CN112539400A - Design method of diffuse transmission free-form curved surface based on collimating lens array - Google Patents
Design method of diffuse transmission free-form curved surface based on collimating lens array Download PDFInfo
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F21—LIGHTING
- F21V—FUNCTIONAL FEATURES OR DETAILS OF LIGHTING DEVICES OR SYSTEMS THEREOF; STRUCTURAL COMBINATIONS OF LIGHTING DEVICES WITH OTHER ARTICLES, NOT OTHERWISE PROVIDED FOR
- F21V5/00—Refractors for light sources
- F21V5/04—Refractors for light sources of lens shape
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Abstract
The invention provides a design method of a diffuse transmission free-form curved surface based on a collimating lens array. Conventional diffusers for LED array lighting are not structurally designed, and use simple planar or hemispherical surfaces. This makes the illumination system using the diffuser less efficient than other designs, since the diffuser scatters light from all directions. Based on the problems, the wide-view-angle light emission of the LED array is corrected by the collimating lens array according to the light emitting characteristics of the LED, and then a mathematical model of a diffuse transmission free form curved surface is established according to the ray path by applying a Bidirectional Transmission Distribution Function (BTDF) of a high-transmittance surface. Due to the complexity of the optical path, a method of pre-setting the collimation plane is proposed to simplify the calculation. Nonlinear algebraic equations are derived from light rays reaching and leaving the preset collimation surfaces, the equations are solved numerically to obtain the contour of the diffuse transmission free-form curved surface, and then the contour is guided into ray tracing software TracePro for simulation.
Description
Technical Field
The invention belongs to the technical field of optics and illumination, and particularly relates to a diffuse transmission free form curved surface for redistributing energy of an LED light source to realize uniform illumination and improve efficiency.
Background
Compared with the traditional light source (such as an incandescent lamp), the LED serving as the fourth generation illumination light source has the advantages of small volume, quick response, long service life, no pollution and the like. Because of these advantages, the LED is increasingly widely used in indoor lighting and outdoor lighting, but because the light intensity distribution of the LED is similar to the cosine distribution of the lambertian body, the LED has the disadvantages of too large divergence angle and uneven illumination distribution, if the LED is directly used for lighting without secondary optical light distribution, a circular light spot with bright center and gradually darkened periphery is formed on the target surface, the uniformity is poor, the light is difficult to irradiate on the target surface, the energy utilization rate is low, glare is easy to generate, and the actual lighting requirements are difficult to meet. Therefore, in practical LED lighting application, secondary light distribution design needs to be performed on the LED again, so that light energy distribution meets the practical lighting requirement. The secondary optical design is to ensure the quality of the emergent light of the whole LED illumination system, and mainly considers how to concentrate the light emitted by the LED light source on the expected target illumination surface through the designed curved surface with refraction or reflection performance so as to meet the actual requirement. In recent years, the diffuse transmission sheet has the advantages of convenient material, convenient processing, uniform light and the like, and is widely applied to array illumination such as street lamps, backlight and the like. However, a diffuse transmission sheet without optical design scatters light in any direction, which leads to a significant reduction in the illumination performance of the system. Based on the problems, the invention provides a method for a diffuse transmission sheet with diffuse transmission free-form curved surface characteristics based on array illumination, which is used for solving the problems of low uniformity and efficiency caused by the traditional diffuse transmission sheet and also provides a method for a free-form curved surface under the conditions of an LED array and a lens array.
Disclosure of Invention
The invention aims to solve the problems of low uniformity and low efficiency of an illumination system caused by a traditional diffuse transmission sheet, provides a method of a diffuse transmission sheet with diffuse transmission free-form curved surface characteristics based on array illumination, and also provides a free-form curved surface method under the conditions of an LED array and a lens array. The invention is realized by the following technical scheme:
(1) and establishing an LED optical system model of a diffuse transmission free-form curved surface based on the collimating lens array.
(2) And establishing a Cartesian coordinate system by taking the annular LED array as a symmetrical point by taking the original point as the symmetrical point.
(3) And constructing a mathematical model of the free-form curved diffuse transmission surface.
(4) And solving a nonlinear equation by using a numerical method to obtain the surface type of the diffusion sheet with the diffuse transmission free-form curved surface. And an annular LED array for emitting light. And the collimating lens collimates the light emitted by the LED. And the free-form curved surface is subjected to diffuse transmission to refract the light rays collimated by the collimating lens array. And the target detection surface receives the light rays passing through the diffuse transmission free-form curved surface.
The diffuse transmission free form curved surface is in a central axis symmetrical shape, and a Cartesian coordinate system is established by taking a section passing through a rotating central axis of the diffuse transmission free form curved surface as a reference surface: the rotation central axis is taken as a z axis, the direction which passes through the origin of the coordinate system and is vertical to the central axis is taken as an x axis, the collimation array and the free-form surface are arranged right above the LED light source, and a target plane to be illuminated is parallel to a light emitting plane of the LED light source and is in the positive direction of the z axis;
in the invention, light rays emitted by the annular LED array light source are collimated by the collimating lens array, are subjected to diffuse transmission by the high-diffuse-transmittance diffuse transmission free-form curved surface, and are equally dispersed to all directions, so that reasonable control and distribution of light intensity are realized, and a uniform illumination light spot is formed on a target detection plane. The annular LED array structure of the illumination system is shown in FIG. 1, the side view of the annular LED array and the structure relation of the collimating lens array is shown in FIG. 2, and the cross section of the system is shown in FIG. 3.
Based on the rotational symmetry principle, the surface type contour of the diffuse transmission free form curved surface based on the collimating lens array can be obtained by rotating a free form curved surface curve obtained by cutting the obtained diffuse transmission free form curved surface on an x0z plane around the normal line of the light emitting surface center of the LED light source, namely the rotation central axis z axis for one circle.
The free curve is determined by programmed calculation through the following steps:
s1, based on the light-emitting characteristics of the LED light source and the light path characteristics of the collimating lens, a mathematical simulation algorithm for horizontally diverging the LED light source light and the collimated light is constructed, a method for presetting a collimating surface is applied, an irradiance distribution function for representing the free-form surface is constructed, and the distribution function is used for solving the unknown coordinates of each discrete point irradiated on the free-form surface.
An ideal single LED light source is a lambertian emitter, and the light intensity of the light emitted from the LED light source to each direction can be expressed as:
(1)
in the formulaβIs the angle between the emitted light and the normal,I 0 （Lm / sr) Is thatβ= 0°The light intensity of the LED. mLighting angle available from the manufacturerβ 1/ 2 Given that, if the LED is Lambertian, thenmCan be considered as 1.
According to the inverse square law of illumination, we can obtain the illumination generated by a single LED in the directionE 0 ，d i Is the distance between the LED and a random point on the pre-set collimation plane.
As shown in fig. 3, the illumination scene represented here is perpendicular to the direction of the optical axis of the LEDs. Therefore, the irradiation distribution of the LED can be obtained by the formulas (1) and (3):
in FIG. 3, the position coordinates of a single LED may be represented as (x)0,y00), therefore, the irradiance distribution impinging on an arbitrary point (x, y, z) is transformed to be represented in a cartesian coordinate system, and then equation (4) is transformed to:
E k （x，y，z) Is the illumination of a single LED chip on a pre-set collimating surface,φis the angle associated with the position of the LED on the annular array, and because each LED chip is independent, the LED arrayE（x，y，z) Is equal to each LED chipE k （x，y，z) So we get:
as can be seen from fig. 3 and 4, the light emitted from the annular LED array is collimated by the collimating lens and then passes through the diffusely transmissive freeform surface. To simplify the analysis, we propose to construct a diffuse transmission freeform surface starting from collimated light on a pre-defined collimating surface. We assume that the light rays arriving at the diffusely transmitting freeform surface from the pre-defined collimating surface are all parallel to the z-axis. Since the light intensities of the light reaching the pre-collimating surface are not equal at different points, we use the average light intensity obtained under collimated conditions: (I s ）。
rIs the radius of the circular array and,lis a variable related to the position of the LED on the ring.
S2 applies a bi-directional transmission distribution function (BTDF) to construct a mathematical simulation algorithm for a diffuse transmission free-form surface to characterize target planar irradiance.
In our study, the diffuse transmission free-form surface is rotationally symmetric. Therefore, we consider a two-dimensional model, and take the x-z plane as an example. FIG. 4 shows, in a two-dimensional plane x-z, a diffuse free-form surfaceThe points and the points on the target plane can be expressed asP（x，0，z) Andt（ xt，0，H). From knowledge of the spatial vector, we can get:
Orepresenting the point from which the light passes after passing through the transmission surfacePTotIs detected by the light source, and the vector of light,Nis a pointPThe normal vector of the transmission curved surface is determined,dxanddzrespectively representing the differential of x and z,ϕis thatOAndNthe angle therebetween.
A diffuse emitter can be seen as a source emitting surface consisting of many lambertian point sources. The irradiance distribution on the target plane can be expressed as:
Ω i is the luminous flux of a single object lighting unit, whereinΩ i = E（x，y，z）·s u 。 s u Representing the unit area divided by the illumination surface. d i ’Is the distance between the free-form surface and the target plane. Bidirectional transmission distribution function (BTDF) To describe the uniformity of light in any direction in space. Given by:
∂ diffuse free-form transmission. To simplify the calculation, several sample points (x) are selected1,0,z1)，(x2,0,z2)，…，(xp,0,zp) To approximate the contour of the free-form surface. Therefore, the transmission irradiance distribution of the target surface given by (9) in the form of a complex integralIs replaced by a simple irradiance superposition of these sampling points, as shown in (11)
ωIs the number of discrete points sampled on the diffuse transmission free-form surface,uis the number of target lighting units.
Then (8) and (10) are substituted into (11), the point on the target surface can be obtainedt（x t ,0,H) The irradiance simulation algorithm of (1) is simplified to (12):
s3 selects the same number of sample points on the object plane in order to match the free form surface sampling points. The set target surface size and distance should be optimal from a result point of view for the purpose of uniform illumination. In order to increase the calculation speed, the differential parameters are replaced by Euler numerical methoddx，dzThe difference equation is calculated. Thus, with unknown points on the free-form surfacep i (x i ,0,z i ) For example, a set of nonlinear algebraic equations can be established as shown in (13):
s4, programming and solving an equation set by using a numerical method, introducing a smoothing algorithm into the obtained discrete point coordinates, and fitting to obtain a free curve; the method specifically comprises the following steps:
setting initial conditions
Setting the initial point coordinate of the free curve as (x 1 ,0,z 1 ) The variation step length of the horizontal coordinate and the vertical coordinate is determined to be optimal through trial and error, and the initial point coordinate depends on the size of the collimating lens array.
② numerical method solving equation set
Obtaining a diffuse transmission free curve based on a diffuse transmission free form curved surface and a target surface limit derivation expression, substituting initial values of coordinates of discrete points on the free curve into a nonlinear algebraic equation set, and carrying out programming iteration solution on the diffuse transmission free curve by taking vertical coordinates of the linear algebraic equation set as unknown variables to obtain coordinate values of a series of discrete pointsp i （x i ,0,z i ）（i=1,2,3 …), whereinx j =x 1 +（ω-1）D x 。ωThe larger the value of (A), the more discrete points of the obtained free curve are, and the more accurate the free curve represented by the points is.
Smooth curve fitting
Performing least square fitting on the coordinate data of the discrete points on the free curve obtained by calculation to obtain a smooth curve; and (4) trial and error are carried out on the initial values of the vertical coordinates of the discrete points on the arc section of the free curve, and the solving process in the step two is repeated until a curved surface contour with the best simulation effect is obtained.
This design provides a method for a diffusely transmissive freeform surface for application to lens array illumination. The light rays emitted by the annular LED light source collimated by the collimating lens are constrained by the diffuse transmission free form curved surface, and the designability of the diffuse transmission sheet is realized. Compared with the traditional diffuse transmission sheet without optical design, the invention has the following advantages and effects: the target surface receives the light rays after diffuse transmission through the diffuse transmission free form curved surface, the irradiation uniformity is high, the light rays are soft, no bad glare exists, and the efficiency is improved to a certain extent. The LED light source can be widely applied to the fields of array light sources and lens array illumination such as road illumination, backlight systems and the like. In addition, in the scheme of the invention, the target is illuminated by the radius of the surfaceRAnd the distance between the target illumination surface phase and the LED light sourceDAs a structural parameter of the diffuse transmission freeform surface model. And appropriate lighting system parameters can be selected according to the requirements of practical application occasions, and the flexibility is higher.
Drawings
FIG. 1 is a block diagram of a ring LED array.
FIG. 2 is a side view of the LED array and the collimating lens array.
Fig. 3 is a schematic diagram of the principle of a diffuse transmission freeform surface.
Fig. 4 is a schematic diagram of the principle of light rays in a collimating lens.
Fig. 5 is a partial view of light rays passing through a collimating lens.
Fig. 6 is a partial view of light rays passing through a diffusely transmissive freeform surface.
Fig. 7 is a 2D profile of a diffuse transmission freeform surface.
Fig. 8 is a 3D profile of a diffuse transmission freeform surface.
Fig. 9 is the simulation result of the irradiation uniformity on the target plane at D =300mm from the light source.
Detailed Description
The following detailed description of the embodiments of the invention is provided in connection with the accompanying drawings. As shown in fig. 2, the collimating lens is located right above the LED light sources, each LED light source corresponds to one collimating lens, and the collimating lens collimates the light emitted from the LED. As shown in fig. 3, a specific case of the design optical path of the present invention based on the diffuse transmission freeform curved surface of the collimator lens array is described. When light emitted from the LED light source enters the collimating lens, the light is emitted to the parallel direction of the z-axis by reflection and refraction to reach the predetermined collimating plane, as shown in fig. 4. At this time, each position of the preset collimation surface can also be regarded as a lambertian body, then the light enters the diffuse transmission free form curved surface, and finally the light which is subjected to diffuse transmission from the diffuse transmission free form curved surface is diffused to other directions or is received by the target surface. After the light rays are refracted or reflected by the collimating lens and are subjected to diffuse transmission by the diffuse transmission free form curved surface with high diffuse transmittance, reasonable control and distribution of light intensity can be realized, and a uniform illumination light spot is formed on a target detection plane. The shape of the diffuse transmission free form curved surface with high diffuse transmission rate is determined by the following scheme:
the present embodiment establishes a cartesian coordinate system with a cross section of a rotation center axis of a diffuse transmission freeform curved surface of high diffuse transmittance based on a collimator lens array as a reference plane, with the above rotation center as a rotation centerThe axis is the z-axis, and the direction passing through the origin of the coordinate system and perpendicular to the central axis is the x-axis. Based on the principle of rotational symmetry, in order to simplify the calculation, three dimensions are converted into two-dimensional conditions, and the solution of the diffuse transmission freeform surface with high diffuse transmittance is shown in fig. 5 and 6. Diffuse transmission freeform surface on two-dimensional plane x-zpPoint on and object planeTThe point on can be expressed asP（x，0，z) Andt（ x t ，0，H) The coordinates of the LED may be expressed as (x)0，y0,0). Selecting a number of sample points (x)1,0,z1），（x2,0,z2），…，（xp,0,zp) To approximate the contour of the free-form surface. Therefore, the transmitted irradiance distribution of the target surface given by equation (10) in the form of a complex integral is replaced by a simple irradiance superposition of these sampling points, expressed as:
in the inventionmIf not less than 1, the LED half-life angle is taken to beβ 1/ 2 =60 °. The light emitted from the collimating lens passes through the preset collimating surface, passes through the diffuse transmission free form curved surface with high diffuse transmittance, and then is uniformly dispersed in all directions. By applying a diffuse transmission free form surface bidirectional transmission distribution function with high diffuse transmittance and regarding all discrete points for defining the free form surface as Lambertian non-correlated secondary point light sources, points on a target surface can be consideredt(x t ,0,H) Irradiance of (c) may be expressed as:
since the high diffuse transmittance freeform curved surfaces are all self-made in the present embodiment, the curved surface diffuse transmittance ∂ =0.9 in the programmed calculation. According to the specific requirements of actual application field for range of illuminated area and its radiation illuminance the target illumination surface can be definedDistance relative to LED light source in lighting systemDAnd the radius of the target illumination surfaceR. To match the freeform surface sampling points, the same number of sample points are selected on the target surface. The set target surface size and distance should be optimal from the result point of view for the purpose of uniform illumination, whenD=300mm，R=100 mm. Thus, with unknown points on the free-form surfacep i （x i , 0,z i ) For example, a set of nonlinear algebraic equations can be established.
The equation for limiting the curve shape of the free-form curved surface established by the invention is a nonlinear algebraic equation, and in order to simplify the calculation difficulty and convert three dimensions into two dimensions, the initial point coordinate (A) is firstly setx 1 ,0,z 1 ) The change step lengths of the horizontal and vertical coordinates are respectively
. Based on a numerical method and by means of MATLAB programming iterative solution, coordinate values of a series of discrete points on the free curve are obtained. And performing smooth fitting on the obtained coordinate data by using a least square method, and correcting the coordinate values of discrete points on the arc segment of the free curve by using a trial and error method if necessary until a diffuse transmission free form curved surface profile with the optimal simulation effect is obtained. In the present embodiment, the coordinates of the initial point of the free curve are set to (0, 0, 4), and the variation step lengths of the horizontal and vertical coordinates are set to (0, 0, 4), respectivelyD x =42，D z =1, number of discrete points defining arc segment of desired free curveω= 63. According to the steps, the equation is iteratively solved by means of MATLAB programming, and the obtained coordinate data are smoothly fitted, so that the coordinate values of a series of discrete points on the arc segment of the free curve are finally obtained, as shown in FIG. 7. The free curve rotates for a circle around the normal line of the light emitting surface center of the LED light source, namely the z axis of the rotating central shaft, and the collimating-based lens array can be obtainedThe profile of the diffuse transmission freeform surface of the column is shown in fig. 8.
The light source system entity model implemented by the invention is led into TracePro for non-sequence ray tracing, and the simulation result is shown in figure 9. FIG. 9 is a simulated view of irradiance distribution within a circular area of the target illumination plane: after the technical scheme is implemented, the irradiation uniformity of the target illumination surface can reach more than 83%, the illumination efficiency can reach more than 9.8%, the irradiation uniformity and the illumination efficiency of the traditional diffuse transmission surface are improved, and the original purpose of the scheme is achieved. The defects caused by the application of the traditional diffuse transmission sheet in array illumination can be improved.
Claims (1)
1. The design method of the diffuse transmission free form curved surface based on the collimating lens array is characterized by comprising the following steps of:
s1, establishing a diffusion sheet with a diffuse transmission free form curved surface for a lens array system, wherein the diffusion sheet comprises an LED light source, a collimating lens array, a diffuse transmission free form curved surface and a target detection surface;
s2 a Cartesian coordinate system is established by taking the annular LED array and the collimating lens as symmetrical points, the diffuse transmission free form curved surface is in a central axis symmetrical shape and can be obtained by rotating a central axis z axis, and a target illumination surface is in the positive direction of the z axis;
s3, constructing a diffusion sheet model with a diffuse transmission free form curved surface, collimating light rays emitted by an LED through a collimating lens to reach a preset collimating surface, then reaching the diffuse transmission free form curved surface through the preset collimating surface, and then forming uniform illumination light spots on a target surface after refraction of the diffuse transmission free form curved surface;
the model of the diffusion sheet with the diffuse transmission freeform curved surface obtained in S4 is a freeform curve obtained on the x0z plane, and the freeform curve is determined by the following calculation steps after the diffusion sheet is rotated for one circle by the z-axis of the central axis of rotation:
s4.1, constructing a mathematical simulation algorithm for horizontally diverging the light rays of the LED light source and the collimated light rays based on the light emitting characteristics of the LED light source and the light path characteristics of the light rays passing through the collimating lens, and constructing an irradiance distribution function for representing the free-form surface by applying a method of presetting a collimating surface, wherein the distribution function is used for solving the unknown coordinates of each discrete point irradiated on the free-form surface;
an ideal single LED light source is a lambertian emitter, and the light intensity of the light emitted from the LED light source to each direction can be expressed as:
in the formulaβIs the angle between the emitted light and the normal,I 0 （Lm / sr) Is thatβ= 0°The intensity of the light of the LED is, mlighting angle available from the manufacturerβ 1/ 2 Given that, if the LED is Lambertian, thenmCan be regarded as 1;
according to the inverse square law of illumination, we can obtain the illumination generated by a single LED in the directionE 0 ，d i Is the distance between the LED and a random point on the preset collimation plane;
the illumination scene represented here is perpendicular to the direction of the optical axis of the LED, and therefore the irradiance distribution of the LED can be obtained from equations (1), (3):
in this lighting system, the position coordinates of the individual LEDs may be expressed as (x)0,y00), therefore, the irradiation distribution irradiated to an arbitrary point (x, y, z) is converted intoExpressed in cartesian coordinates, then equation (4) is transformed into:
E k （x，y，z) Is the illumination of a single LED chip on a pre-set collimating surface,φis the angle associated with the position of the LED on the annular array, and because each LED chip is independent, the LED arrayE（x，y，z) Is equal to each LED chipE k （x，y，z) So we get:
the light emitted by the annular LED array is collimated by the collimating lens and then passes through the diffuse transmission free-form surface, and in order to simplify the analysis, we propose to construct the diffuse transmission free-form surface from the collimated light on the preset collimating surface, and we assume that the light rays reaching the diffuse transmission free-form surface from the preset collimating surface are all parallel to the z-axis, and since the light intensity of the light reaching the preset collimating surface is not equal at different points, we use the average light intensity obtained under the collimating condition (the average light intensity is the light intensity obtained under the collimating condition: (the light intensity is theI s ）；
rIs the radius of the circular array and,lis a variable related to the position of the LED on the ring;
s4.2, constructing a mathematical simulation algorithm of the diffuse transmission free-form surface for representing the target plane irradiance by applying a Bidirectional Transmission Distribution Function (BTDF);
in the design method, the diffuse transmission free form curved surface is rotationally symmetrical and is positioned on a two-dimensional planePoints on the diffuse free-form surface and points on the target plane can be represented as on x-zP（x，0，z) Andt（ x t ，0，H) From knowledge of the spatial vector, one can obtain:
Orepresenting the point from which the light passes after passing through the transmission surfacePTotIs detected by the light source, and the vector of light,Nis a pointPThe normal vector of the transmission curved surface is determined,dxanddzrespectively representing the differential of x and z,ϕis thatOAndNthe angle therebetween;
a diffuse emitter can be viewed as a source emitting surface consisting of many lambertian point sources, and the irradiance distribution on the target plane can be expressed as:
Ω i is the luminous flux of a single object lighting unit, whereinΩ i = E（x，y，z）·s u ，s u Representing the unit area divided by the illumination surface,d i ’is the distance between the free form surface and the target plane, the bi-directional transmission distribution function (BTDF) is used to describe the uniformity of light in any direction in space, given by:
∂ is the diffuse transmission of the free-form surface, and several sample points (x) are selected to simplify the calculation1,0,z1），（x2,0,z2），…，（xp,0,zp) To approximate the contour of a free-form surface, the transmitted irradiance distribution of the target surface given by (9) in complex integral form is therefore replaced by a simple irradiance superposition of these sampling points, as shown in (11):
ωis the number of discrete points sampled on the diffuse transmission free-form surface,uis the number of target lighting units;
then (8) and (10) are substituted into (11), the point on the target surface can be obtainedt（x t ,0,H) The irradiance simulation algorithm of (1) is simplified to (12):
s4.3 in order to match the sampling points of the free-form surface, the same number of sample points are selected on the target surface, the set size and distance of the target surface are optimal from the viewpoint of the result based on the purpose of uniform illumination, and in order to improve the calculation speed, an Euler numerical method is used for replacing a differential parameterdx，dzTo calculate a difference equation, so that the unknown point on the free-form surface is usedp i （x i ,0,z i ) For example, a set of nonlinear algebraic equations can be established as shown in (13):
s4.4, programming and solving an equation set by using a numerical method, introducing a smoothing algorithm into the obtained discrete point coordinates, and fitting to obtain a free curve; the method specifically comprises the following steps:
initial condition setting
Setting the initial point coordinate of the free curve as (x 1 ,0,z 1 ) The variation step length of the horizontal coordinate and the vertical coordinate is determined to be optimal through trial and error, and the initial point coordinate depends on the size of the collimating lens array;
numerical method for solving equation set
Obtaining a diffuse transmission free curve based on a diffuse transmission free form curved surface and a target surface limit derivation expression, substituting initial values of coordinates of discrete points on the free curve into a nonlinear algebraic equation set, and programming and iterating to solve the diffuse transmission free curve by taking vertical coordinates of the linear algebraic equation set as unknown variables;
fitting of smooth curves
Performing least square fitting on the coordinate data of the discrete points on the free curve obtained by calculation to obtain a smooth curve; and (4) trial and error are carried out on the initial values of the vertical coordinates of the discrete points on the arc section of the free curve, and the solving process in the step two is repeated until a curved surface contour with the best simulation effect is obtained.
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