JP7475591B2 - Information processing method, information processing device, program, and recording medium - Google Patents

Description

Patent Law Article 30, Paragraph 2 applied. Disclosed at the 45th VCAD System Research Meeting on February 24, 2017 (Method of generating physical surfaces using algorithm functions)

This 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.

Conventionally, three-dimensional parts and the like are designed and analyzed using the computational, memory, and analytical capabilities 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 required 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).

JP 2004-021474 A

One method for drawing curves in the above-mentioned CAD and the like is the B-spline curve. However, because a B-spline curve is expressed using multiple control points and B-spline basis functions, it is difficult to modify it using a physically meaningful shape expression or control method.

This technology was proposed in light of the current 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.

As a result of extensive research, the inventors of the present application have found that the above-mentioned problems can be solved by the following technique, which uses functions introduced to discretize differential equations having physical meaning and express local relationships regarding physical phenomena.

[1]
A computer converts the B-spline surface into a triangular polygon, and covers the triangular polygon with lattice points obtained by dividing a surface z=f(x, y) in a three-dimensional space into i equal parts in the x direction and j equal parts in the y direction;
For each lattice point, a quadratic polynomial P is calculated using the vertices of the triangular polygon having a radius of R or less and centered on the lattice point. _i.j. Approximate by (i, j),
The quadratic polynomial P _i.j. A function G shown in the following formula (A) multiplied by the second derivative of (i, j) in the X direction and the Y direction _i.j. An information processing method that uses (i, j) to make small corrections to smooth the second derivative and reproduce a surface having the second derivative.
[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 ), curvature k(x _i ), and the second derivative of the specified point P _i Using ", 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.
[3]
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 of the specified point P _i Using ", 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 Let 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 α is calculated as shown in the following formula (H) so as to pass through
An information processing method for correcting a parabola into a straight line from a specified point to its end.
[4]
The computer determines the starting point P ₀ From the specified point P _k A parabola up to a specified point P _k From end point P _n For a straight line up to x, divide the straight 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.
[5]
A computer calculates z by using a function shown in the following formula (P) obtained by discretizing the following formula (N) obtained by dividing the Poisson equation shown in the following formula (M) into i equal parts in the x direction and j equal parts in the y direction using the following formula (O-1) and the following formula (O-2) by a finite difference method. _i,j An information processing device that repeats the process of calculating and determines the converged result as the solution.
[6]
The computer divides the heat conduction equation shown in the following formula (Q) into i equal parts in the x direction and j equal parts in the y direction into the following formula (R): _i,j This is an information processing method in which the process of calculating the above is repeated and the converged result is regarded as the solution.
[7]
An information processing method in which a computer determines the velocity and pressure related to a point of interest using relational equations relating to the velocity and pressure at 12 points around the point of interest, using functions shown in the following equations (X), (Y), and (Z) obtained by discretizing the Navier-Stokes equations shown in the following equations (U), (V), and (W) using a finite difference method, and determines the minimum value of a total of 14 relational equations as a solution.
[8]
The B-spline surface is converted into a triangular polygon, and the triangular polygon is covered with lattice points obtained by dividing the three-dimensional surface z=f(x, y) into i equal parts in the x direction and j equal parts in the y direction;
For each lattice point, a quadratic polynomial P is calculated using the vertices of the triangular polygon having a radius of R or less and centered on the lattice point. _i.j. Approximate by (i, j),
The quadratic polynomial P _i.j. A function G shown in the following formula (A) multiplied by the second derivative of (i, j) in the X direction and the Y direction _i.j. An information processing device that uses (i, j) to perform minute corrections to smooth the second derivative and reproduce a curved surface having the second derivative.
[9]
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 Using ", 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.
[10]
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 of the specified point P _i Using ", 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 Let 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 α is calculated as shown in the following formula (H) so as to pass through
An information processing device that corrects a parabola into a straight line from a specified point to its end.
[11]
The computer determines the starting point P ₀ From the specified point P _k A parabola up to a specified point P _k From end point P _n For a straight line up to x, divide the straight 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.
[12]
The following formula (N) is obtained by dividing the Poisson equation shown in the following formula (M) into i equal parts in the x direction and j equal parts in the y direction. The following formula (O-1) and formula (O-2) are used to discretize the equation by the finite difference method to obtain the following formula (P): _i,j An information processing device that repeats the process of calculating and determines the converged result as the solution.
[13]
The following equation (R) is obtained by dividing the x direction of the heat conduction equation shown in the following equation (Q) into i equal parts and the y direction into j equal parts. The following equation (S) and the following equation (S') are discretized by the difference method to obtain the function shown in the following equation (T), and the function is calculated by using the past data in terms of time. _i,j An information processing device that repeats the process of calculating and determines the converged result as the solution.
[14]
An information processing method in which the Navier-Stokes equations shown in the following formulas (U), (V), and (W) are discretized by a finite difference method to determine the velocity and pressure related to a point of interest using relational equations related to the velocity and pressure at 12 points around the point of interest, and the minimum values of a total of 14 relational equations are used as solutions.
[15]
A program for causing a computer to execute the information processing method according to any one of claims [1] to [7].
[16]
A recording medium on which the program according to claim [15] is recorded.

According to the present technology, since a function is used to express a local relationship regarding a physical phenomenon, correction can be made using a physically meaningful shape expression and control method .

FIG. 1 is a block diagram showing an outline of the configuration of an information processing device. FIG. 2 is a flowchart showing a method for designing an aspheric lens. FIG. 3 is a diagram for explaining an example of calculation of an algorithm function in the first surface generating step. FIG. 4 is a diagram for explaining an example of calculation of the algorithm function in the second surface generating step. FIG. 5A is a diagram showing the cross-sectional shape of an aspherical lens that realizes a focal length of 10, and FIG. 5B is a diagram showing the results of tracing rays through the aspherical lens. FIG. 6A is a diagram showing the cross-sectional shape of an aspheric lens that realizes a focal length of 5, and FIG. 6B is a diagram showing the results of tracing rays through the lens. FIG. 7A is a diagram showing the XY plane of the sphere and the rectangular prism, and FIG. 7B is a diagram showing the XZ plane of the sphere and the rectangular prism. FIG. 8 is a diagram showing an example of a program for a surface having a Gaussian curvature. FIG. 9 is a diagram showing a lattice specifying a Gaussian curvature and a lattice specifying a mean curvature. FIG. 10A is a graph showing the minimum curvature and maximum curvature when the boundary line is a straight line, and FIG. 10B is a graph showing the Gaussian curvature and mean curvature when the boundary line is a straight line. 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 mean curvature when the boundary line is an arc. FIG. 12 is a diagram for explaining a correction method for moving the end point _Pn of a broken line by Δy in the y direction. 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. FIG. 14 is a diagram for explaining a method of moving a specified point _Pk by Δy in the y direction on a parabola with both ends fixed. FIG. 15A is a graph showing the calculation results of a parabola, and FIG. 15B is a graph showing the calculation results of the radius of curvature of the parabola. FIG. 16 is a diagram for explaining a method of correcting a curve to make it linear from a specified point to its end point. 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. 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.

Hereinafter, embodiments of the present technology will be described in detail in the following order.
1. Overview of the Technology 2. Information Processing Device 3. Specific Examples

1. Overview of the technology
This technology discretizes differential equations (hereinafter also referred to as "equations with physical meaning") that are expressed by 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.

First, let's organize the symbol system. We define the difference operator, derivatives, partial derivatives, etc. as shown in the formula below. There is no need for complicated discussions as there are no limit operations.

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 constructing the theory.

Next, we derive algorithmic functions. Algorithmic functions are functions introduced to express local relationships in physical phenomena. In a general definition, for each vertex of a broken line or polyhedron, multiple relational equations that must be satisfied by the data associated with each vertex and its adjacent vertices are specified. This is based on the finding that many physical phenomena can be recognized when local relationships are in harmony as a whole.

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

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.

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

Below, we explain some examples of equations that have physical meaning and are replaced with algorithmic functions.

[Differential equation of curvature]
When the curvature κ is expressed using a curve expressed by an 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).

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 representation.

[Poisson equation]
In the Poisson equation expressed as formula (5), when the x direction is divided into i equal parts and the y direction is divided into j equal parts, it is expressed as formula (6). When formula (6) is discretized and expressed using the central difference of formulas (7-1) and (7-2), it becomes an algorithm function as formula (8).

In the normal 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 solution is taken 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/destruction exists.

[Heat conduction equation]
In the heat conduction equation expressed as 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 as formula (10). When formula (10) is discretized and expressed using the central difference of formulas (11-1) and (11-2), it becomes an algorithm function as formula (12).

This algorithm function repeats a process of calculating using past data in terms of 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.

[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 differencing, the resulting algorithm functions are shown in equations (16), (17), and (18).

This algorithm function only needs to satisfy equations (16), (17), and (18) at each point. Here, 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 surrounding the point of interest. Therefore, the solution of the algorithm function is found as the minimum value of a total of 14 relational equations.

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

Algorithm functions are also easy to handle, and can be calculated using a computer in a simple manner. In particular, the calculation process of algorithm functions relies only on local information, so there is no need to allocate large-capacity memory or virtual memory, and no specialized library is required to solve large-scale linear equations.

Algorithm functions can also be applied to inverse analysis problems, in which the coefficients and boundary conditions of differential equations are changed to find the optimal solution to a problem.Also, since algorithm functions are defined by the adjacent relationships of information at each point, they have the ability to be expanded, such as by reducing the grid spacing and, when convergence occurs, resulting in a curved surface with the specified curvature.

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.

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 generation 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 generation 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 the light ray is the same as a pre-specified focal length and generate the second curved surface.

The storage device 12 is composed of a primary storage device such as a random access memory (RAM) and a secondary storage device such as a hard disk drive (HDD) or a solid state drive (SSD), 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.

The input unit 13 is a user interface, such as a touch panel or a keyboard, for inputting various data. 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.

The output device 14 is, for example, a display or a printer, and is capable of displaying, printing, outputting, and other processing of the calculation results of the algorithm function.

The function of calculating the algorithm function is not limited to the configuration of the information processing device described above, and can be configured, for example, by multiple hardware devices such as terminal devices and server devices, and can 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 to implement this function and implementing it on a computer. It is also possible to provide a computer-readable recording medium on which such a computer program is stored. Examples of the recording medium include a magnetic disk, optical disk, magneto-optical disk, and flash memory. In addition, the computer program can be distributed, for example, via a network, without using a recording medium.

<Specific examples>
Next, a specific example of information processing using the above-mentioned algorithm function will be described.

[Aspherical 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 generation step (S1) for generating a first curved surface using a function obtained by discretizing a differential equation of curvature, and a second curved surface generation 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.

When the curvature κ = C at grid point i is specified, the function obtained by discretizing the curvature differential equation is the algorithm function expressed by the following equation (4).

In the first curved surface generation step (S1), the curvature of the first curved surface of the aspherical lens is specified to generate the first curved surface using an algorithm function expressed by equation (4). If the aspherical lens is a symmetrical optical system in which the lens shape is symmetrical on a cross section perpendicular to the lens and 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 aspherical lens and generate the first curved surface by rotating about the optical axis. This makes it possible to reduce the amount of calculation of the algorithm function that generates the first curved surface.

Figure 3 is a diagram for explaining an example of the calculation of the algorithm function in the first curved surface generation process. In Figure 3, in the first curved surface generation process (S1), the curvature C of each lattice point i is specified from x = 0 to x = a, and the value of yi is repeatedly updated to obtain a curve. Then, by inverting it about the optical axis (y-axis), the cross section of the first curved surface of the aspheric lens can be generated. In addition, by rotating it about the optical axis (y-axis), the first curved surface of the aspheric lens can be generated.

In the second curved surface generation step (S2), the curvature of the second curved surface is calculated using an algorithm function expressed by equation (4) so that the intersection of 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 generation step (S1), if 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 that generates the second curved surface.

Figure 4 is a diagram for explaining an example of the calculation of the algorithm function in the second curved surface generation step. In Figure 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 is the same as the 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.

In addition, in the second curved surface generation step (S2), an evaluation function expressed as the sum of the intersection point between the optical axis of the aspheric lens and the 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.

Figure 5(A) shows the cross-sectional shape of an aspherical lens that achieves a focal length of 10, and Figure 5(B) shows the results of tracing rays through the aspherical lens. Also, Figure 6(A) shows the cross-sectional shape of an aspherical lens that achieves a focal length of 5, and Figure 6(B) shows the results of tracing rays through the lens.

This method of designing an aspheric lens uses an algorithm function, making it easy to calculate. In addition, partial corrections and smoothing can be performed by controlling the curvature, which is a physical shape expression. The above-mentioned method of designing an aspheric lens can also be applied to the design of a Paulel lens, which generates a uniformly distributed line light by diverging collimated light in a fan shape in one direction. In addition, by using the design data obtained by this design method to process the lens using a pressing method using a mold or a direct processing method using grinding and polishing, it is possible to manufacture a lens by controlling the curvature.

[How to generate a specified Gaussian surface]
The Gaussian curvature and the mean curvature are expressed by equations (19-1) and (19-2), respectively, and E, F, and G are equations that do not include z _{i and j} , so they can be expressed by equation (20). If α, β, and γ are expressed by equations (21-1), (21-2), and (21-3), respectively, we obtain equation (22), which can be solved as a quadratic equation in z _{i and j} .

In equation (22), _k1 ( _xi , _yi ) _k2 ( _xi , _yi ) are the specified Gaussian curvature values, and ^∂2zi _,j / _∂xi∂yi is a value _determined by ( _zi+1,j+1 -zi _+1,j-1 -zi _-1,j+1 -zi _-1,j-1 )/4ΔxΔy. Since there is no term for zi _,j , it can be treated as a constant.

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. Figure 7(A) is a diagram showing the XY plane of the sphere and rectangular prism, and Figure 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 20 x 20 division, the maximum error was 0.0 and the minimum error was -0.0014. Furthermore, in the case of 200 x 200 division, the maximum error was 0.0 and the minimum error was -0.00042.

Figure 8 shows an example 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 using a simple program.

Next, we consider the creation of a surface with a 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 black circle a specifies the Gaussian curvature, and the white circle b specifies the mean curvature to find the algorithm function.

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

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 and converging, a surface with the specified curvature is obtained.

[How to correct broken lines]
Next, the correction of a two-dimensional broken line will be considered. 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.

The vertices of a two-dimensional broken line are defined as _Pi ( _xi , _yi ), 0≦i≦n. The intervals between x _i are equal and Δx. Consider the problem of finding a smooth curve b by moving the end point _Pn of the broken line a by Δy in the y direction. Also, assume that there is no change from _P0 to _Pk .

If we regard the broken line x _i as a function f(x _i ) on x _i , then we can define the discrete second derivative f''(x _i ) at x _i . If Δx is equally spaced, we obtain equation (23). The new algorithm function that function g(x _i ) on x _i must satisfy is defined by equations (24-1) and (24-2), and α is found to satisfy equation (25). Using the formulas of discrete differentiation and geometry, α can be calculated, resulting in equation (26). This operation multiplies the weight by 1 up to x _k , and by a weight of (1 + α(x - x _k )) from x _k onwards, so it guarantees the continuity of the second derivative.

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

[Modification method by moving a specified point of a parabola with both ends fixed]
Next, the movement of a designated point of a parabola with both ends fixed will be considered. 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. In Fig. 14, a is the parabola before the movement, and b is the parabola after the movement.

First, a curve satisfying equations (27-1), (27-2), and (27-3) is obtained. The new algorithm function that function g(x _i ) on x _i should satisfy is defined by equations (28-1), (28-2), and (28-2), and α and β are determined so as to obtain equations (29-1) and (29-2). α and β can be calculated specifically, and are given by equations (30-1) and (30-2).

Figure 15(A) is a graph showing the calculation results of a parabola, and Figure 15(B) is a graph showing the calculation results of the radius of curvature of a parabola. In Figure 15(A), a is a parabola where the value is 100 between P0 and P1 and -10 at position 50, and b is a parabola that has been moved -2 in the y direction at position 60 relative to curve a. Also, in Figure 15(B), a shows the radius of curvature of parabola a shown in Figure 15(A), and b shows the radius of curvature of parabola b shown in Figure 15(B). It can be seen from Figure 15(B) that the radius of curvature changes significantly at the specified point, position 60.

[How to modify a curve to make it a straight line 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. First, a curve that satisfies the formulas (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 modified portion 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 the formulas (32-1), (32-2), and (32-3). The K(x) function is defined by α in formula (34), and this modification is possible by adjusting α so that it passes through the specified point at the end point.

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

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

Specifically, a quadratic polynomial approximation is performed using STL vertices with a radius of R or less, centered on the lattice 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 lattice points.

Next, G _ij (i, j) is calculated by multiplying by the second derivative in the X direction and the Y direction, and the algorithm function is calculated with the relational expression in the lattice being the expression (35).

Figure 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 Figure 17(B) is a graph showing the calculation results of the second-order differential in the Y direction using the grid points before smoothing. Also, Figure 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 Figure 18(B) is a graph showing the calculation results of the second-order differential in the Y direction using the grid points after smoothing. The second-order differential was smoothed by making small corrections using an algorithm function to the B-spline surface generated for a body part measuring 1400 mm in the horizontal (X) direction and 1000 mm in the vertical (Y) direction.

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.

（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。 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 of 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 by a finite difference method.
(3)
The information processing method according to claim 2, wherein the difference method is a central difference method.
(4)
The information processing method according to any one of (1) to (3), in which, in the second curved surface generation step, a curvature C of the second curved surface is specified to be the same as a curvature C of the first curved surface, and then the curvature C of the second curved surface is changed, and a solution of the function is converged so that an intersection of an 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), in the second surface generation process, after the curvature C of the second surface is specified to be the same as the curvature C of the first surface, the curvature C of the second surface is changed, and the second surface is generated 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.

(7)
a first curved surface generating unit that uses a function obtained by discretizing a differential equation of curvature expressed using a curve expressed by an explicit function, specifies a curvature C for all lattice points i except for boundaries, updates the value of the curve at the lattice point i, and repeats this process to obtain a curve of a first curved surface, and rotates the curve about the optical axis to generate the first curved surface of the aspheric lens;
and a second curved surface generation unit that uses the function to specify a curvature C for all lattice points i except for 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, updates the value of the curve at the lattice point i, and repeats this process to obtain a curve of a second curved surface, and rotates the curve about the optical axis to generate the second curved surface of the aspherical lens.
(8)
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 curvature differential equation 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 of a first curved surface, and rotating the curve about the optical axis to generate the first curved surface of the aspheric lens;
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, obtaining a curve of a second curved surface by repeating this process, and rotating the curve about the optical axis to generate the second curved surface of the aspherical lens.
(9)
A recording medium having the program described in (8) recorded thereon.

10 Information processing device, 11 CPU, 12 Storage device, 13 Input unit, 14 Output device, 21 First curved surface, 22 Second curved surface

Claims (4)

A computer converts the B-spline surface into a triangular polygon, and covers the triangular polygon with lattice points obtained by dividing a surface z=f(x, y) in a three-dimensional space into i equal parts in the x direction and j equal parts in the y direction;
For each lattice point, approximation is performed using a quadratic polynomial P _ij (i, j) using the vertices of the triangular polygon having a radius of R or less and centered on the lattice point;
The information processing method uses a function G _ij (i, j) shown in the following equation (A) obtained by multiplying the second-order differentials of the quadratic polynomial P _ij (i, j) in the X and Y directions to perform minute correction to smooth the second-order differentials, thereby reproducing a curved surface having second-order differentials.

The B-spline surface is converted into a triangular polygon, and the triangular polygon is covered with lattice points obtained by dividing the three-dimensional surface z=f(x, y) into i equal parts in the x direction and j equal parts in the y direction;
For each lattice point, approximation is performed using a quadratic polynomial P _ij (i, j) using the vertices of the triangular polygon having a radius of R or less and centered on the lattice point;
The information processing device smoothes the second-order differential by minutely correcting it using a function G _ij (i, j) shown in the following equation (A) obtained by multiplying the second-order differential of the quadratic polynomial P _ij (i, j) in the X direction and the Y direction, and reproduces a curved surface having the second-order differential.

A program for causing a computer to execute the information processing method according to claim 1 . A recording medium on which the program according to claim 3 is recorded.

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