A method for spherical multi-resolution discretization representation

Abstract

The present invention discloses a spherical multi-resolution discretization representation method, which is applicable to the field of three-dimensional construction of atomic sphere models. A spherical solid figure and an inscribed regular dodecahedron model in the sphere are established. Then, the projection points of the center points of all the faces of the regular dodecahedron on the sphere and the vertices of all the regular pentagons form a spherical point set, and each face projection point and each vertex of the regular pentagon are connected to form a set of plane triangles. After that, the projection points of the center points of the longest sides of each plane triangle on the sphere are connected with the three vertices of the plane triangle to form two plane triangles. After traversing the set, the newly obtained projection points are added to the spherical point set, and this step is repeated. Finally, the set of plane triangles and the spherical point set are the spherical discretization representation method. The triangular mesh formed by this method is more uniform, and the overall discretization division roughness is consistent. It can approximate the original continuous spherical surface with any specified precision, so as to accurately simulate the atomic sphere model.

Description

Technical Field

[0001] The present invention relates to a method for spherical multi - resolution discretization representation, especially a method for spherical multi - resolution discretization representation used in the study of atomic sphere models, and belongs to the field of constructing three - dimensional space models of molecules and atoms in computers. Background Art

[0002] Currently, during the experiment on molecules and atoms, since physical operations cannot be carried out, computers must be used for simulation. When visualizing molecules and atoms, atoms are usually represented by spheres, which requires converting the spherical surface structure of the sphere into a discretized representation form that can be read by a computer. The expected discretization method should be two - way: that is, the atomic sphere is discretely represented by specifying the atomic center and radius, and conversely, the center of gravity coordinates, radius and other information of the atomic sphere should also be calculable from the discretized data. However, most of the existing discretization methods for atomic sphere models use a method similar to the longitude and latitude of the earth to divide layers. This method is simple for discretization, but it is difficult to directly and quickly calculate the original center coordinates and radius of the sphere from the discretized representation data because it lacks the discretization framework. In addition, there is also the problem of uneven discretization.

[0003] In 3D modeling, triangular patches or polygon patches are usually used to approximately represent the continuous modeling surface. In the field of computer vision representation, triangular patches can be regarded as "Butter and Bread", because triangular patches can approximately represent the shape surface and have many advantages such as being simple, intuitive and easy to segment. Although there may be a disadvantage of a relatively large approximation error compared with polygon patches when the number of segmentation times is the same. The discretization representation for an atomic sphere as a model is more complex than that of a polyhedron. Due to the regular and symmetric bending form of the spherical surface, when directly applying triangular patches for discretization, it involves the problem of the segmentation scale of the spherical surface. In other words, how close the discretized representation result of the spherical surface is to the smooth spherical surface. There are many solutions for the discretization representation of the spherical surface. An intuitive and simple method is to use polar coordinates. Each time, an angle is opened for the Z - axis angle, so that the sphere is divided into ring slices similar to dimensions. Then, each ring is further divided by the angle of rotation around the Z - axis, that is, similar to the division of longitude lines. This processing method is simple and intuitive, but the size difference of the triangular meshes is relatively large. The pole segmentation is more delicate, but the segmentation around the equator line is relatively rough. The method of using a similar longitude - latitude segmentation inevitably brings the problem of a large difference in the segmentation uniformity. Summary of the Invention

[0004] Aiming at the deficiencies of the prior art, a spherical multi-resolution discretization representation method is provided to overcome the problem of uneven triangular patches after atomic spherical discretization segmentation using longitude and latitude, and to achieve approximation of the original continuous atomic sphere surface with any specified precision, making the triangular mesh more uniform and the overall discretization roughness more consistent.

[0005] To achieve the above technical objectives, the present invention provides a spherical multi-resolution discretization representation method. First, a spherical solid graph representing the atomic sphere is established in a computer, and an inscribed regular dodecahedron model is established within the spherical solid graph. Each face of the regular dodecahedron model is a regular pentagon, and the maximum number of spherical divisions is set. Then, the projection points of the center points of all faces of the regular dodecahedron model on the sphere, and all the vertices of the regular pentagons form a spherical point set. The five vertices of each face are respectively connected to the projection points on the sphere to form a set of planar triangles. The projection points of the center points of the longest sides of each planar triangle in the set of planar triangles on the sphere, and the three vertices of the planar triangle are connected to form two new planar triangles. Traverse the set of planar triangles to form a new set of planar triangles with double the number, add the new planar triangles to the set of planar triangles to increase the length of the set of planar triangles, add the projection points of the center points of the longest sides of all planar triangles in the set of planar triangles with increased length on the sphere to the spherical point set to increase the length of the spherical point set. Repeat the coordinate calculation of the projection points of the center points of the longest sides on the sphere in this step, continuously divide new planar triangles to expand the length of the set of planar triangles, and expand the length of the spherical point set through the projection points of the center points of the longest sides of the new planar triangles until the requirements of the preset maximum number of spherical divisions S are met. Obtain the final set of planar triangles and the final spherical point set, which are used as the representation method of spherical discretization, so as to realize the discretization representation of the spherical body of the atomic sphere model.

[0006] Furthermore, the specific steps of representing the atomic sphere model by using the discretization method are as follows:

[0007] Step 1) Generate a spherical solid graph with a radius of r by a computer to simulate the atomic sphere, establish a right-handed coordinate system O with the center of the sphere as the coordinate origin xyz , set the maximum number of spherical divisions S, and initialize the number of divisions s = 0. Use the inscribed regular dodecahedron in the spherical solid graph as the basic segmentation model, where each face of the regular dodecahedron is a planar regular pentagon, with a total of 20 vertices. At the same time, define and initialize the spherical point set P, which contains the 20 vertices where the regular dodecahedron is connected to the sphere and the coordinates in the O xyz coordinate system.

[0008] Step 2) Calculate the center points of each planar regular pentagon of the regular dodecahedron in O xyzThe coordinates in the coordinate system are used to determine the coordinates of the projection points of the center points of each planar regular pentagon on the spherical solid figure. A total of 12 projection points can be obtained. These 12 projection points are added to the spherical point set in step 1) to obtain a new spherical point set P'. For the convenience of subsequent description, let P = P'.

[0009] Step 3) Connect the projection points of the center points of each face of the regular dodecahedron on the sphere with the 5 vertices of each plane to form 60 planar triangles. Define and initialize the planar triangle set T, where each element in T is the coordinate combination of the three vertices of each planar triangle in the O xyz coordinate system, that is, the elements in T are the x, y, z coordinates and vertex indices of the three vertices forming the triangle, and the spherical surface can be discretized by the planar triangle set T.

[0010] Step 4) Use the vertex coordinate combinations of the planar triangle set T to calculate the midpoint coordinates of the longest side of each planar triangle, then determine the coordinates of the projection points of the midpoints of the longest sides of each planar triangle on the spherical solid figure and form a projection point set. Then add the obtained projection point set to the spherical point set P to increase the length of the spherical point set P, obtaining a new spherical point set P''. For the convenience of algorithm description, let P = P''.

[0011] Step 5) Connect the spherical projection points of the midpoints of the longest sides of the planar triangles in the planar triangle set T with the three vertices of the planar triangles to form two planar triangles. Traverse all the planar triangles in the planar triangle set T to obtain a new planar triangle set T'. At the same time, replace the previous planar triangle set T with the obtained new planar triangle set T', that is, let T = T'. After completion, make the segmentation times s = s + 1.

[0012] Step 6) Use the spherical point set P and the planar triangle set T obtained in step 4) and step 5), and loop through step 4) and step 5) until the segmentation times s of the planar triangles is equal to the preset maximum spherical segmentation times S, then stop the loop and output the finally obtained spherical point set P and planar triangle set T. The larger the maximum spherical segmentation times S, the higher the resolution of the spherical surface segmentation. The spherical point set and the planar triangle set are used to represent the way of discretizing the spherical surface of the atomic sphere model, which directly provides support for the subsequent analysis of the ball-and-stick model of the molecule composed of atoms.

[0013] Furthermore, the specific process of step 1) is as follows:

[0014] Step 11) For the spherical solid figure with a radius of r generated by the computer, establish a right-handed coordinate system O with the center of the sphere as the coordinate origin xyz ;

[0015] Step 12) Set the maximum number of spherical divisions S, where S is a positive real number greater than 1, and initialize the division number s = 0;

[0016] Step 13) Define and initialize the spherical point set with the 20 vertices of the regular dodecahedron inscribed in the spherical solid figure in Step 11), and obtain:

[0017] P = {p1, p2,..., p i ,..., p 20} (1)

[0018] In the formula: i ∈ (1, 20), p i is the vertex of the regular dodecahedron on the sphere, and p i has coordinates (x xyz , y i , z i ) in O i .

[0019] Furthermore, the specific process of Step 2) is as follows:

[0020] Step 21) The regular dodecahedron inscribed in the spherical solid figure has twelve faces, and each face is a regular pentagon. Then the vertices of each regular pentagon are represented as follows:

[0021]

[0022] In the formula: represents the vertex of the regular pentagon on each face of the regular dodecahedron, and this vertex belongs to the spherical point set P, that is represents the three-dimensional coordinates of the vertex of each regular pentagon in the coordinate system O xyz , j ∈ (1, 12) represents the index value of the twelve faces of the regular dodecahedron, and k ∈ (1, 5) represents the index value of the five vertices of each face of the regular dodecahedron;

[0023] Step 22) Given the coordinates of the five vertices of each regular pentagon of the regular dodecahedron inscribed in the spherical solid figure, obtain the coordinates of the center point of each regular pentagon as:

[0024]

[0025] In the formula: represents the coordinates of the center point of the regular pentagon on the jth face of the regular dodecahedron, represents the coordinates of the kth vertex of the regular pentagon on the jth face of the regular dodecahedron;

[0026] Step 23) After obtaining the center points of each regular pentagon of the regular dodecahedron inscribed in the spherical solid figure Then, its projection point on the sphere can be obtained for:

[0027]

[0028] Where: is the coordinate of the projection point of the center point of the plane regular pentagon of the jth face of the regular dodecahedron on the sphere, r is the radius of the sphere, Represents the center point of each planar regular pentagon The distance to the coordinate origin;

[0029] Step 24) Traverse each face of the inscribed regular dodecahedron to obtain a set of projection points P 投 :

[0030]

[0031] Where: Represents the projection point of the center point of the plane regular pentagon of the jth face of the regular dodecahedron on the sphere;

[0032] Step 25) The projection point set P 投 Adding it to the spherical point set P, we get a new spherical point set P′ with 32 elements, which is expressed as follows:

[0033] P′={p1, p2, ..., p i ,…,p 32} (6) Where: i∈(1, 32). In order to facilitate the representation of the new spherical point set later, it is still represented by P, that is, P=P′.

[0034] Further, the specific process of step 3) is as follows:

[0035] Step 31) Connect the projection points of the center points of each regular pentagon of the inscribed regular dodecahedron on the spherical surface and the five vertices of a regular pentagon in the plane Each projection point can form five plane triangles, traversing the projection point set P in step 24) 投 There are twelve projection points in the image, and finally 60 plane triangles can be obtained;

[0036] Step 32) Record and combine the three vertices of the plane triangle formed by each projection point and the plane pentagon, define and initialize the plane triangle set T, expressed as:

[0037] T={ <p l , p m , p n >|p l , p m , p n∈P} (7)

[0038] Where: p l , p m , p n are the three vertices forming a plane triangle, <p l , p m , p n > represents the plane triangle formed by these three vertices, where P is the set of spherical points obtained in step 25), and l, m, n are the index values of the spherical points in the set of spherical points P.

[0039] Furthermore, the specific process of step 4) is as follows:

[0040] Step 41) For each plane triangle <p l , p m , p m > in the set of plane triangles T, where p l , p m , p n The three vertices are elements in the set of spherical points P, and determine the midpoint coordinates of the longest side among the three sides of each plane triangle, specifically as follows:

[0041]

[0042] Where: β ∈ (1, h), h is the number of elements in the set of plane triangles T, is the midpoint of the longest side of the β-th plane triangle in the set of plane triangles T in the coordinate system O xyz . The coordinates in it are (x l , y l , z l ), (x m , y m , z m ), (x n , y n , z n ) are the coordinates of the three vertices p l , p m , p n in the coordinate system O xyz . is the plane triangle <p l , p m , p n > The magnitude of the vector formed by the three sides , that is, the lengths of the three sides, represents the longest value among the three sides;

[0043] Step 42) Get each plane triangle in the plane triangle set T <p l , p m , p n The midpoint of the longest side of Afterwards, obtain Projection point of a point on a spherical solid for:

[0044]

[0045] Where: The midpoint of the longest side of the βth plane triangle in the plane triangle set T The coordinates of the projection point on the sphere, r is the radius of the sphere, Represents the midpoint of the longest side of each plane triangle The distance to the coordinate origin;

[0046] Step 43) Traverse the plane triangle set T to obtain the projection point set P′ composed of the projection points of the longest side midpoint of each plane triangle on the spherical surface 投 :

[0047]

[0048] Where: h is the number of elements in the plane triangle set T, It is the projection point of the midpoint of the longest side of the β-th plane triangle in the plane triangle set T on the sphere;

[0049] Step 44) Projection point set P′ 投 Add to the spherical point set P to obtain a new spherical point set P″, which is expressed as follows:

[0050] P″=P+P′ 投 ={p1, p2, ..., p i ,…,p g} (11)

[0051] Where: i∈(1,g), g is the projection point set P′ 投 The sum of the number of elements in and the number of elements in the spherical point set P. For the needs of subsequent algorithms, the new spherical point set is still represented by P, that is, P=P".

[0052] Further, the specific process of step 5) is as follows:

[0053] Step 51) Connect the projection points of the longest side midpoint of each plane triangle in the plane triangle set T on the sphere and the three vertices p of the corresponding plane triangle l , p m , p n, two planar triangles can be formed, and by traversing the set of projected points P' obtained in step 43) 投 , a new set of planar triangles T' can be obtained:

[0054] T' = {<p l , p m , p n >|p l , p m , p <m:math xmlns:m="http: / / schemas.openxmlformats.org / officeDocument / 2006 / math"><m:mrow><m:mrow><m:mn>0< / m:mn><m:mn>0< / m:mn><m:mn>0< / m:mn><m:mn>0< / m:mn><m:mn>0< / m:mn><m:mn>7< / m:mn><m:mn>3< / m:mn>< / m:mrow>< / m:mrow>< / m:math> n ∈ P} (12)

[0055] In the formula: p l , p m , p n are the three vertices that make up the planar triangle, <p l , p m , p n > represents the planar triangle formed by these three vertices, where P is the set of spherical points obtained in step 44), and l, m, n are used as the index values of the spherical points in the set of spherical points P;

[0056] In step 52), for the need to facilitate the algorithm loop, the obtained new set of planar triangles T' is still represented by T, that is, T = T';

[0057] In step 53), one division is completed, and the number of division times value is incremented by one, that is, s = s + 1.

[0058] Furthermore, the specific process of step 6) is as follows:

[0059] In step 61), it is judged whether the number of division times s is equal to the maximum number of division times S. If s = S, then step 63) is performed; otherwise, step 62) is performed;

[0060] In step 62), based on the set of spherical points P obtained in step 44) and the set of planar triangles T obtained in step 51), steps 41) to 44), steps 51) to 53), and step 61) are sequentially performed;

[0061] In step 63), the set of spherical points P obtained in step 44) and the set of planar triangles T obtained in step 51) are output. The spherical solid figure is represented by all the spherical points in the set of spherical points and the planar triangles in the set of planar triangles formed between the spherical points, achieving the purpose of spherical discretization.

[0062] Beneficial effects:

[0063] This method uses a regular pentagon polyhedron as the segmentation skeleton of the atomic sphere model. On this basis, further subdivision and triangulation representation of the atomic spherical model are carried out, and the simplicity of discrete calculation is maintained during segmentation, and the area sizes of the triangular patches after division are similar, so that the discrete representation is relatively uniform, and it is possible to approximate the spherical surface with any specified precision, achieving the purpose of evenly dividing the spherical surface. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 is a schematic flow chart of the spherical multi-resolution discretization representation method of the present invention;

[0065] Figure 2 is a wireframe diagram of the regular pentagon polyhedron of the spherical multi-resolution discretization representation method of the present invention.

[0066] Figure 3 is the circumscribed sphere of the regular pentagon polyhedron of the spherical multi-resolution discretization representation method of the present invention

[0067] Figure 4 is the triangle formed by the regular pentagon of the spherical multi-resolution discretization representation method of the present invention.

[0068] Figure 5 is the sphere formed by using Matlab sphere(100) of the spherical multi-resolution discretization representation method of the present invention.

[0069] Figure 6 is the sphere constructed by the spherical multi-resolution discretization representation method of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0070] The present invention will be further described below with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solutions of the present invention and cannot be used to limit the protection scope of the present invention.

[0071] Figure 5 What is shown is the drawback that the representative atomic sphere model drawn by using the sphere method of matlab in the prior art has fine pole segmentation but rough equatorial segmentation.

[0072] This application is directed to a spherical multi-resolution discretization representation method for an atomic sphere model. Similarly, it can also be directed to a spherical multi-resolution discretization representation method for other sphere models by visually reading. Only some relatively simple parameters of the actual sphere need to be obtained to image the spherical simulation of the sphere in the computer, with fast boundaries. At the same time, it also provides a method for imaging a sphere object in the computer through measurement and participating in other related simulation experiments.

[0073] ]>Such as Figure 1As shown in the figure, a spherical multi-resolution discretization representation method of the present invention. Using a computer to generate a spherical solid graph of an atomic sphere model, taking the regular dodecahedron inscribed in the sphere as the basic model, and dividing and discretizing the sphere; first, establish a mathematical model mainly composed of the spherical solid graph and the regular dodecahedron inscribed in the sphere, and set the maximum number of divisions of the sphere; then, use the projection points of the center points of all the faces of the regular dodecahedron on the sphere and all the vertices of the regular pentagons to form a set of spherical points, and connect the projection point of each face and each vertex of the regular pentagon to form a set of plane triangles; then, use the projection points of the center points of the longest sides of each plane triangle in the set of plane triangles on the sphere, and connect the three vertices of the plane triangle to form two plane triangles, traverse the set of plane triangles, and then obtain a new set of plane triangles, and add the newly obtained projection points to the set of spherical points to obtain a new set of spherical points. Repeat this step until the maximum number of divisions of the sphere is reached; finally, obtain the set of plane triangles and the set of spherical points, which are used as the marking method for spherical discretization. The advantages of this method compared with other segmentation and discretization methods: the formed triangular mesh is more uniform, the overall roughness of the discrete division is relatively consistent, and the original continuous spherical surface can be approximated with any specified accuracy. See specifically Figure 6 。

[0074] As Figure 1 shown in the figure, the present invention adopts a spherical multi-resolution discretization representation method as follows:

[0075] Step 1) For the spherical solid graph with a radius of r generated by a computer, establish a right-handed coordinate system O with the center of the sphere as the coordinate origin xyz , set the maximum number of divisions S of the sphere, and initialize the number of divisions s = 0; use the regular dodecahedron inscribed in the spherical solid graph as the basic division model, where each face of the regular dodecahedron (as shown in Figure 2 and 3 ) is a regular pentagon on a plane, with a total of 20 vertices. At the same time, define and initialize the set of spherical points P, where P contains the 20 vertices where the regular dodecahedron is connected to the sphere and the coordinates in the O xyz coordinate system.

[0076] The specific process is as follows:

[0077] Step 11) For the spherical solid graph with a radius of r generated by a computer, establish a right-handed coordinate system O with the center of the sphere as the coordinate origin xyz 。

[0078] Step 12) Set the maximum number of divisions S of the sphere, S is a positive real number greater than 1, and initialize the number of divisions s = 0.

[0079] Step 13) Use the 20 vertices of the regular dodecahedron inscribed in the spherical solid graph in step 11) on the sphere to define and initialize the set of spherical points, and obtain:

[0080] P = {p1, p2,..., p i ,..., p 20} In formula (1): i ∈ (1, 20), p i is the vertex of the regular dodecahedron on the spherical surface, and p i has the coordinates (x xyz , y i , z i ) in O i .

[0081] Step 2) Calculate the coordinates of the center point of each regular pentagon on the plane of the regular dodecahedron in the O xyz coordinate system, determine the coordinates of the projection point of this center point on the spherical solid figure, a total of 12 projection points can be obtained, add these 12 projection points to the spherical point set in Step 1) to get a new spherical point set P', for the convenience of description later, let P = P'.

[0082] The specific process is as follows:

[0083] Step 21) The inscribed regular dodecahedron of the spherical solid figure has twelve faces, and each face is a regular pentagon on the plane, then the vertices of each regular pentagon on the plane are represented as follows:

[0084]

[0085] In the formula: represents the vertex of the regular pentagon on each face of the regular dodecahedron, and this vertex belongs to the spherical point set P in Step 13), that is At the same time determines the coordinates of each vertex of the regular pentagon on the plane in the coordinate system O xyz , j ∈ (1, 12), represents the index value of the twelve faces of the regular dodecahedron, k ∈ (1, 5), represents the index value of the five vertices of each face of the regular dodecahedron,

[0086] Step 22) Given the coordinates of the five vertices of each regular pentagon on the inscribed regular dodecahedron of the spherical solid figure, find the coordinates of the center point of each regular pentagon on the plane as:

[0087]

[0088] In the formula: represents the coordinates of the center point of the regular pentagon on the j-th face of the regular dodecahedron, represents the coordinates of the k-th vertex of the regular pentagon on the j-th face of the regular dodecahedron.

[0089] Step 23) After obtaining the center point of each regular pentagon on the inscribed regular dodecahedron of the spherical solid figure After that, the projection point on the spherical surface can be obtained as follows:

[0090]

[0091] In the formula: are the coordinates of the projection point on the spherical surface of the center point of the regular pentagon on the j-th face of the regular dodecahedron, r is the radius of the spherical surface, represents the center point of each regular pentagon on the plane to the distance from the origin of coordinates.

[0092] Step 24) Traverse each face of the inscribed regular dodecahedron, and a set P of projection points can be obtained 投 :

[0093]

[0094] In the formula: represents the projection point on the spherical surface of the center point of the regular pentagon on the j-th face of the regular dodecahedron.

[0095] Step 25) Add the set P of projection points 投 to the set P of spherical surface points, and a new set P' of spherical surface points with 32 elements is obtained, which is expressed as follows:

[0096] P' = {p1, p2,..., p i ,..., p 32} (6)

[0097] In the formula: i ∈ (1, 32). For the convenience of subsequent representation, the new set P of spherical surface points is still denoted as P, that is, P = P'.

[0098] Step 3) As Figure 4 shown, connect the projection points of the center points of each regular pentagon on the plane of the regular dodecahedron and the 5 vertices of each regular pentagon on the plane, 5 plane triangles can be formed. Traverse all the faces of the regular dodecahedron, and 60 plane triangles can be obtained. Define and initialize the set T of plane triangles, where each element in T is the coordinate combination of the three vertices of each plane triangle in the O xyz coordinate system, indicating that all the plane triangles in the set of plane triangles can be used to discretize the spherical surface.

[0099] The specific process is as follows:

[0100] Step 31) Connect the projection points on the spherical surface of the center points of each regular pentagon on the plane of the inscribed regular dodecahedron of the spherical solid figure and the five vertices of the regular pentagon on the plane Then each projection point can form five plane triangles. Traverse the set P of projection points in Step 24) 投Among the twelve projection points, 60 planar triangles can be obtained finally.

[0101] Step 32) Record and combine the three vertices of the planar triangle formed by each projection point and the planar pentagon, define and initialize the planar triangle set T, which is expressed as:

[0102] T = {<p l , p m , p n >|p l , p m , p n ∈P} (7)

[0103] In the formula: p l , p m , p n are the three vertices of the planar triangle, <p l , p m , p n > represents the planar triangle formed by these three vertices, where P is the spherical point set obtained in Step 25), and l, m, n are the index values of the spherical points in the spherical point set P.

[0104] Step 4) For each combination of vertex coordinates of the planar triangles in the planar triangle set T, calculate the midpoint coordinates of the longest side of the planar triangle, determine the projection point coordinates of the midpoint of the longest side of the planar triangle on the spherical solid figure, traverse all the planar triangles in the planar triangle set T, form a projection point set with the newly obtained projection points, add the obtained projection point set to the spherical point set P to obtain a new spherical point set P″. For the convenience of algorithm description, let P = P″.

[0105] The specific process is as follows:

[0106] Step 41) For each planar triangle <p l , p m , p n > in the planar triangle set T, where p l , p m , p n the three vertices are elements in the spherical point set P, and determine the midpoint coordinates of the longest side among the three sides of each planar triangle, which is specifically shown as follows:

[0107]

[0108] In the formula: β ∈ (1, h), h is the number of elements in the planar triangle set T, is the midpoint of the longest side of the β-th planar triangle in the planar triangle set T in the coordinate system O xyz and (xl , y l , z l ), (x m , y m , z m ), (x n , y n , z n ) are the coordinates of the three vertices p l , p m , p n in the coordinate system O xyz . is the magnitude of the vectors formed by the three sides of the planar triangle <p l , p m , p n >, that is, the lengths of the three sides, and represents the value of the longest length among the three sides. represents the numerical value of the longest side among the three sides of the planar triangle.

[0109] Step 42) Obtain the midpoint of the longest side l , p m , p n > of each planar triangle in the set T of planar triangles , and then obtain the projection point of the point on the spherical solid figure as:

[0110]

[0111] In the formula: is the coordinate of the projection point of the midpoint of the longest side of the β-th planar triangle in the set T of planar triangles on the spherical surface, r is the radius of the spherical surface, represents the distance from the midpoint of the longest side of each planar triangle to the origin of the coordinates.

[0112] Step 43) Traverse the set T of planar triangles, and a set P' of projection points formed by the projection points of the midpoints of the longest sides of each planar triangle on the spherical surface can be obtained 投 :

[0113]

[0114] In the formula: h is the number of elements in the set T of planar triangles, is the projection point of the midpoint of the longest side of the β-th planar triangle in the set T of planar triangles on the spherical surface.

[0115] Step 44) Add the set P' of projection points 投 to the set P of spherical surface points to obtain a new set P'' of spherical surface points, which is expressed as follows:

[0116] P″ = P + P′ 投 = {p1, p2, …, p i , …, p g} (11)

[0117] Where: i ∈ (1, g), g is the sum of the number of elements in the projected point set P′ 投 and the number of elements in the spherical point set P. For the needs of subsequent algorithms, the new spherical point set is still denoted as P, that is, P = P″.

[0118] Step 5) Connect the projection points of the midpoints of the longest sides of each plane triangle in the plane triangle set T to the three vertices of the plane triangle, two plane triangles can be formed. Traverse all the plane triangles in the plane triangle set, a new plane triangle set T′ can be obtained. At the same time, replace the previous plane triangle set T with the obtained new plane triangle set T′, that is, let T = T′, and after completion, make the segmentation times s = s + 1.

[0119] The specific process is as follows:

[0120] Step 51) Connect the projection points on the sphere of the midpoints of the longest sides of each plane triangle in the plane triangle set T and the three vertices p l , p m , p n of the corresponding plane triangle, two plane triangles can be formed. Traverse the projected point set P′ obtained in step 43) 投 , a new plane triangle set T′ can be obtained:

[0121] T′ = {<p l , p m , p n >|p l , p m , p n ∈ P} (12)

[0122] Where: p l , p m , p n are the three vertices that make up the plane triangle, <p l , p m , p n > represents the plane triangle formed by these three vertices, where P is the spherical point set obtained in step 44), and l, m, n are the index values of the spherical points in the spherical point set P.

[0123] Step 52) For the need to facilitate the algorithm loop, denote the obtained new plane triangle set T′ as T, that is, T = T′.

[0124] Step 53) Complete one segmentation, increment the segmentation count value by one, i.e., s = s + 1.

[0125] Step 6) Using the spherical point set P and the planar triangle set T obtained in Step 4) and Step 5), loop through Step 4) and Step 5). If the segmentation count s is equal to the maximum spherical segmentation count S, stop the loop and output the finally obtained spherical point set P and planar triangle set T. Otherwise, continue to loop through Step 4) and Step 5). Here, the magnitude of the maximum spherical segmentation count controls the resolution of the spherical segmentation, and the spherical point set and the planar triangle set represent the marking method of the spherical discretization.

[0126] The specific process is as follows:

[0127] Step 61) Determine whether the segmentation count s is equal to the maximum segmentation count S. If s = S, proceed to Step 63); otherwise, proceed to Step 62).

[0128] Step 62) Based on the spherical point set P obtained in Step 44) and the planar triangle set T obtained in Step 51), sequentially perform Step 41) to Step 44), Step 51) to Step 53), and Step 61).

[0129] Step 63) Output the spherical point set P obtained in Step 44) and the planar triangle set T obtained in Step 51). The spherical solid figure is represented by all the spherical points in the spherical point set and the planar triangles in the planar triangle set formed between the spherical points, achieving the purpose of spherical discretization.

Claims

1. A method for spherical multi - resolution discretization representation, characterized in that: First, establish a spherical solid figure representing an atomic sphere in a computer, and establish an inscribed regular dodecahedron model within the spherical solid figure. Each face of the regular dodecahedron model is a regular pentagon, and the maximum number of spherical divisions is set. Then, form a set of spherical points with the projection points on the sphere of the center points of all the faces of the regular dodecahedron model and the vertices of all the regular pentagons. Connect the five vertices of each face to the projection points on the sphere respectively to form a set of planar triangles. Use the projection points on the sphere of the midpoints of the longest sides of each planar triangle in the set of planar triangles, and connect them to the three vertices of the planar triangle to form two new planar triangles. Traverse the set of planar triangles to form a new set of planar triangles with double the quantity. Add the new planar triangles to the set of planar triangles to increase the length of the set of planar triangles. Add the projection points on the sphere of the midpoints of the longest sides of all the planar triangles in the set of planar triangles with increased length to the set of spherical points to increase the length of the set of spherical points. Repeat the coordinate calculation of the projection points of the midpoints of the longest sides on the sphere in this step, continuously divide to generate new planar triangles to expand the length of the set of planar triangles, and expand the length of the set of spherical points through the projection points on the sphere of the midpoints of the longest sides of the new planar triangles until the requirements of the preset maximum number of spherical divisions S are met. Obtain the final set of planar triangles and the final set of spherical points, which are used as the representation of spherical discretization, thereby realizing the discrete representation of the spherical surface of the atomic sphere model.

2. The spherical multi-resolution discretization representation method according to claim 1, characterized in that The specific steps for representing the atomic sphere model by using the discretization method are as follows: Step 1) Generate a spherical solid figure with a radius of r by computer to simulate an atomic sphere, and establish a right-handed coordinate system O with the center of the sphere as the coordinate origin xyz , set the maximum number of spherical divisions S, and initialize the division number s = 0; use the regular dodecahedron inscribed in the spherical solid figure as the basic division model, where each face of the regular dodecahedron is a regular pentagon in a plane, with a total of 20 vertices. At the same time, define and initialize the spherical point set P, which includes the 20 vertices where the regular dodecahedron is in contact with the sphere and O xyz coordinates in the coordinate system; Step 2) Calculate the coordinates of the center point of each regular pentagon on the regular dodecahedron in the O xyz coordinate system, determine the coordinates of the projection points of the center points of each regular pentagon on the spherical solid figure, and a total of 12 projection points can be obtained. Add these 12 projection points to the spherical point set in Step 1) to obtain a new spherical point set P′. For the convenience of subsequent description, let P = P′; Step 3) Connect the projection points of the center points of each face of the regular dodecahedron on the spherical surface with the 5 vertices of each plane to form 60 plane triangles. Define and initialize the plane triangle set T, where each element in T is the coordinate combination of the three vertices of each plane triangle in the O xyz coordinate system, that is, the elements in T are the x, y, z coordinates and vertex indices of the three vertices forming the triangle, and the spherical surface can be discretized by the plane triangle set T; Step 4) Use the vertex coordinate combinations of the set of planar triangles T respectively to calculate the midpoint coordinates of the longest sides of each planar triangle, then determine the projection point coordinates of the midpoints of the longest sides of each planar triangle on the spherical solid figure and form a set of projection points, and then add the obtained set of projection points to the set of spherical points P to increase the length of the set of spherical points P, obtaining a new set of spherical points P″. For the convenience of algorithm description, let P = P″. Step 5) Connect the spherical projection points of the midpoints of the longest sides of the planar triangles in the set of planar triangles T to the three vertices of the planar triangle to form two planar triangles. Traverse all the planar triangles in the set of planar triangles T to obtain a new set of planar triangles T′. At the same time, replace the previous set of planar triangles T with the obtained new set of planar triangles T′, that is, let T = T′. After completion, make the number of divisions s = s + 1. Step 6) Use the set of spherical points P and the set of planar triangles T obtained in Step 4) and Step 5), and loop through Step 4) and Step 5) until the number of divisions s of the planar triangles is equal to the preset maximum number of spherical divisions S, then stop the loop and output the finally obtained set of spherical points P and the set of planar triangles T. The larger the maximum number of spherical divisions S, the higher the resolution of the spherical division. The set of spherical points and the set of planar triangles are used as the representation of the spherical discretization of the atomic sphere model, which directly provides support for the subsequent analysis of the ball-and-stick model of molecules composed of atoms.

3. The spherical multi - resolution discretization representation method according to claim 2, characterized in that, The specific process of the said Step 1) is as follows: Step 11) For a computer-generated spherical solid figure with a radius of r, establish a right-handed coordinate system O with the center of the sphere as the origin of coordinates xyz ; Step 12) Set the maximum number of spherical divisions S, where S is a positive real number greater than 1, and initialize the division number s = 0; Step 13) Define and initialize the spherical point set with the 20 vertices of the regular dodecahedron inscribed in the spherical solid figure in Step 11), and obtain: P = {p1, p2, …, p i , …, p 20} (1) Where: i ∈ (1, 20), p i is the vertex of the regular dodecahedron on the spherical surface, and p i has coordinates (x xyz , y i , z i ) in O i .

4. The spherical multi-resolution discretization representation method according to claim 2, wherein The specific process of the said Step 2) is as follows: Step 21) The regular dodecahedron inscribed in the spherical solid figure has twelve faces, and each face is a regular pentagon on a plane. Then the vertices of each regular pentagon on the plane are represented as follows: In the formula: represents the vertex of the regular pentagon on each face of the regular dodecahedron, and this vertex belongs to the spherical point set P, that is represents the three-dimensional coordinates of the vertex of each regular pentagon in the coordinate system O xyz where j ∈ (1, 12) represents the index value of the twelve faces of the regular dodecahedron, and k ∈ (1, 5) represents the index value of the five vertices of each face of the regular dodecahedron; Step 22) Given the five vertex coordinates of each planar pentagon of the regular dodecahedron inscribed in the spherical solid figure, the center point of each planar regular pentagon is obtained The coordinates are as follows: Wherein: represents the central point coordinates of the regular pentagon on the j-th face of the regular dodecahedron, represents the coordinates of the k-th vertex of the regular pentagon on the j-th face of the regular dodecahedron; Step 23) After obtaining the center points of each planar regular pentagon of the inscribed regular dodecahedron of the spherical solid figure its projection points on the spherical surface can be obtained as follows: In the formula: are the coordinates of the projection point on the spherical surface of the center point of the j-th plane regular pentagon of the regular dodecahedron, r is the radius of the spherical surface, represents the center point of each plane regular pentagon the distance to the origin of coordinates; Step 24) Traverse each face of the inscribed regular dodecahedron to obtain a set P of projection points 投 : In the formula: represents the projection point on the spherical surface of the center point of the regular pentagon on the plane of the j-th face of the regular dodecahedron; Step 25) Add the set of projection points P 投 to the set of spherical points P to obtain a new set of spherical points P' with 32 elements, which is represented as follows: P′ = {p1, p2, …, p i , …, p 32} (6) Where: i ∈ (1, 32). For the convenience of representing the new spherical point set later, it is still denoted as P, that is, P = P'.

5. The spherical multi - resolution discretization representation method according to claim 2, wherein The specific process of the said Step 3) is as follows: Step 31) Connect the projection points on the sphere of the central points of each regular pentagon on the inscribed regular dodecahedron of the spherical solid figure and the five vertices of the regular pentagon on the plane Then each projection point can form five plane triangles. Traverse the twelve projection points in the projection point set P in step 24) 投 Finally, 60 plane triangles can be obtained; Step 32) Record and combine the three vertices of the plane triangle formed by each projection point and the regular pentagon on the plane, define and initialize the plane triangle set T, and represent it as: T = {<p l , p m , p n 〉 | p l , p m , p n ∈ P} (7) where: p l , p m , p n are the three vertices forming a planar triangle, <p l , p m , p n > represents the planar triangle formed by these three vertices, where P is the set of spherical points obtained in step 25), and l, m, n are the index values of the spherical points in the set of spherical points P.

6. The spherical multi - resolution discretization representation method according to claim 2, characterized in that, The specific process of the said Step 4) is as follows: Step 41) For each planar triangle <p l , p m , p n > in the set T of planar triangles, where the three vertices of p l , p m , p n are elements of the set P of spherical points, and determine the coordinates of the midpoint of the longest side among the three sides of each planar triangle, specifically as follows: ​ where: β ∈ (1, h), h is the number of elements in the set T of planar triangles, is the midpoint of the longest side of the β-th planar triangle in the set T of planar triangles in the coordinate system O xyz coordinates, (x l , y l , z l ), (x m , y m , z m ), (x n , y n , z n ) are the coordinates of the three vertices p l , p m , p n in the coordinate system O xyz , is the magnitude of the vector formed by the three sides of the planar triangle <p l , p m , p n >, that is, the lengths of the three sides, and represents the value of the longest length among the three sides; ​ Step 42) After obtaining the midpoint of the longest side of each planar triangle <p l , p m , p n > in the set T of planar triangles, the projection point of the point on the spherical solid figure is obtained as: ​​ In the formula: is the midpoint of the longest side of the β-th plane triangle in the set T of plane triangles coordinates of the projection point on the spherical surface, r is the radius of the spherical surface represents the midpoint of the longest side of each plane triangle distance to the origin of coordinates; Step 43) Traverse the set T of planar triangles, and a set P' of projection points formed by the projection points on the sphere of the midpoints of the longest sides of each planar triangle can be obtained. 投 : where: h is the number of elements in the set T of planar triangles, is the projection point on the spherical surface of the midpoint of the longest side of the β-th planar triangle in the set T of planar triangles; Step 44) Add the set of projection points P′ 投 to the set of spherical points P to obtain a new set of spherical points P″, which is expressed as follows: P" = P + P' 投 = {p1, p2, …, p i , …, p g} (11) where: i ∈ (1, g), and g is the sum of the number of elements in the set of projection points P′ 投 and the number of elements in the set of spherical surface points P. For the needs of subsequent algorithms, the new set of spherical surface points is still denoted as P, i.e., P = P″.

7. The spherical multi-resolution discretization representation method according to claim 2, wherein The specific process of the said Step 5) is as follows: Step 51) Connect the projection points on the sphere of the midpoints of the longest sides of each planar triangle in the set T of planar triangles and the three vertices p l ,p m ,p n of the corresponding planar triangle, and two planar triangles can be formed. Traverse the set P' of projection points obtained in step 43) 投 , and a new set T' of planar triangles can be obtained: T' = {<p l , p m , p n >| p l , p m , p n ∈ P} (12) Where: p l , p m , p n are the three vertices forming a planar triangle, <p l , p m , p n > represents the planar triangle formed by these three vertices, where P is the set of spherical points obtained in step 44), and l, m, n are used as the index values of the spherical points in the set of spherical points P that form the vertices of the triangle; Step 52) For the need of facilitating the algorithm loop, the obtained new plane triangle set T' is still denoted as T, that is, T = T'; Step 53) Complete one division, and increase the value of the division number by one, that is, s = s + 1.

8. The spherical multi - resolution discretization representation method according to claim 2, wherein The specific process of the said Step 6) is as follows: Step 61) Judge whether the division number s is equal to the maximum division number S. If s = S, then proceed to Step 63); otherwise, proceed to Step 62); Step 62) Based on the spherical point set P obtained in Step 44) and the plane triangle set T obtained in Step 51), successively perform Steps 41) to 44), Steps 51) to 53), and Step 61); Step 63) Output the spherical point set P obtained in Step 44) and the plane triangle set T obtained in Step 51). The spherical solid figure is represented by all the spherical points in the spherical point set and the plane triangles in the plane triangle set formed by the spherical points, achieving the purpose of spherical discretization.

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