CN112036006A - Illumination light source design method based on multiple diffuse reflection mathematical model - Google Patents

Abstract

The invention provides a design method of an LED uniform illumination light source based on a mathematical model of multiple diffuse reflection of a diffuse reflection free surface. Diffuse reflection lenses have been widely used because of their unique optical properties, but the efficiency of diffuse reflection lenses is low, so their efficiency and uniformity need to be improved. Based on the above problems, according to the luminous characteristics of the LED light source, a bidirectional reflection distribution function (BRDF) is applied to establish a primary diffuse reflection mathematical model for the diffuse reflection surface. Similarly, when the LED light is reflected to the diffuse reflection surface, the diffuse reflection surface is also a Lambertian body, which considers the situation of multiple diffuse reflections and establishes a corresponding mathematical model. The diffuse surface profile is obtained by numerically solving the nonlinear algebraic equations. Finally, the simulation results show that when the radius of the target surface and the distance between the light source and the target lighting surface are 1:4, the effect is the best. Compared with other methods, the uniformity and efficiency are significantly improved, which can effectively broaden the practical application of indirect lighting. prospects in application.

Description

Lighting source design method based on multiple diffuse reflection mathematical model

technical field

The invention belongs to the field of optical technology and lighting technology, and in particular relates to a multiple diffuse reflection free-form surface for redistributing LED light source energy to achieve uniform lighting and high efficiency.

Background technique

As the fourth-generation lighting source, light-emitting diodes (LEDs) have many advantages such as small size, fast response, long life, and no pollution. also increasingly widespread. However, since the spatial light intensity of the LED is similar to the Lambertian cosine distribution, directly using the LED for illumination without secondary light distribution will form a circular light spot with a bright center and gradually darkening around the target surface, with poor uniformity and large It is difficult for the light to reach the target surface with a large field of view, the energy utilization rate is low, and it is easy to generate glare, which is difficult to meet the actual lighting needs. Therefore, in the actual application of LED lighting, it is necessary to carry out a secondary light distribution design for the LED, so that the light energy distribution can meet the actual lighting needs. In recent years, the research of LED lighting has made great progress, especially in the field of reflective lighting and refraction lighting. Catadioptric redistribution of emitted light is an effective way to improve lighting performance. However, conventional mirrors or refractive lenses have high requirements on machining accuracy and materials. The diffuse reflector has the advantages of convenient material, high uniformity, and convenient processing, and has been widely used in daily life, but the efficiency of the diffuse reflector without any design is significantly lower than other designs. Therefore, improving the efficiency and uniformity of diffuse free-form surfaces is still an important research. However, some lighting systems with diffuse reflection free-form surfaces always sacrifice lighting efficiency in order to obtain higher uniformity. In addition, if a large amount of light is reflected or refracted by the mirror surface, it may also cause the phenomenon of glare and discomfort. In severe cases, a series of traffic accidents will occur in some road lighting. Based on the above problems, the present invention proposes a method for designing a free-form surface with multiple diffuse reflections, which not only improves the illumination uniformity in a specific area of the target plane, but also improves the illumination efficiency to a certain extent.

SUMMARY OF THE INVENTION

The invention aims to overcome many defects and constraints existing in the prior art, design a multiple diffuse reflection free-form surface for redistributing the energy of the LED light source, and realize the requirement of uniform illumination in a specific target area. The present invention is achieved through the following technical solutions:

(1) Establish an LED optical system model based on diffuse indirect illumination.

(2) Use a single LED as a point light source and use it as the central origin to establish a Kadir coordinate system.

(3) Construct the mathematical model of the free-form surface diffuse reflection surface for secondary light distribution.

(4) Using numerical methods to solve equations, the surface profiles of free-form surfaces with multiple diffuse reflections were obtained.

CCD camera for capturing images in color vision systems. A single LED light source is used to emit light in different wavelength bands. Multiple diffuse reflection free-form surfaces for diffuse reflection of light emitted by a single LED light source. The target detection surface is used to receive the light after the diffuse reflection of the free-form surface.

The multiple diffuse reflection free-form surface for redistributing light source energy according to the present invention has a central axis-symmetric shape, and the cross section passing through the rotation center axis of the multiple diffuse reflection free-form surface is used as the reference plane to establish a Cartesian coordinate system: the above-mentioned The central axis of rotation is the Y axis, and the direction passing through the origin of the coordinate system and perpendicular to the central axis is the X axis. The double free-form surface is placed directly above the LED light source, and the target plane to be illuminated is parallel to the light emitting plane of the LED light source. in the negative direction of the axis;

In the present invention, the light emitted by a single LED light source will be equally dispersed to all directions after multiple diffuse reflections from a high diffuse reflection free-form surface, so as to realize the reasonable control and distribution of the light intensity, and form on the target detection plane. an evenly illuminated area. The cross-section of the lighting system is shown in Figures 1 and 2.

Based on the principle of rotational symmetry, the present invention can obtain the desired value by rotating the free-form surface curve obtained by the obtained multiple diffuse reflection free-form surface on the XOZ plane around the center normal of the light-emitting surface of the LED light source, that is, the rotation center axis Z-axis. The surface profile of a multi-diffuse free-form surface with high diffuse reflectance.

The free curve is determined by the following steps of programming calculation:

1. Based on the characteristics of LED light source and specular reflection surface, construct a mathematical simulation algorithm for constructing specular reflection free-form surface to horizontally diverge LED light source light, and apply bidirectional reflection distribution function ( BRDF ) to construct diffuse transmission free-form surface to characterize the target plane The mathematical simulation algorithm of irradiance, both algorithms take the coordinates of each discrete point on the desired free curve as an unknown quantity.

The irradiance distribution of an ideal Lambertian LED point light source can be expressed by a cosine function:

为辐射照度，

为LED光源沿其光轴方向上的出射光强，d为LED光源与检测目标之间的距离，

为实际出射光线与光轴间的夹角，m为与LED半衰角

有关的数值，由下式给出：in

is the irradiance,

is the outgoing light intensity of the LED light source along its optical axis, d is the distance between the LED light source and the detection target,

is the angle between the actual outgoing light and the optical axis, m is the half-dead angle with the LED

The relevant values are given by:

由LED芯片生产厂商提供，定义为当出射光线的辐射强度为沿LED光源光轴方向出射强度一半时，该出射光线与光轴之间的夹角。half-life angle

Provided by the LED chip manufacturer, it is defined as the angle between the outgoing light and the optical axis when the radiant intensity of the outgoing light is half of the outgoing intensity along the optical axis of the LED light source.

A Lambertian surface can be defined as a surface with constant illumination in any direction, with a hemispherical reflectivity of 1. Since a hemisphere is a highly diffuse reflective surface, it can be approximated as a Lambertian surface. For Lambertian surfaces, there are:

是在一定发射角∂下的出射光强度。因此，目标平面上的辐照度分布可由以下公式给出：In the formula,

is the outgoing light intensity at a certain emission angle ∂. Therefore, the irradiance distribution on the target plane can be given by:

是由式(3)给出的自由曲面的接收辐照度，

是自由曲面和目标照明面之间的距离，

表示自由曲面上的面积元素。BRDF是双向反射分布函数，用来表示自由曲面的反射特性。可通过以下方式给出：in the formula

is the received irradiance of the free-form surface given by Eq. (3),

is the distance between the free-form surface and the target illumination surface,

Represents an area element on a free-form surface. BRDF is a bidirectional reflection distribution function, which is used to represent the reflection characteristics of free-form surfaces. It can be given by:

是自由曲面的漫反射比。为了简化计算，用所有这些二次源的简单辐照度叠加代替由式(3)给出的目标平面的复积分反射辐照度分布：in

is the diffuse reflectance of the free-form surface. To simplify the calculation, the complex integral reflected irradiance distribution of the target plane given by equation (3) is replaced by the simple irradiance superposition of all these secondary sources:

where N is the number of discrete points in the intersection.

和

。根据空间向量的知识，我们可以得到：When the light exits from the LED light source and reaches the diffuse reflection free-form surface, it reaches the target lighting surface through diffuse reflection again. At this time, only one diffuse reflection is considered. The schematic diagram of the light is shown in Figure 1. A point p on the diffuse free-form surface and a point T on the target plane can be further expressed as xz on the two-dimensional plane

and

. With knowledge of space vectors, we can get:

。利用以上式子建立基于一次漫反射间接照明的辐照分布数学模型，目标检测平面上任意位置

的辐照度可由下一式子进一步表示为：After the ray is diffusely reflected by the free-form surface, a certain outgoing ray intersects the target detection plane at a point

. Using the above formula to establish a mathematical model of the irradiance distribution based on a diffuse indirect illumination, any position on the target detection plane

The irradiance of can be further expressed as:

,光线示意图如2所示。此时，点p类似于一个朗伯体光源，等同地分散到该漫反射自由曲面的各个方向（假设每个辐射点辐射到两个不相邻点），与漫反射自由曲面再次相交于点

，某条出射光线与目标检测平面交于点

。我们可以得到：Similarly, when two diffuse reflections occur, it is assumed that the light is emitted by the LED light source and intersects the inner surface of the high-diffuse free-form surface at the point

, the schematic diagram of the light is shown in 2. At this point, the point p is similar to a Lambertian light source, equally dispersed to all directions of the diffuse free-form surface (assuming that each radiant point radiates to two non-adjacent points), and intersects the diffuse free-form surface again at point

, an outgoing ray intersects the target detection plane at a point

. We can get:

的辐照度可由下一式子进一步表示为：Use the above formula to establish a mathematical model of irradiance distribution based on secondary diffuse reflection indirect illumination, any position on the target detection plane

The irradiance of can be further expressed as:

的辐照度可以由一下式子进一步表示为：In the same way, when three diffuse reflections occur, any position on the target detection plane

The irradiance can be further expressed as:

为未知变量的方程。2. According to the irradiation distribution requirements of the target detection plane in the actual LED optical system, a set of nonlinear algebraic equations for defining the shape of the diffuse reflection free-form curve are established by using the above-mentioned mathematical simulation algorithm. According to the specific requirements for the scope of the illuminated area and its irradiance intensity in practical applications, the distance H of the target detection plane relative to the LED light source and the radius R of the illuminated area in this lighting system are determined. At the same time, the same number of sampling points as the discrete points fixed on the free-form surface are selected on the detection plane. In order to achieve the goal of uniform illumination, let the irradiance distribution expressions (8), (10) and (11) at each sampling point be ) takes the value of 1, so that a set of coordinates of discrete points on the free curve is constructed.

is the equation for the unknown variable.

3. Use the numerical method to program to solve the equation system, and introduce a smoothing algorithm to the obtained discrete point coordinates, and fit a free curve; the specific steps include the following:

① Set initial conditions

，其横、竖坐标变化步长分别为

；初始点坐标及其步长的取值取决于本发明所述的漫反射自由曲面的尺寸大小，因此需要根据实际应用场合中对目标检测平面照明范围及辐照度值的具体要求确定该自由曲面在XOY平面和Z轴上的投影范围，进而完成初始点的设置。Set the coordinates of the initial point of the free curve as

, the horizontal and vertical coordinate change steps are respectively

; The value of the initial point coordinate and its step size depends on the size of the diffuse reflection free-form surface of the present invention, so it is necessary to determine the free surface according to the specific requirements of the target detection plane illumination range and irradiance value in practical applications. The projection range of the surface on the XOY plane and the Z axis, and then complete the setting of the initial point.

② Solve the system of equations by numerical method

，其中

。N的取值越大，得到的自由曲线的离散点也就越多，由这些离散点表征的自由曲线也就越精确。Based on the numerical method, the specular reflection free curve is obtained from the expression, and then the initial value of the coordinates of each discrete point on the free curve is brought into the above non-linear algebraic equation system, and the vertical coordinate is used as the unknown variable to solve iteratively by programming, and a series of Coordinate values of discrete points

,in

. The larger the value of N , the more discrete points of the free curve obtained, and the more accurate the free curve represented by these discrete points.

③ Fitting a smooth curve

Fit the coordinate data of discrete points on the free curve obtained by calculation to obtain a smooth curve; considering that the diffuse reflection free surface is affected by a series of technological levels, economic indicators, etc. during actual processing, a trial-and-error method is used if necessary. , correct the initial value of the vertical coordinates of each discrete point on the arc segment of the free curve, and repeat the solution process in step ② until a surface profile that is convenient for processing and manufacturing is obtained.

The invention provides a design method for a multiple diffuse reflection free-form surface with high diffuse reflectivity for realizing uniform illumination. The light emitting direction from a single LED light source is constrained by the designed diffuse reflection free-form surface diffuse reflection surface to realize the redistribution of its emitted energy. Compared with some existing lighting systems, the present invention has the following advantages and effects: the target lighting surface receives the light after the diffuse reflection of the multiple diffuse reflection free-form surfaces with high diffuse reflectivity, the irradiation uniformity is high, the light is soft, and there is no defect. glare, and the efficiency has been improved to a certain extent. It can be widely used in color image visual measurement, indoor home and many other fields that require uniform lighting conditions. In addition, in the design scheme of the present invention, the radius R of the target lighting surface and the distance H between the target lighting surface and the LED light source are used as the structural parameters of the multiple diffuse reflection free-form surface lighting model. Appropriate lighting system parameters can be selected according to actual application requirements, which expands the prospect of actual indirect lighting.

Description of drawings

Figure 1 is a schematic diagram of the optical path design of multiple diffuse reflection free-form surfaces.

FIG. 2 is a schematic diagram of a diffuse reflection free-form surface with one reflection.

FIG. 3 is a schematic diagram of a diffuse reflection free-form surface with secondary reflection.

FIG. 4 is a schematic diagram of a two-dimensional contour of a diffuse reflection free-form surface and a three-dimensional diffuse reflection free-form surface with one diffuse reflection.

FIG. 5 is a schematic diagram of a two-dimensional contour of a diffuse reflection free-form surface and a three-dimensional diffuse reflection free-form surface with secondary diffuse reflection.

FIG. 6 is a schematic diagram of a two-dimensional outline of a diffuse reflection free-form surface and a three-dimensional diffuse reflection free-form surface with three diffuse reflections.

FIG. 7 is a simulation diagram of the irradiation distribution in the circular illumination area of the target illumination surface.

Detailed ways

The specific implementation of the present invention will be described in detail below with reference to the accompanying drawings. As shown in FIG. 1 , the specific situation of the light path of the diffuse reflection free-form surface when multiple diffuse reflections occur in the present invention is introduced. When the light emitted from the LED light source is incident on the diffuse reflection surface with high diffuse reflectivity, the light will be reflected again to all directions of the diffuse reflection surface. The primary body, diffuse reflection occurs again and exits to the high diffuse reflectance diffuse reflection in other directions or the target receiving surface. In the same way, repeat this step, and then analyze the light outgoing, consider the situation of multiple diffuse reflections, and establish a corresponding mathematical model. After the diffuse reflection free-form surface with high diffuse reflectivity has multiple diffuse reflections, the reasonable control and distribution of light intensity can be realized, and a uniform illumination area can be formed on the target detection plane. The shape of the multiple diffuse reflection free-form surface with high diffuse reflectivity is determined by the following scheme:

为出射夹角的光线由单颗LED光源发出，与高漫反射率的漫反射自由曲面内表面相交于点p,坐标记为

，该光线经过所述的漫反射自由曲面漫反射之后，某条出射光线与目标照明面相交于点T，坐标记为

。为了简化计算，用上式所有的二次源的简单辐照度叠加代替有由式子（4）给出的目标平面的复积分反射辐照度分布可表示为：In this embodiment, a Cartesian coordinate system is established based on the cross section of the rotation center axis of the multiple diffuse reflection free-form surface with high diffuse reflectivity as the reference plane, and the above-mentioned rotation center axis is the Z axis, passing through the origin of the coordinate system and perpendicular to the center axis The direction is the X axis. Based on the principle of rotational symmetry, in order to simplify the calculation problem, the three-dimensional case is converted into a two-dimensional case. Suppose with

The light at the exit angle is emitted by a single LED light source, and intersects with the inner surface of the diffuse reflection free-form surface with high diffuse reflectance at point p , and the coordinates are marked as

, after the ray is diffusely reflected by the diffuse reflection free-form surface, a certain outgoing ray intersects the target illumination surface at point T, and the coordinates are marked as

. In order to simplify the calculation, the complex integral reflected irradiance distribution of the target plane given by equation (4) can be expressed as:

=30°，则m=4.82。由LED光源发出的光线经过高漫反射率的多次漫反射自由曲面后会均匀地分散到各个方向。应用高漫反射率的多次漫反射自由曲面双向反射分布函数，将用于定义自由曲面的所有离散点看作是朗伯非相关次级点光源，目标照明面上任意位置的

，

，

处的辐射照度可以表示为：In the present invention, the LED half-life angle is taken as

=30°, then m =4.82. The light emitted by the LED light source will be uniformly dispersed in all directions after passing through the multiple diffuse reflection free-form surfaces with high diffuse reflectivity. Applying the bidirectional reflectance distribution function of multiple diffuse reflection free-form surfaces with high diffuse reflectance, all discrete points used to define the free-form surface are regarded as Lambertian non-correlated secondary point light sources, and any position on the target illumination surface is used.

,

,

The irradiance at can be expressed as:

。根据实际应用场合中对被照明区域的范围和其辐射照度的具体要求，确定目标照明面在本照明系统中相对于LED光源的距离H和目标照明面半径R的距离。本实施方式中上述两个仿真参数分别取为H=600 mm，R=200 mm。同时，在被照目标照明面上选取与定义自由曲面的离散点相同数量的采样点，另采样点处的辐照分布表达式（16）、（17）和（18）均取值为1，如此构建三组以自由曲线上各离散点的坐标。Since the free-form surfaces with high diffuse reflectance in this embodiment are all self-made, the diffuse reflectance of the surface is calculated by programming.

. According to the specific requirements for the scope of the illuminated area and its irradiance in practical applications, determine the distance H of the target lighting surface relative to the LED light source and the distance R of the target lighting surface radius in this lighting system. In this embodiment, the above two simulation parameters are respectively taken as H =600 mm and R =200 mm . At the same time, the same number of sampling points as the discrete points defining the free-form surface are selected on the illuminated surface of the illuminated target, and the irradiance distribution expressions (16), (17) and (18) at the other sampling points are all taken as 1, In this way, three groups are constructed with the coordinates of each discrete point on the free curve.

以及横竖坐标变化步长

。基于数值方法，并且借助MATLAB编程迭代求解，得到自由曲线上一系列离散点的坐标值。利用最小二乘法对所求得的坐标数据进行平滑拟合，必要的时候采用试错法，修正自由曲线弧段上各离散点的坐标值，重复以上求解过程，直到获得一个便于加工制造的多次漫反射自由曲面的轮廓。The equation established in the present invention for defining the shape of the free-form curve is a nonlinear algebraic equation. In order to simplify the calculation difficulty, the three-dimensional situation is converted into a two-dimensional situation. First, the initial point coordinates of the free-form curve are set.

And the horizontal and vertical coordinate change step size

. Based on the numerical method and iterative solution with the help of MATLAB programming, the coordinate values of a series of discrete points on the free curve are obtained. Use the least squares method to smoothly fit the obtained coordinate data. If necessary, use the trial-and-error method to correct the coordinate values of each discrete point on the arc segment of the free curve. Repeat the above solution process until you obtain a multi-component that is convenient for processing and manufacturing. The outline of the secondary diffuse free-form surface.

，用于定义所求自由曲线弧段的离散点数目N=57。根据以上步骤，借助MATLAB编程迭代求解方程组并且平滑拟合所得的坐标数据，最终得到自由曲线弧段上一系列的离散点的坐标值：（1，0，5.2），（2，0，5.1），（3，0，5.0）,......,（57，0，1.7）。将该自由曲线绕LED光源发光面中心法线即旋转中心轴Z轴旋转一周，即可得到本发明所述的高漫反射率的多次漫反射自由曲面的面型轮廓（如图4、图5、图6所示）。In this implementation method, the coordinates of the initial point of the free curve are set as: (1, 0, 5.2), and the change steps of the horizontal and vertical coordinates are respectively

, which is used to define the number of discrete points N = 57 for the arc segment of the free curve. According to the above steps, use MATLAB programming to iteratively solve the equation system and smoothly fit the obtained coordinate data, and finally obtain the coordinate values of a series of discrete points on the arc segment of the free curve: (1, 0, 5.2), (2, 0, 5.1 ), (3, 0, 5.0), ..., (57, 0, 1.7). The surface profile of the multiple diffuse reflection free-form surface with high diffuse reflectivity according to the present invention can be obtained by rotating the free-form curve around the center normal of the light-emitting surface of the LED light source, that is, the rotation center axis Z-axis (as shown in Fig. 4 and Fig. 5, as shown in Figure 6).

The solid model of the light source system designed by the present invention is imported into the optical simulation software TracePro for non-sequential ray tracing, and the simulation result is shown in FIG. 7 . Figure 7: The simulation diagram of the irradiation distribution in the circular lighting area of the target lighting surface: it can be seen that after implementing this technical solution, the irradiation uniformity of the target lighting surface can reach 85.36%, compared with 79.89% of the radiation using the traditional diffuse reflection plane The illumination uniformity is obviously improved, and the reasonable control and distribution of the light energy distribution is realized. It can meet the requirements of high uniformity lighting in many fields such as color vision measurement and indoor lighting.

Claims (1)

1. the design method based on the establishment of the multiple diffuse reflection mathematical model of the diffuse reflection free-form surface, is characterized in that, comprises: Build an LED optical system model based on diffuse reflection indirect illumination, including CCD camera, LED light source, diffuse reflection free-form surface and target detection surface; A single LED is regarded as a point light source and a Cartesian coordinate system is established with it as the central origin. The double free-form surface used to redistribute the energy of the light source has a central axis-symmetric shape, and the cross-section passing through the central axis of rotation of the free-form surface is used as the reference plane. Taking the z-axis of the above-mentioned rotation central axis, the direction passing through the origin of the coordinate system and perpendicular to the central axis is the x-axis, the double free-form surface is placed directly above the LED light source, and the target plane to be illuminated is parallel to the light-emitting plane of the LED light source, and in the negative direction of the z-axis; The mathematical model of the diffuse reflection free-form surface used for secondary light distribution is constructed. First, the specular reflection free-form surface diffuses the LED light horizontally and then passes through the diffuse-reflection free-form surface and then equally disperses to all directions of the diffuse-reflection free-form surface to achieve light intensity. Reasonable control and distribution, uniform illumination is formed on the target detection surface, and the free curve intercepted by the double free-form surface on the XOZ plane is rotated around the normal line of the center of the LED light-emitting surface, that is, the z-axis of the central axis of rotation to obtain high diffuse reflectance. The surface profile of the free-form surface, the free-form curve is calculated and determined by the following steps: 1) Based on the characteristics of the LED light source and specular reflection surface, construct a mathematical simulation algorithm for constructing a specular reflection free-form surface to horizontally diverge the light of the LED light source, and apply a bidirectional reflection distribution function (BTDF) to construct a diffuse transmission free-form surface to characterize the target plane. Mathematical simulation algorithm of irradiance, both algorithms take the coordinates of discrete points on the desired free curve as unknown quantities; The irradiance distribution of an ideal Lambertian LED point light source can be expressed by a cosine function:

in

is the irradiance,

is the outgoing light intensity of the LED light source along its optical axis, d is the distance between the LED light source and the detection target,

is the angle between the actual outgoing light and the optical axis, m is the half-dead angle with the LED

the relevant value; A Lambertian surface can be defined as a surface with constant illumination in any direction, and its hemispherical reflectivity is 1. Since a hemisphere is a highly diffuse reflective surface, it can be approximately regarded as a Lambertian surface. For a Lambertian surface, Have:

In the formula,

is the outgoing light intensity at a certain emission angle ∂, so the irradiance distribution on the target plane can be given by:

in the formula

is the received irradiance of the free-form surface given by Eq. (3),

is the distance between the free-form surface and the target illumination surface,

Represents the area element on the free-form surface, BRDF is the bidirectional reflection distribution function, which is used to represent the reflection characteristics of the free-form surface; it can be given by:

in

is the diffuse reflectance of the free-form surface; to simplify the calculation, the complex integral reflected irradiance distribution of the target plane given by equation (3) is replaced by the simple irradiance superposition of all these secondary sources:

where N is the number of discrete points in the intersection; 2) Assumption

The light at the exit angle is emitted by the LED light source and intersects with the inner surface of the high-diffuse reflectance free-form surface at the point

, after the ray is diffusely reflected by the free-form surface, a certain outgoing ray intersects the target detection plane at a point

, using the above formula to establish a mathematical model of the irradiance distribution based on a diffuse reflection indirect illumination, the irradiance at any position on the target detection plane can be further expressed as:

Similarly, when two diffuse reflections occur, it is assumed that the light is emitted by the LED light source and intersects the inner surface of the high-diffuse free-form surface at the point

, at this time, the point p is similar to a Lambertian light source, equally dispersed to all directions of the diffuse reflection free-form surface (assuming that each radiant point radiates to two non-adjacent points), and the diffuse reflection free-form surface again intersects at point

, an outgoing ray intersects the target detection plane at a point

;Using the above formula to establish a mathematical model of irradiance distribution based on secondary diffuse reflection indirect illumination, the irradiance at any position on the target detection plane can be further expressed as:

Similarly, when three diffuse reflections occur, the irradiance at any position on the target detection plane can be further expressed as:

3) Use numerical method programming to solve the equation system, introduce a smoothing algorithm to the obtained discrete point coordinates, and obtain a free curve by fitting; the specific steps are as follows: ① Set initial conditions Set the coordinates of the initial point of the free curve as

, the horizontal and vertical coordinate change steps are respectively

; The value of the initial point coordinate and its step size depends on the size of the diffuse reflection free-form surface of the present invention; ② Solve the system of equations by numerical method Based on the numerical method, the specular reflection free curve is obtained from the expression, and then the initial value of the coordinates of each discrete point on the free curve is brought into the above non-linear algebraic equations, and the vertical coordinate is used as the unknown variable to program iteratively to solve the diffuse reflection free curve ; ③ Fitting a smooth curve Fit the coordinate data of discrete points on the free curve obtained by calculation to obtain a smooth curve; if necessary, use the trial-and-error method to correct the initial value of the vertical coordinates of each discrete point on the arc segment of the free curve, and repeat step ②. Solve the process until a surface profile that is easy to manufacture is obtained.

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