A method for predicting the extreme features of spheres and ellipsoids

Abstract

This application provides a method for predicting the pole features of spheres and ellipsoids. The method first analyzes the scattering mechanism of spheres and ellipsoids; then establishes a resonance equation; then determines the primary and secondary radii of curvature corresponding to the crawling trajectory points; and finally, predicts the poles using the resonance equation. This forward prediction helps eliminate false poles generated during actual extraction, simplifies the data volume when establishing a pole library for radar recognition, and facilitates the establishment of a radar pole library and improves its accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of radar target recognition, and in particular to a method for predicting extreme point features of a sphere and an ellipsoid. Background Art

[0002] The rapid development of ultra-wideband radar technology and the increasing demand for evaluating radar characteristics of low-observable targets have made scattering mechanisms, such as creeping wave diffraction, increasingly important. In radar target stealth research, the contribution of creeping wave diffraction has become non-negligible, and in certain configurations, it has even become the primary source of target scattering. For target detection, military targets in the high-frequency band (3-30 MHz) are generally located in the resonant region. The contribution of creeping wave diffraction in this resonant region is greater than in the high-frequency region. The target scattering signature in this resonant region contains more information about the target, facilitating target identification. For target identification, target pole features, as physical properties of the target, are directional and polarization invariant. Accurately acquiring the target poles lays the foundation for target identification.

[0003] Existing radar recognition methods rely on calculating the natural poles of an object using matrix bundle methods or Cauchy methods on radar echo data. However, these methods essentially perform matrix numerical operations, and even after filtering the resulting poles, false poles still exist. Therefore, a physically meaningful and reliable method for predicting the pole features of typical convex surfaces, such as spheres and ellipsoids, is urgently needed. Summary of the Invention

[0004] The purpose of the present invention is to solve the technical problems in the above background and to provide a method for predicting the extreme point features of a sphere and an ellipsoid, comprising the following steps:

[0005] S1. Analyze the scattering mechanism of spheres and ellipsoids;

[0006] S2. Establish the resonance equation;

[0007] S3, determining the main curvature radius and the secondary curvature radius corresponding to the crawling trajectory point;

[0008] S4. Predict the poles through the resonance equation.

[0009] In a preferred embodiment, the scattering mechanism of the sphere and ellipsoid obtained by analysis in step S1 is used to clarify the physical meaning of the corresponding poles of the sphere and ellipsoid, and through a specific physical process, the physical meaning is converted into a mathematical formula for calculation to deduce the pole feature data;

[0010] The specific scattering mechanism of spheres and ellipsoids is as follows: the resonance process of creeping wave diffraction is formed by the coherent superposition of the primary diffraction field and the diffraction field contributed to the observation point again after the creeping wave orbits the sphere. After the incident field excites the creeping wave diffraction field, it continuously crawls along the short-range line on the curved surface and radiates energy outward along the tangent direction of the surface. The creeping wave propagation process follows the generalized Fermat principle, that is, the creeping wave can only propagate along the geodesic of the curved surface.

[0011] In the preferred scheme, when the creeping wave crawls to the exit point position that can be received by a single-station radar, the phases of electromagnetic waves of different frequencies at this point are different, and coherent superposition or cancellation will occur, forming a resonance phenomenon. The creeping wave multi-loop UTD diffraction field is calculated based on the resonance phenomenon. When the incident point and the exit point remain unchanged, the creeping wave diffraction field at the exit point changes in a geometric progression with the number of crawling circles, and the common ratio is the attenuation coefficient α accumulated during one circle. The resonance equation 1-α is constructed, and then the parameter k in the corresponding attenuation coefficient is solved to obtain the theoretical pole predicted by the resonance equation.

[0012] In a preferred embodiment, step S2 includes: the creeping wave excited by each polarization component of the incident wave is composed of an infinite number of creeping wave modes, where m represents the order of the mode, and each mode has its own diffraction coefficient and attenuation constant. When the surface is a closed surface, each mode will propagate around the surface infinitely many times to form a high-order diffraction field, where l represents the number of creeping circles around the closed surface. The diffraction field of the field point is expressed as:

[0013]

[0014] Where, L = ∮dt′, represents the phase jump caused by the caustic region that the surface diffraction ray may pass through when propagating from Q1 to Q2, It represents the phase jump caused by all the caustic regions passed by when the closed surface is circled, and it can be sorted out as follows:

[0015]

[0016] The dyadic transfer function is:

[0017]

[0018] In a preferred solution, step S2 further includes: when the creeping wave circulates on the curved surface to form a closed path, selecting an exit point on the closed path, the creeping wave diffraction field at the exit point changes in a geometric progression with the number of creeping circles, and the common ratio is R, and the total creeping wave field value at the exit point is the accumulation of the above geometric progression:

[0019]

[0020] in:

[0021]

[0022] Where L is a closed path that completes one circle.

[0023] In a preferred solution, the resonance equation in step S2 is:

[0024]

[0025] Among them, r n represents the principal radius of curvature, r b represents the minor curvature radius.

[0026] In a preferred solution, the main curvature radius and the secondary curvature radius corresponding to the crawling trajectory point in step S3 are determined by the propagation direction vector corresponding to the crawling wave trajectory point and the secondary direction vector perpendicular thereto.

[0027] In a preferred solution, the direction vector corresponding to the trajectory point is a vector that is along the propagation direction of the creeping wave and is tangent to the surface as the propagation direction vector;

[0028] The secondary direction vector perpendicular to the propagation direction vector is obtained by cross-producting the normal vector corresponding to the surface trajectory point with the propagation direction vector;

[0029] The specific curvature radius is calculated by the propagation direction and the sub-direction;

[0030] The main curvature radius is calculated by changing the surface curvature along the propagation direction, and the secondary curvature radius is calculated by changing the vector along the secondary direction.

[0031] In a preferred solution, on a sphere, the main curvature and the minor curvature are both the size of the sphere radius. When calculating the pole corresponding to the closed crawling trajectory of the ellipsoid parallel to the major axis, the main curvature radius and the minor curvature radius corresponding to each trajectory point on the crawling path are calculated;

[0032] When calculating the main curvature radius, the curvature radius change along the propagation direction can be calculated directly using the ellipse curvature radius calculation formula:

[0033]

[0034] The parameter a is the length of the minor axis of the ellipse, the parameter b is the length of the major axis of the ellipse, and θ is the angle formed by the line connecting the point on the ellipse and the origin and the negative direction of the x-axis.

[0035] In the preferred scheme, for the secondary curvature radius corresponding to the trajectory point, the specific calculation is to calculate the elliptical curve formed by the intersection of the plane formed by the secondary direction vector of the trajectory point and the corresponding normal vector and the ellipsoid surface, calculate the semi-major axis and semi-minor axis of the elliptical curve, and then calculate the curvature radius at the vertex of the semi-major axis as the secondary curvature radius of the trajectory point through the elliptical curvature radius calculation formula.

[0036] In the preferred solution, in the Cartesian rectangular coordinate system, the ellipsoid equation is:

[0037]

[0038] The intersection line of the ellipsoid and the YOZ plane is used as the closed path for the specified calculation, that is, the meridian. The subsequent calculation is performed by expressing the ellipsoid equation in the spherical coordinate system. The ellipsoid equation in the spherical coordinate system is:

[0039]

[0040] The trajectory points on the meridian are

[0041] According to the gradient formula Calculate the normal vector of any point on the ellipsoid as:

[0042]

[0043] The secondary direction vector corresponding to any trajectory point on the meridian is defined as

[0044] In the preferred solution, the propagation direction vector corresponding to the trajectory point is the cross product of the secondary direction vector corresponding to the trajectory point and the normal vector corresponding to the trajectory point:

[0045]

[0046] The plane equation composed of the normal vector and the secondary direction vector of the trajectory point is calculated by the point normal equation of the plane using the trajectory point coordinates and the propagation direction vector:

[0047]

[0048] The radius of curvature corresponding to the vertex of the major axis of the ellipse formed by the intersection of the plane and the ellipsoid is the minor curvature radius required in the formula.

[0049] In the preferred solution, the space ellipse equation is obtained by combining the plane equation with the ellipsoid equation:

[0050]

[0051] is the angle parameter of the corresponding trajectory point in the spherical coordinate system;

[0052] The length of the semi-major axis of the space ellipse is:

[0053]

[0054] The length of the semi-minor axis of the space ellipse is:

[0055]

[0056] According to the formula, the secondary curvature radius of the trajectory point is calculated as:

[0057] In the preferred solution, in step S4, the calculated main curvature radius r n and the minor curvature radius r b Substitute into the resonance equation to predict the poles.

[0058] The beneficial effects of the present invention are as follows: In practical applications, this patent effectively proposes corresponding pole laws for the selection of poles in the field of radar recognition by combining diffraction theory with physical meaning, and performs forward prediction of sphere and ellipsoid poles, providing a theoretical reference for recognition. Forward prediction can help eliminate false poles generated in actual extraction, and the amount of data can be correspondingly streamlined when establishing a pole library for radar recognition, thereby facilitating the establishment of radar recognition pole libraries and improving accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 It is a schematic diagram of the diffraction mechanism.

[0060] Figure 2 It is a schematic diagram of the direction vector definition.

[0061] Figure 3 This is a schematic diagram for calculating the principal radius of curvature of an ellipse.

[0062] Figure 4 It is a schematic diagram of the intersection of the ellipsoid and the plane.

[0063] Figure 5 This is a schematic diagram of the ellipsoid extraction results.

[0064] Figure 6 This is a schematic diagram of the resonance phenomenon. DETAILED DESCRIPTION

[0065] like Figures 1 to 6 The present invention is further described below with reference to the accompanying drawings.

[0066] 1. Purpose of the Invention

[0067] The current approach to acquiring the pole characteristics of radar targets is through data extraction: using the matrix bundle method and its improved algorithm to obtain pole data for time-domain echo data, and using the Cauchy method to obtain pole data for frequency-domain radar echo data. Both methods can provide target-corresponding pole data. However, both extraction methods perform numerical calculations on radar echo data, resulting in unclear physical meaning in the acquisition process and the corresponding physical meaning in the results. Therefore, the present invention clarifies the physical meaning of the corresponding poles of typical convex surfaces based on the resonant scattering mechanism and provides a derivation process for these poles.

[0068] The principle of clarifying the physical meaning: through the specific physical process, the physical meaning is converted into mathematical formulas for calculation and the extreme characteristic data is deduced. Figure 1 As shown in the figure, when a creeping wave reaches the exit point where it can be received by a single-station radar, the phases of electromagnetic waves of different frequencies differ, resulting in coherent superposition or cancellation, forming a resonance phenomenon. The resonance phenomenon is used to calculate the creeping wave multi-turn UTD diffraction field. With the incident and exit points unchanged, the creeping wave diffraction field at the exit point varies in a geometric progression with the number of creeping turns, with the common ratio being the accumulated attenuation coefficient α over one turn. This constructs the resonance equation 1-α, and then solves the parameter k corresponding to the attenuation coefficient to obtain the theoretical pole predicted by the resonance equation.

[0069] 2. Specific plan

[0070] 2.1 Scattering mechanism of convex surface

[0071] When a plane wave strikes a convex surface, it produces various scattering and reflection phenomena. Creeping wave diffraction is a type of scattering phenomenon. The scattering effect recurs after a certain delay, and the scattering response that repeats after a certain delay is called a post-scattering response. This process occurs infinitely, with the energy decreasing each time. This effect is called resonant scattering.

[0072] Taking a sphere as a typical convex surface as an example, creeping wave diffraction is the source of spherical resonant scattering. Figure 1As shown in the figure, when an electromagnetic wave impinges on the surface of a sphere, apart from the electromagnetic wave reflected by the surface, part of the electromagnetic wave continues to propagate tangentially to the incident sphere, while part of the electromagnetic wave creeps along the geodesic lines of the curved surface. The excitation point of the creeping wave diffraction field is generally at the point where the convex surface is tangent to the incident electromagnetic field, called the shadow boundary. The electromagnetic wave only "creeps" after passing through the shadow boundary. The resonant process of creeping wave diffraction is formed by the coherent superposition of the primary diffraction field and the multi-turn diffraction field. The multi-turn diffraction field is the diffraction field that the creeping wave contributes to the observation point after circumnavigating the sphere. After the incident field excites the creeping wave diffraction field, it continuously creeps along the geodesic lines of the curved surface while radiating energy outward along the surface tangent. The creeping wave propagation process follows the generalized Fermat principle, which states that the creeping wave can only propagate along the geodesic lines of the curved surface.

[0073] 2.2 Establishment of resonance equation

[0074] From the high-frequency asymptotic solution of the scattered field in the shadow area, it can be seen that the creeping wave excited by each polarization component of the incident wave is composed of an infinite number of creeping wave modes, with m representing the order of the mode. Each mode has its own diffraction coefficient and attenuation constant. If the surface is a closed surface, each mode will propagate around the surface infinitely many times, forming a high-order diffraction field, with l representing the number of circles around the closed surface. Due to the influence of the attenuation constant, the high-order diffraction of the creeping wave is usually not considered. Based on this, the diffraction field at the field point can be expressed as:

[0075]

[0076] Where, L = ∮dt′, represents the phase jump caused by the caustic region that the surface diffraction ray may pass through when propagating from Q1 to Q2, Represents the phase jump caused by all the caustic regions passed by when the closed surface is circling around. Rearranged as:

[0077]

[0078] The dyadic transfer function is:

[0079]

[0080] When a creeping wave forms a closed path on a curved surface, an exit point is selected on the selected path. The creeping wave diffraction field at the exit point changes in a geometric progression with the number of creeping circles, and the common ratio is R. The total creeping wave field value at the exit point is the accumulation of the above geometric progression:

[0081]

[0082] in:

[0083]

[0084] Where L is a closed path that completes one circle.

[0085] The resonance equation can be obtained by sorting:

[0086]

[0087] where r n represents the principal radius of curvature, r b represents the minor curvature radius.

[0088] 2.3 Determination of the main and minor curvature radii

[0089] The main curvature radius and the minor curvature radius corresponding to the creeping wave trajectory point are mainly determined by the propagation direction vector corresponding to the creeping wave trajectory point and the minor direction vector perpendicular to it. Figure 2 As shown, the direction vector corresponding to the trajectory point is a vector that is tangent to the surface and along the propagation direction of the creeping wave. The secondary direction vector perpendicular to the propagation direction vector is obtained by cross-producting the normal vector corresponding to the surface trajectory point with the propagation direction vector. The specific curvature radius corresponding to the propagation direction and the secondary direction are calculated. Figure 2 For point P in the image, the main curvature radius is calculated by changing the surface curvature along the propagation direction, and the secondary curvature radius is calculated by changing the vector along the secondary direction.

[0090] On a sphere, the principal curvature and the minor curvature are both the size of the sphere's radius, such as Figure 2 When calculating the pole corresponding to the closed crawling trajectory of the ellipsoid parallel to the major axis, it is necessary to calculate the main curvature radius and the secondary curvature radius corresponding to each trajectory point on the crawling path. When calculating the main curvature radius, the curvature radius change along the propagation direction can be calculated directly using the ellipse curvature radius calculation formula:

[0091]

[0092] The parameters a, b, and θ have the following meanings: Figure 4 shown.

[0093] Depend on Figure 3 The wavy arrow in the middle also shows the corresponding direction of curvature change.

[0094] For the minor curvature radius corresponding to the trajectory point, it is also necessary to consider that it changes with the direction indicated by the minor direction vector. In the specific calculation, the elliptical curve formed by the intersection of the plane formed by the minor direction vector of the trajectory point and the corresponding normal vector and the ellipsoid surface is calculated, and the semi-major axis and semi-minor axis of the elliptical curve are calculated. Then, the curvature radius at the vertex of the semi-major axis is calculated by the elliptical curvature radius calculation formula as the minor curvature radius of the trajectory point. The specific calculation formula is as follows:

[0095] In the Cartesian coordinate system, the equation of the ellipsoid is:

[0096]

[0097] In order to facilitate the calculation of the specified closed path (the intersection line of the ellipsoid and the YOZ plane is used as the specified closed path, which is named the meridian), the ellipsoid equation is expressed in the spherical coordinate system for subsequent calculations. The ellipsoid equation in the spherical coordinate system is:

[0098]

[0099] The trajectory points on the meridian are

[0100] According to the gradient formula Calculate the normal vector of any point on the ellipsoid as:

[0101]

[0102] The secondary direction vector corresponding to any trajectory point on the meridian is defined as

[0103] The propagation direction vector corresponding to the trajectory point is the cross product of the secondary direction vector corresponding to the trajectory point and the normal vector corresponding to the trajectory point:

[0104]

[0105] The plane equation composed of the normal vector and the secondary direction vector of the trajectory point can be calculated from the point normal equation of the plane using the trajectory point coordinates and the propagation direction vector:

[0106]

[0107] The radius of curvature corresponding to the vertex of the major axis of the ellipse formed by the intersection of the plane and the ellipsoid is the minor curvature radius required in the formula, such as Figure 4 As shown in the figure, point A corresponds to the vertex of the major axis.

[0108] The space ellipse equation can be obtained by combining the plane equation and the ellipsoid equation:

[0109] ( is the angle parameter of the corresponding trajectory point in the spherical coordinate system)

[0110] The length of the semi-major axis of the space ellipse is:

[0111]

[0112] The length of the semi-minor axis of the space ellipse is:

[0113]

[0114] According to the formula, the secondary curvature radius of the trajectory point is calculated as:

[0115] The calculated principal curvature radius r n and the minor curvature radius r b Substituting this into the resonance equation allows for pole prediction.

[0116] 3. Results presentation

[0117] 3.1 Comparison between sphere analysis results and prediction results

[0118] Table 1: Comparison of spherical pole analysis results and resonance equation prediction results

[0119]

[0120]

[0121] It can be concluded from Table 1 that the numerical errors of the attenuation angular frequency and the attenuation factor of the resonance equation prediction results are less than 2% when compared with the analytical results, which proves that the resonance equation prediction method is feasible and effective.

[0122] 3.2 Comparison of ellipsoid extraction results and prediction results

[0123] There are no analytical results available for the pole data corresponding to the crawling trajectory on the ellipsoid, so we used CST software simulation results for comparison. We selected an ellipsoid with a semi-major axis of 0.15 m and a semi-minor axis of 0.075 m, and set the frequency range to 0.1 GHz to 4.5 GHz.

[0124] Table 2: Comparison of 0.15m-0.075m ellipsoid pole extraction results and phase matching results

[0125]

[0126] Table 3: Comparison of 0.15m-0.075m ellipsoid pole extraction results and resonance equation prediction results

[0127]

[0128] Comparing Tables 2 and 3, the resonance equation method reduces the numerical error of the attenuation angular frequency by approximately 5.48% for the ellipsoidal shape compared to the phase matching method. Because the attenuation factor in the actual extracted results may include attenuation perpendicular to the predicted path, the attenuation factors are not completely consistent. However, the attenuation factor predicted by the resonance equation is numerically closer to the simulated extraction result than the phase matching result.

[0129] The above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit the same. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A method for predicting extreme features of a sphere and an ellipsoid, characterized by: The following steps are involved: S1. Analyze the scattering mechanism of spheres and ellipsoids; S2. Establish the resonance equation; S3, determining the main curvature radius and the secondary curvature radius corresponding to the crawling trajectory point; S4, predict the poles by resonance equation; The resonance equation in step S2 is: Among them, r n represents the principal radius of curvature, r b represents the minor curvature radius; The main curvature radius and the minor curvature radius corresponding to the crawling trajectory point in step S3 are determined by the propagation direction vector corresponding to the crawling wave trajectory point and the minor direction vector perpendicular thereto; The direction vector corresponding to the trajectory point is a vector that is along the propagation direction of the creeping wave and is tangent to the surface as the propagation direction vector; The secondary direction vector perpendicular to the propagation direction vector is obtained by cross-producting the normal vector corresponding to the surface trajectory point with the propagation direction vector; The specific curvature radius is calculated by the propagation direction and the sub-direction; The main curvature radius is calculated by the change of the surface curvature along the propagation direction, and the minor curvature radius is calculated by the change of the vector along the minor direction; In step S4, the calculated main curvature radius r n and the minor curvature radius r b Substitute into the resonance equation to predict the poles.

2. The method for predicting extreme features of a sphere and an ellipsoid according to claim 1, wherein: The scattering mechanism of the sphere and ellipsoid obtained by analysis in step S1 is used to clarify the physical meaning of the corresponding poles of the sphere and ellipsoid, and through a specific physical process, the physical meaning is converted into a mathematical formula for calculation to deduce the pole feature data; The specific scattering mechanism of spheres and ellipsoids is as follows: the resonance process of creeping wave diffraction is formed by the coherent superposition of the primary diffraction field and the diffraction field contributed to the observation point again after the creeping wave orbits the sphere. After the incident field excites the creeping wave diffraction field, it continuously crawls along the short-range line on the curved surface and radiates energy outward along the tangent direction of the surface. The creeping wave propagation process follows the generalized Fermat principle, that is, the creeping wave can only propagate along the geodesic of the curved surface.

3. The method for predicting the extreme features of a sphere and an ellipsoid according to claim 2, wherein: When the creeping wave crawls to the exit point position that can be received by a single-station radar, the phases of electromagnetic waves of different frequencies at this point are different, and coherent superposition or cancellation will occur, forming a resonance phenomenon. The multi-loop UTD diffraction field of the creeping wave is calculated based on the resonance phenomenon. When the incident point and exit point remain unchanged, the creeping wave diffraction field at the exit point changes in a geometric progression with the number of crawling circles. The common ratio is the attenuation coefficient α accumulated during one circle. The resonance equation 1-α is constructed, and then the parameter k in the corresponding attenuation coefficient is solved to obtain the theoretical pole predicted by the resonance equation.

4. The method for predicting extreme features of a sphere and an ellipsoid according to claim 1, wherein: step S2 includes: the creeping wave excited by each polarization component of the incident wave is composed of an infinite number of creeping wave modes, with m representing the order of the mode. Each mode has its own diffraction coefficient and attenuation constant. When the surface is a closed surface, each mode will propagate around the surface infinitely many times, forming a high-order diffraction field. l represents the number of creeping circles around the closed surface. The diffraction field at the field point is expressed as: Where, L = ∮dt′, represents the phase jump caused by the caustic region that the surface diffraction ray may pass through when propagating from Q1 to Q2, It represents the phase jump caused by all the caustic regions passed by when the closed surface is circled, and it can be sorted out as follows: The dyadic transfer function is:

5. The method for predicting extreme features of a sphere and an ellipsoid according to claim 1, wherein: Step S2 further includes: when the creeping wave circulates on the curved surface to form a closed path, selecting an exit point on the closed path, the creeping wave diffraction field at the exit point changes in a geometric progression with the number of creeping circles, and the common ratio is R. The total creeping wave field value at the exit point is the accumulation of the above geometric progression: in: Where L is a closed path that completes one circle.

6. The method for predicting extreme features of a sphere and an ellipsoid according to claim 1, wherein: On a sphere, the main curvature and the minor curvature are both the size of the sphere radius. When calculating the pole corresponding to the closed crawling trajectory of the ellipsoid parallel to the major axis, the main curvature radius and the minor curvature radius corresponding to each trajectory point on the crawling path are calculated; When calculating the main curvature radius, the curvature radius change along the propagation direction can be calculated directly using the ellipse curvature radius calculation formula: Where parameter a is the length of the minor axis of the ellipse, parameter b is the length of the major axis of the ellipse, and θ is the angle formed by the line connecting the point on the ellipse and the origin and the negative direction of the x-axis.

7. The method for predicting extreme features of a sphere and an ellipsoid according to claim 1, wherein: For the minor curvature radius corresponding to the trajectory point, the specific calculation is to calculate the elliptical curve formed by the intersection of the plane formed by the minor direction vector of the trajectory point and the corresponding normal vector and the ellipsoid surface, calculate the semi-major axis and semi-minor axis of the elliptical curve, and then calculate the curvature radius at the vertex of the semi-major axis using the elliptical curvature radius calculation formula as the minor curvature radius of the trajectory point.

8. The method for predicting extreme features of a sphere and an ellipsoid according to claim 7, wherein: In the Cartesian coordinate system, the equation of the ellipsoid is: The intersection line of the ellipsoid and the YOZ plane is used as the closed path for the specified calculation, that is, the meridian. The subsequent calculation is performed by expressing the ellipsoid equation in the spherical coordinate system. The ellipsoid equation in the spherical coordinate system is: The trajectory points on the meridian are According to the gradient formula Calculate the normal vector of any point on the ellipsoid as: The secondary direction vector corresponding to any trajectory point on the meridian is defined as 9. The method for predicting extreme point features of a sphere and an ellipsoid according to claim 1, wherein: The propagation direction vector corresponding to the trajectory point is the cross product of the secondary direction vector corresponding to the trajectory point and the normal vector corresponding to the trajectory point: The plane equation composed of the normal vector and the secondary direction vector of the trajectory point is calculated by the point normal equation of the plane using the trajectory point coordinates and the propagation direction vector: The radius of curvature corresponding to the vertex of the major axis of the ellipse formed by the intersection of the plane and the ellipsoid is the minor curvature radius required in the formula.

10. The method for predicting extreme point features of a sphere and an ellipsoid according to claim 1, wherein: By combining the plane equation and the ellipsoid equation, we can obtain the space ellipse equation: is the angle parameter of the corresponding trajectory point in the spherical coordinate system; The length of the semi-major axis of the space ellipse is: The length of the semi-minor axis of the space ellipse is: According to the formula, the secondary curvature radius of the trajectory point is calculated as:

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