintained and corrected.

[Correction method by moving the specified point of a parabola with both ends fixed]
Next, consider the movement of a specified point of a parabola with both ends fixed. FIG. 14 is a diagram for explaining a method of moving the designated point _Pk by Δy in the y direction with respect to a parabola whose both ends are fixed. In FIG. 14, a is the parabola before movement, and b is the parabola after movement.

First, curves satisfying the formulas (27-1), (27-2) and (27-3) are obtained. A new algorithm function to be satisfied by the function g(x _i ) on x _i is defined by the formulas (28-1), (28-2) and (28-2), and the formulas (29-1) and (29 -2) Determine α and β so as to satisfy the formula. α and β can be specifically calculated, and are given by equations (30-1) and (30-2).

FIG. 15A is a graph showing the calculation result of the parabola, and FIG. 15B is a graph showing the calculation result of the radius of curvature of the parabola. In FIG. 15(A), a is a parabola with 100 between P0 and P1 and -10 at position 50, and b is a -2 shift in the y direction at position 60 with respect to curve a. is a parabola. 15B, a indicates the radius of curvature of the parabola a shown in FIG. 15A, and b indicates the radius of curvature of the parabola b shown in FIG. 15B. From FIG. 15B, it can be seen that the radius of curvature greatly changes at the position of 60, which is the designated point.

[Method of modifying a curve to make a straight line from the specified point to the end]
16A and 16B are diagrams for explaining a method of correcting a curve to form a straight line from a specified point to an end. First, curves satisfying the formulas (31-1), (31-2) and (31-3) are obtained. The curve is corrected linearly from the specified point _k1 . In order for the curvature to be continuous, there must be a connecting portion of the curvature. For this reason, the correction part is divided into three parts, and the function values of formulas (32-1), (32-2), and (32-3) are obtained for the second-order differentiation of the original point sequence. Multiply by the curvature weighting function K(x). The K(x) function is defined in (34) by α, and adjusting α to pass through specified points at the endpoints allows this modification.

[Method for smoothing B-spline curved surface]
A method for smoothing a B-spline curved surface, which is shown as a specific example, regards the B-spline curved surface as a manifold, performs second-order polynomial approximation at each point, and calculates a second-order differential. These curved surfaces with second-order derivatives are reproduced by algorithm functions.

First, the B-spline curved surface is converted into polygons (STL conversion) with an error of, for example, 0.001 mm, and covered with, for example, 121×121 lattice points. Second-order polynomial approximation is then performed at the grid points.

Specifically, a second-order polynomial approximation is performed using STL vertices having a radius R or less centering on a lattice point. Since the STL vertices lie on the original curved surface, a relatively accurate approximation can be given. Considering the B-spline curved surface as a manifold, it is covered with a quadratic polynomial at the lattice points.

Subsequently, G _ij (i, j) is calculated by multiplying the second-order differentials in the X direction and the Y direction, and the algorithm function is calculated using the relational expression in the lattice as the formula (35).

FIG. 17A is a graph showing the calculation result of the second derivative in the X direction using the grid points before smoothing, and FIG. It is a graph which shows the calculation result of differentiation. Further, FIG. 18A is a graph showing the calculation result of the second derivative in the X direction using the grid points after smoothing, and FIG. It is a graph which shows the calculation result of the second derivative of . A body part with a size of 1400 mm in the horizontal (X) direction and 1000 mm in the vertical (Y) direction, which was generated by a B-spline curved surface, was finely corrected using an algorithm function to smooth the second derivative.

As can be seen from the graphs shown in FIGS. 17 and 18, there is noise before the correction, but it becomes smooth after the smoothing. Also, the amount of movement of the vertex is as small as 0.402 mm, and it can be seen that the second derivative can be smoothed by a small correction.

（７）
陽関数で表される曲線を用いて表した曲率の微分方程式を離散化して得られる関数を用い、境界を除くすべての格子点ｉについて曲率Ｃを指定して、格子点ｉでの曲線の値を更新し、それを繰り返すことにより第１曲面の曲線を得て、当該曲線を光軸に対して回転させて非球面レンズの第１曲面を生成する第１曲面生成部と、
前記関数を用い、前記非球面レンズの光軸と光線との交点と、予め指定された焦点距離とが同じとなるように、境界を除くすべての格子点ｉについて曲率Ｃを指定して、格子点ｉでの曲線の値を更新し、それを繰り返すことにより第２曲面の曲線を得て、当該曲線を光軸に対して回転させて非球面レンズの第２曲面を生成する第２曲面生成部とを備える情報処理装置。
（８）
陽関数で表される曲線を用いて表した曲率の微分方程式を離散化して得られる関数を用い、境界を除くすべての格子点ｉについて曲率Ｃを指定して、格子点ｉでの曲線の値を更新し、それを繰り返すことにより第１曲面の曲線を得て、当該曲線を光軸に対して回転させて非球面レンズの第１曲面を生成する第１曲面生成工程と、
前記関数を用い、前記非球面レンズの光軸と光線との交点と、予め指定された焦点距離とが同じとなるように、境界を除くすべての格子点ｉについて曲率Ｃを指定して、格子点ｉでの曲線の値を更新し、それを繰り返すことにより第２曲面の曲線を得て、当該曲線を光軸に対して回転させて非球面レンズの第２曲面を生成する第２曲面生成工程とを有する処理をコンピュータに実行させるプログラム。
（９）
（８）に記載のプログラムを記録した記録媒体。 In addition, the present technology can be expressed as (1) to (29) below.
(1)
Using the function obtained by discretizing the differential equation of the curvature expressed using the curve represented by the explicit function, specifying the curvature C for all grid points i except the boundary, the value of the curve at the grid point i a first curved surface generation step of obtaining a curve by repeating the update and rotating the curve with respect to the optical axis to generate a first curved surface of the aspherical lens;
Using the function, the curvature C is specified for all grid points i excluding the boundary so that the intersection of the optical axis of the aspherical lens and the light ray is the same as the focal length specified in advance, and the grid a second curved surface generation step of updating the value of the curve at the point i, repeating it to obtain a curve, and rotating the curve about the optical axis to generate the second curved surface of the aspherical lens. Information processing methods.
(2)
The information processing method according to (1), wherein the function is obtained by discretizing the differential equation of the curvature by a finite difference method.
(3)
The information processing method according to (2), wherein the difference method is central difference.
(4)
In the second curved surface generation step, after specifying the curvature C of the second curved surface to be the same as the curvature C of the first curved surface, the curvature C of the second curved surface is changed so that the optical axis of the aspherical lens and the light ray The information according to any one of (1) to (3), wherein the solution of the function converges to generate the second curved surface so that the intersection point of and the pre-specified focal length are the same Processing method.
(5)
Using an evaluation function represented by the sum of the intersection of the optical axis of the aspherical lens and the light ray and a pre-specified focal length,
In the second curved surface generation step, after specifying the curvature C of the second curved surface to be the same as the curvature C of the first curved surface, the curvature C of the second curved surface is changed so that the evaluation function is minimized. The information processing method according to any one of (1) to (3), wherein the second curved surface is generated.
(6)
The information processing method according to any one of items (1) to (5), which is represented by the following equation (1) when the function specifies curvature κ=C at grid point i.

(7)
Using the function obtained by discretizing the differential equation of the curvature expressed using the curve represented by the explicit function, specifying the curvature C for all grid points i except the boundary, the value of the curve at the grid point i a first curved surface generation unit for generating a first curved surface of an aspherical lens by updating and repeating the process to obtain a curve of a first curved surface, and rotating the curve with respect to the optical axis;
Using the function, the curvature C is specified for all grid points i excluding the boundary so that the intersection of the optical axis of the aspherical lens and the light ray is the same as the focal length specified in advance, and the grid Generate a second curved surface by updating the value of the curve at the point i, repeating it to obtain a curve of the second curved surface, and rotating the curve about the optical axis to generate the second curved surface of the aspherical lens. An information processing device comprising:
(8)
Using the function obtained by discretizing the differential equation of the curvature expressed using the curve represented by the explicit function, specifying the curvature C for all grid points i except the boundary, the value of the curve at the grid point i a first curved surface generation step of obtaining a curve of a first curved surface by updating and repeating the process, and rotating the curve with respect to the optical axis to generate the first curved surface of the aspherical lens;
Using the function, the curvature C is specified for all grid points i excluding the boundary so that the intersection of the optical axis of the aspherical lens and the light ray is the same as the focal length specified in advance, and the grid Generate a second curved surface by updating the value of the curve at the point i, repeating it to obtain a curve of the second curved surface, and rotating the curve about the optical axis to generate the second curved surface of the aspherical lens. A program that causes a computer to execute a process having steps.
(9)
A recording medium recording the program according to (8).

REFERENCE SIGNS LIST 10 information processing device 11 CPU 12 storage device 13 input unit 14 output device 21 first curved surface 22 second curved surface

Claims (7)

The computer divides the x direction into i equal parts and the y direction into j equal parts on the curved surface z = f (x, y) in the three-dimensional space, and uses the quadratic equation of Z _{i, j} shown in the following formula (A) Information for generating a curved surface based on Gaussian curvature κ ₁ ( _{xi ,} yi ₎ κ ₂ (xi , _yi ) specified for all lattice points i ( xi _, yi ₎ excluding the boundary of the curved surface Processing method.

2. The information processing method according to claim 1, wherein the grid is narrowed and converged to generate a curved surface having the specified Gaussian curvature. 3. The information processing method according to claim 1, wherein said boundary is a straight line. 3. The information processing method according to claim 1, wherein said boundary is an arc. In a curved surface z = f (x, y) in a three-dimensional space, using the quadratic equation of Z i, j shown in the following equation (A) in which the x direction is divided into i and the y direction is divided into j _, An information processing device that generates _a curved surface based on Gaussian curvature κ ₁ ( _{xi , yi} ) κ ₂ (xi _{, yi} ) specified for all lattice points i ( xi, yi ) excluding boundaries.

In a curved surface z = f (x, y) in a three-dimensional space, using the quadratic equation of Z i, j shown in the following equation (A) in which the x direction is divided into i and the y direction is divided into j _, A process for generating a curved surface based on the Gaussian curvature κ ₁ ( _{xi , yi} ) κ ₂ (xi _{, yi} ) specified for all grid points i ( xi, yi ₎ excluding the boundary is performed by a computer. program to run.

A recording medium on which the program according to claim 6 is recorded.

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