CN112328972A - Method and system for evaluating crystal grain structure - Google Patents

Method and system for evaluating crystal grain structure Download PDF

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CN112328972A
CN112328972A CN202011251581.8A CN202011251581A CN112328972A CN 112328972 A CN112328972 A CN 112328972A CN 202011251581 A CN202011251581 A CN 202011251581A CN 112328972 A CN112328972 A CN 112328972A
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陈曦
兰金明
陈昊
邬冠华
敖波
吴伟
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Nanchang Hangkong University
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Abstract

The invention relates to a method and a system for evaluating a grain structure, wherein the method is used for acquiring a plurality of structural characteristics of grains in a reference test block; determining fusion characteristics according to the structural characteristics by a circle-like mapping method; fitting a plurality of fusion characteristics corresponding to a plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model; and evaluating the grain structure of the reference test block to be evaluated through the spiral visual evaluation model. According to the invention, a plurality of structural features of the crystal grain are determined as a fusion feature through a circle-like mapping method, the crystal grain structure is evaluated according to the fusion feature containing the plurality of structural features, the accuracy of the evaluation of the crystal grain structure is improved, in addition, a plurality of fusion features corresponding to a plurality of reference test blocks are fitted into a Fermat spiral, the Fermat spiral is taken as an evaluation model, the closer the fusion feature is to the origin, the better the crystal grain structure is, and the visualization of the evaluation of the crystal grain structure is improved.

Description

Method and system for evaluating crystal grain structure
Technical Field
The invention relates to the technical field of high-temperature alloys, in particular to a method and a system for evaluating a grain structure.
Background
The high-temperature alloy has good high-temperature strength and oxidation and corrosion resistance, can bear large complex stress at the high temperature of more than 600 ℃, and is widely applied to hot end parts of turbine discs, blades and the like of aircraft engines. With the development of high thrust-weight ratio of aeroengines, the load of hot-end parts is increased, which puts more severe requirements on the materials. The performance of the high-temperature alloy is obviously influenced by the structural characteristics of the grain size, roundness, long and short axes and the like, so that the comprehensive evaluation of the grain structure has important significance for evaluating the material performance.
However, the evaluation of the grain structure of the existing high-temperature alloy mostly only remains the improvement of the grain size evaluation method. Since the grain structure has a plurality of quantization parameters, the grain structure cannot be accurately characterized only by using single characteristics such as grain size and roundness.
Disclosure of Invention
Based on the above, the invention aims to provide a method and a system for evaluating a grain structure, which improve the evaluation accuracy and improve the reliability of high-temperature alloy application.
In order to achieve the purpose, the invention provides the following scheme:
a method of evaluating a grain structure, the method comprising:
acquiring a plurality of structural characteristics of crystal grains in a reference test block;
determining fusion characteristics according to the structural characteristics by a circle-like mapping method;
fitting a plurality of fusion characteristics corresponding to a plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model;
and evaluating the grain structure of the reference test block to be evaluated through the spiral visual evaluation model.
Optionally, the structural features include size, roundness, and long-to-short axis ratio of grains.
Optionally, before determining the fusion feature by the circle-like mapping method according to the plurality of structural features, the method specifically includes:
and normalizing each structural feature.
Optionally, the determining the fusion feature according to the plurality of structural features by a circle-like mapping method specifically includes:
setting projection points on a plane in sequence by taking a set original point as a center according to a set included angle weight and a set radius weight; the projection point of each structural feature is expressed as
Figure BDA0002771758240000021
The distance from each projection point to the set origin is cjrjProjection point
Figure BDA0002771758240000022
A line connecting with the set origin and a projection point
Figure BDA0002771758240000023
The included angle between the line and the set origin is thetaj，，θjTo set the weight of the included angle, cjFor each of said structural characteristic values, riSetting the radius weight;
and sequentially connecting the projection points to form a polygon, wherein the geometric center of the polygon is a fusion feature.
Optionally, the Fermat spiral is expressed by
Figure BDA0002771758240000024
Taking the central point of the Fermat spiral as the origin of a rectangular coordinate system, wherein SθRepresents the included angle between the connecting line of the point on the Fermat spiral and the origin and the positive semi-axis in the horizontal direction, SRRepresenting the distance of a point on the fermat spiral from said origin, and λ is the coefficient of the spiral.
The invention also discloses a crystal grain structure evaluation system, which comprises:
the structural feature acquisition module is used for acquiring a plurality of structural features of the crystal grains in the reference test block;
the fusion characteristic determining module is used for determining fusion characteristics according to the structural characteristics by a circle-like mapping method;
the spiral visual evaluation model determining module is used for fitting a plurality of fusion characteristics corresponding to a plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model;
and the evaluation module is used for evaluating the grain structure of the reference test block to be evaluated through the spiral visual evaluation model.
Optionally, the structural features include size, roundness, and long-to-short axis ratio of grains.
Optionally, the system further comprises:
and the normalization module is used for normalizing the structural features acquired by the structural feature acquisition module.
Optionally, the fusion feature determining module specifically includes:
the projection unit is used for sequentially arranging projection points on a plane by taking the set original point as the center according to the set included angle weight and the set radius weight; the projection point of each structural feature is expressed as
Figure BDA0002771758240000025
The distance from each projection point to the set origin is cjrjProjection point
Figure BDA0002771758240000026
A line connecting with the set origin and a projection point
Figure BDA0002771758240000031
The included angle between the line and the set origin is thetaj，，θjTo set the weight of the included angle, cjFor each of said structural characteristic values, riSetting the radius weight;
and the fusion characteristic determining unit is used for sequentially connecting the projection points to form a polygon, and the geometric center of the polygon is a fusion characteristic.
Optionally, the Fermat spiral is expressed by
Figure BDA0002771758240000032
Taking the central point of the Fermat spiral as the origin of a rectangular coordinate system, wherein SθRepresents the included angle between the connecting line of the point on the Fermat spiral and the origin and the positive semi-axis in the horizontal direction, SRRepresenting the distance of a point on the fermat spiral from said origin, and λ is the coefficient of the spiral.
According to the specific embodiment provided by the invention, the invention discloses the following technical effects:
the invention discloses a method and a system for evaluating a crystal grain structure, wherein a plurality of structural characteristics of a crystal grain are determined as a fusion characteristic through a quasi-circle mapping method, the crystal grain structure is evaluated according to the fusion characteristic comprising the structural characteristics, the accuracy of the evaluation of the crystal grain structure is improved, in addition, a plurality of fusion characteristics corresponding to a plurality of reference test blocks are fitted into a Fermat spiral, the Fermat spiral is taken as an evaluation model, the closer the fusion characteristics are to the original point, the better the crystal grain structure is, and the visualization of the evaluation of the crystal grain structure is improved.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the embodiments will be briefly described below, and it is obvious that the drawings in the following description are only some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings without inventive exercise.
FIG. 1 is a schematic flow chart of a method for evaluating a crystal structure according to the present invention;
FIG. 2 is a schematic diagram of a circle-like space and a projected polygon of the present invention;
FIG. 3 is a schematic diagram of an evaluation model of the grain structure fusion characteristics according to the present invention;
FIG. 4 is a schematic metallographic structure diagram of GH4169 superalloy with different forging temperatures and forging deformation amounts;
FIG. 5 is a schematic view of a spiral visual evaluation model according to the present invention;
FIG. 6 is a schematic error plot of the radius of the fused feature and the fitted radius of the present invention;
FIG. 7 is a schematic diagram of a system for evaluating a grain structure according to the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The invention aims to provide a method and a system for evaluating a grain structure, which improve the evaluation accuracy and improve the reliability of high-temperature alloy application.
In order to make the aforementioned objects, features and advantages of the present invention comprehensible, embodiments accompanied with figures are described in further detail below.
Fig. 1 is a schematic flow chart of a method for evaluating a crystal grain structure according to the present invention, and as shown in fig. 1, the method for evaluating a crystal grain structure includes the following steps:
step 101: a plurality of structural features of the grains in the reference test block are obtained. The structural characteristics include the size, roundness and long-to-short axis ratio of the crystal grains.
Wherein, step 101 specifically includes: and performing metallographic phase sample preparation on the reference test block, and extracting the structural characteristics of the high-temperature alloy grains, such as size, roundness, length-to-axial ratio and the like.
Step 102: and determining fusion characteristics by a circle-like mapping method according to the plurality of structural characteristics.
Before step 102, the method specifically includes: and normalizing each structural feature.
Wherein, step 102 specifically includes:
taking a set origin as a center, and sequentially arranging the structural features in two according to a set included angle weight and a set radius weightSetting projection points on the dimensional plane; the projection point of each structural feature is expressed as
Figure BDA0002771758240000041
The distance from each projection point to the set origin is cjrjProjection point
Figure BDA0002771758240000042
A line connecting with the set origin and a projection point
Figure BDA0002771758240000043
The included angle between the line and the set origin is thetaj，，θjTo set the weight of the included angle, cjFor each of said structural characteristic values, riAnd setting the radius weight.
And sequentially connecting the projection points to form a polygon, wherein the geometric center of the polygon is a fusion feature.
Before the circle-like mapping, a circle-like mapping parameter is set, wherein the circle-like mapping parameter comprises a radian Q and a radius R. Radian Q ═ θ1,θ2,...,θM)，
Figure BDA0002771758240000044
Radius R ═ R1,r2,...,rM)。
Step 103: and fitting a plurality of fusion characteristics corresponding to the plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model.
Wherein, the expression of the Fermat spiral in step 103 is
Figure BDA0002771758240000051
Taking the central point of the Fermat spiral as the origin of a rectangular coordinate system, wherein SθRepresents the included angle between the connecting line of the point on the Fermat spiral and the origin and the positive semi-axis in the horizontal direction, SRRepresenting the distance of a point on the fermat spiral from said origin, and λ is the coefficient of the spiral.
Step 104: and evaluating the grain structure of the reference test block to be evaluated through the spiral visual evaluation model.
Wherein, in step 104, the method specifically comprises: and setting a threshold point on the spiral visual evaluation model, wherein the threshold point corresponds to a threshold radius, the threshold radius is the distance from the threshold point to the Fermat spiral original point, if the fusion characteristic radius corresponding to the fusion characteristic of the crystal grains of the reference block to be evaluated is smaller than the threshold radius, the crystal grain structure of the reference block to be evaluated is qualified, and if the fusion characteristic radius corresponding to the fusion characteristic of the crystal grains of the reference block to be evaluated is larger than or equal to the threshold radius, the crystal grain structure of the reference block to be evaluated is unqualified.
The following is a detailed procedure of a crystal structure evaluation method of the present invention:
s1, carrying out metallographic phase sample preparation on the reference test block, and extracting the structural characteristics of the high-temperature alloy grains, wherein the structural characteristics comprise grain size, roundness and major-minor axis ratio.
S2, normalizing the grain structure characteristics, setting a circle-like mapping parameter, projecting the grain structure characteristics to a 2-dimensional circular space by using a circle-like mapping method, constructing a projection polygon and extracting the fusion characteristics of the grain structure, and specifically comprises the following steps:
s21, sorting the grain structure features according to the projection order, and expressing as X ═ X (X) in the form of a variable1,x2,...,xM) And normalized to obtain
Figure BDA0002771758240000052
M represents the dimension of the structural feature, i.e. the number of structural features.
S22, calculating the radian of the corresponding dimension anchor point (point corresponding to the structural feature) on the arc according to the set angle weight, and recording the set angle weight as Q ═ θ1,θ2,...,θM-1)，
Figure BDA0002771758240000053
Setting the distance weight of a projection point on a connecting line from a dimension anchor point to the center of a circle, namely setting the radius weight, and recording the result of setting the radius weight as the radius R ═ R (R)1,r2,...,rM). Radian Q and radius R are circle-like mapping parameters.
And S23, taking the origin O as the center of a circle and the radius R as the radius in the rectangular coordinate system to form a circle-like space, as shown in FIG. 2. Will be provided with
Figure BDA0002771758240000054
The dimension anchor points are sequentially distributed on the circular arc C, and the coordinates of the dimension anchor points are expressed as follows:
Ej＝(Ejx,Ejy)＝(rjcos(αj),rjsin(αj)) (1)
in the formula, EjxRepresenting a dimension anchor EjAbscissa of (a), EjyRepresenting a dimension anchor EjThe ordinate of (a) is,
Figure BDA0002771758240000061
αjand representing the included angle between the connection line from the dimension anchor point to the origin and the positive half shaft of the rectangular coordinate system.
S24, mixing
Figure BDA0002771758240000062
Sequentially projecting the image to a quasi-circular space to form projection points
Figure BDA0002771758240000063
At a distance of O from the center of the circle
Figure BDA0002771758240000064
The specific calculation formula is as follows:
Figure BDA0002771758240000065
projection point
Figure BDA0002771758240000066
Is expressed as:
Figure BDA0002771758240000067
in the formula (I), the compound is shown in the specification,
Figure BDA0002771758240000068
are respectively projection points
Figure BDA0002771758240000069
The abscissa and the ordinate of (a).
S25, as shown in FIG. 2, sequentially connecting the projection points to form a polygon
Figure BDA00027717582400000610
The geometric center S of the polygon is the fusion feature covering the effective information of the grain structure, and the specific calculation formula of the geometric center of the polygon is as follows:
Figure BDA00027717582400000611
Figure BDA00027717582400000612
Figure BDA00027717582400000613
in the formula, Sx、SyIs the abscissa and ordinate of the projected point S, Sθ、SRThe radian and radius of the projection point S.
S3, using Fermat spiral to fit the radian and radius of the grain structure fusion feature obtained in the step S2, wherein the specific expression of Fermat spiral is as follows:
Figure BDA00027717582400000614
where λ is the undetermined coefficient of the helix, SθIn order to blend in the radians of the features,
Figure BDA00027717582400000615
to obtain for fittingThe radius of the fused feature of (a).
S4, setting the fitting error as an optimization target, and optimizing the target by adjusting the circle-like mapping parameter using GWO (Grey Wolf Optimizer), where the optimization target set in the step specifically is:
Figure BDA0002771758240000071
in the formula (I), the compound is shown in the specification,
Figure BDA0002771758240000072
to average relative error, SRFor the true radius of the fused feature to be,
Figure BDA0002771758240000073
and N is the number of test pieces for the fusion characteristic radius obtained by fitting.
S5, when the optimization target in the step S4 reaches the minimum, determining the optimal spiral coefficient and the circle-like mapping parameter, and determining a spiral visualization evaluation model based on the circle-like mapping, wherein the optimization target problem formula is specifically as follows:
Figure BDA0002771758240000074
in the formula, lambda represents a spiral coefficient, R and Q represent circle-like mapping parameters,
Figure BDA0002771758240000075
the average relative error in the fitting process is indicated.
S6, as shown in FIG. 3, making a spiral line in the circle-like space by using the spiral visual evaluation model, and marking the position of the fusion feature. The closer the position of the fusion feature is to the circle center, the better the grain structure of the reference block to be evaluated is.
The specific embodiment of the invention selects 32 test pieces of the nickel-based superalloy GH4169 processed by the forging process to explain the evaluation method of the invention.
By adopting the method, firstly, metallographic sampling is carried out in the step S1 to obtain the microstructure morphology of 32 reference test blocks as shown in FIG. 4, and the grain size, the roundness and the length-to-diameter ratio are calculated, wherein the grain size is represented by the average grain diameter. Then, the fusion feature S of the grain structure is extracted using step S2. And then, the radius and radian of the characteristic S are fused by using the Fermat spiral fitting of the step S3, and a spiral visualization evaluation model based on the quasi-circle mapping is preliminarily constructed. Optimizing the quasi-circle mapping parameters and the helix undetermined coefficients through the steps S4 and S5, determining a comprehensive visual evaluation model of the GH4169 grain structure,
Figure BDA0002771758240000076
the average relative error for this model is only 0.0046. The circle-like mapping parameters are shown in table 1, the evaluation model image is shown in fig. 5, and the error between the radius of the fusion feature and the evaluation model output radius is shown in fig. 6.
TABLE 1 circle-like mapping parameters of spiral visual evaluation model
Figure BDA0002771758240000077
As can be seen from fig. 5, the arrangement of the fused features of the reference test block in the circle-like space has no overlapping phenomenon, and all the fused features fall on the spiral line output by the evaluation model. The enlarged portion of the area in the upper left corner is clearly visible, and the fit degree of the fusion characteristic and the spiral line is excellent. Meanwhile, the determination of the evaluation model can simply observe the position relation of the fusion characteristics of each reference test block. As can be seen from fig. 6, the error between the real value and the fitting value of the fused feature of the 32 test blocks is very small, and no singular point appears. Obviously, compared with the method for representing the grain structure by using only a single characteristic, the method provided by the invention has the advantages that the range of the high-temperature alloy which can be evaluated is wider, meanwhile, the comprehensive visual evaluation can be realized, and the method has higher accuracy and robustness.
According to the invention, the evaluation method based on the existing grain structure is only limited to a single feature, a circle-like mapping method is provided, the fusion feature covering the global grain structure information is extracted from a plurality of features, and a spiral visual evaluation model of the high-temperature alloy grain feature is finally established by combining the radian and the radius of the fusion feature with Fermat spiral fitting. The method effectively avoids the shortage of information content in the use of single characteristics, and solves the problems of high error and poor robustness of the conventional evaluation model of the grain structure. The method provided by the invention can be used for visually evaluating the crystal grain structures of different high-temperature alloys under the condition of keeping higher accuracy, and has higher practicability and popularization.
Fig. 7 is a schematic structural diagram of a system for evaluating a grain structure according to the present invention, and as shown in fig. 7, the system for evaluating a grain structure includes:
a structural feature obtaining module 201, configured to obtain a plurality of structural features of a grain in a reference block;
a fusion feature determining module 202, configured to determine a fusion feature according to the plurality of structural features by a circle-like mapping method;
the spiral visual evaluation model determining module 203 is used for fitting a plurality of fusion characteristics corresponding to a plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model;
and the evaluation module 204 is configured to evaluate the grain structure of the reference block to be evaluated through the spiral visual evaluation model.
The structural characteristics include the size, roundness and long-to-short axis ratio of the crystal grains.
The system further comprises:
a normalization module, configured to normalize each structural feature acquired by the structural feature acquisition module 201.
The fusion feature determining module 202 specifically includes:
the projection unit is used for sequentially arranging projection points on a plane by taking the set original point as the center according to the set included angle weight and the set radius weight; the projection point of each structural feature is expressed as
Figure BDA0002771758240000091
Each of the projection points to the set originA distance of ciriProjection point
Figure BDA0002771758240000092
A line connecting with the set origin and a projection point
Figure BDA0002771758240000093
The included angle between the line and the set origin is thetai，θiTo set the weight of the included angle, ciFor each of said structural characteristic values, riSetting the radius weight;
and the fusion characteristic determining unit is used for sequentially connecting the projection points to form a polygon, and the geometric center of the polygon is a fusion characteristic.
The expression of the Fermat spiral is
Figure BDA0002771758240000094
Taking the central point of the Fermat spiral as the origin of a rectangular coordinate system, wherein SθRepresents the included angle between the connecting line of the point on the Fermat spiral and the origin and the positive semi-axis in the horizontal direction, SRRepresenting the distance of a point on the fermat spiral from said origin, and λ is the coefficient of the spiral.
According to the method and the system for evaluating the grain structure, a circle-like mapping method is adopted, the circle-like mapping method can map a data set from a high-dimensional space to a low-dimensional space, different mapping methods are set by changing the radius, the angle and the like, and a reliable technical means is provided for extracting the fusion characteristics of the grain structure. The difference of the grain structures of different high-temperature alloys is large, so that the arrangement rules of the fusion features in the circle space are different, and the difficulty of identifying the quality of the grain structure features is increased. The input of the Fermat spiral is radian, the output is radius, the relation between the radian and the radius of the fusion characteristic can be fitted by utilizing the property, and the problem that the fusion characteristic arrangement is difficult to judge in a circle space is solved.
The embodiments in the present description are described in a progressive manner, each embodiment focuses on differences from other embodiments, and the same and similar parts among the embodiments are referred to each other. For the system disclosed by the embodiment, the description is relatively simple because the system corresponds to the method disclosed by the embodiment, and the relevant points can be referred to the method part for description.
The principles and embodiments of the present invention have been described herein using specific examples, which are provided only to help understand the method and the core concept of the present invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, the specific embodiments and the application range may be changed. In view of the above, the present disclosure should not be construed as limiting the invention.

Claims (10)

1. A method for evaluating a crystal grain structure, the method comprising:
acquiring a plurality of structural characteristics of crystal grains in a reference test block;
determining fusion characteristics according to the structural characteristics by a circle-like mapping method;
fitting a plurality of fusion characteristics corresponding to a plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model;
and evaluating the grain structure of the reference test block to be evaluated through the spiral visual evaluation model.
2. The grain structure evaluation method according to claim 1, wherein the structural characteristics include a size, a roundness, and an axial length ratio of the grains.
3. The grain structure evaluation method according to claim 1, wherein before determining the fusion feature by a circle-like mapping method based on the plurality of structural features, the method further comprises:
and normalizing each structural feature.
4. The grain structure evaluation method according to claim 1, wherein the determining of the fusion feature from the plurality of structural features by a circle-like mapping method specifically comprises:
setting projection points on a plane in sequence by taking a set original point as a center according to a set included angle weight and a set radius weight; the projection point of each structural feature is expressed as
Figure FDA0002771758230000011
The distance from each projection point to the set origin is cjrjProjection point
Figure FDA0002771758230000012
A line connecting with the set origin and a projection point
Figure FDA0002771758230000013
The included angle between the line and the set origin is thetaj，，θjTo set the weight of the included angle, cjFor each of said structural characteristic values, riSetting the radius weight;
and sequentially connecting the projection points to form a polygon, wherein the geometric center of the polygon is a fusion feature.
5. The grain structure evaluation method according to claim 1, wherein the fermat spiral is expressed by
Figure FDA0002771758230000014
Taking the central point of the Fermat spiral as the origin of a rectangular coordinate system, wherein SθRepresents the included angle between the connecting line of the point on the Fermat spiral and the origin and the positive semi-axis in the horizontal direction, SRRepresenting the distance of a point on the fermat spiral from said origin, and λ is the coefficient of the spiral.
6. A system for evaluating a grain structure, the system comprising:
the structural feature acquisition module is used for acquiring a plurality of structural features of the crystal grains in the reference test block;
the fusion characteristic determining module is used for determining fusion characteristics according to the structural characteristics by a circle-like mapping method;
the spiral visual evaluation model determining module is used for fitting a plurality of fusion characteristics corresponding to a plurality of reference test blocks by adopting a Fermat spiral line to determine a spiral visual evaluation model;
and the evaluation module is used for evaluating the grain structure of the reference test block to be evaluated through the spiral visual evaluation model.
7. The grain structure evaluation system according to claim 6, wherein the structural characteristics include a size, roundness, and a long-to-short axis ratio of the grains.
8. The grain structure evaluation system according to claim 6, further comprising:
and the normalization module is used for normalizing the structural features acquired by the structural feature acquisition module.
9. The grain structure evaluation system according to claim 6, wherein the fusion characteristic determination module specifically comprises:
the projection unit is used for sequentially arranging projection points on a plane by taking the set original point as the center according to the set included angle weight and the set radius weight; the projection point of each structural feature is expressed as
Figure FDA0002771758230000021
The distance from each projection point to the set origin is cjrjProjection point
Figure FDA0002771758230000022
A line connecting with the set origin and a projection point
Figure FDA0002771758230000023
The included angle between the line and the set origin is thetaj，，θjTo set the weight of the included angle, cjFor each of said structural characteristic values, riSetting the radius weight;
and the fusion characteristic determining unit is used for sequentially connecting the projection points to form a polygon, and the geometric center of the polygon is a fusion characteristic.
10. The grain structure evaluation system of claim 6, wherein the Fermat spiral is expressed by
Figure FDA0002771758230000024
Taking the central point of the Fermat spiral as the origin of a rectangular coordinate system, wherein SθRepresents the included angle between the connecting line of the point on the Fermat spiral and the origin and the positive semi-axis in the horizontal direction, SRRepresenting the distance of a point on the fermat spiral from said origin, and λ is the coefficient of the spiral.
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