Efficient cutting simulation method for Tri-dexel model based on octree acceleration

Abstract

The invention relates to the technical field of intelligent numerical control machining, in particular to a Tri-dexel model efficient cutting simulation method based on octree acceleration, and the method comprises the steps: firstly selecting a three-dimensional model needing Tri-dexel processing; performing grid division on the three-dimensional model to obtain a plurality of triangular grid units, and generating a Dexcel ray according to the three-dimensional model; triangular patches in the triangular mesh are stored in an octree form; grouping the Dexcel rays according to the depth of the octree, and searching patches possibly intersected with the Dexcel rays in the octree by using the Dexcel rays for intersection; performing de-duplication processing on the repeated intersection points; the intersection points of the same Dexel ray are connected in pairs, and finally a Tri-dexel model is generated; and then the Tri-dexel model of the workpiece cut by the tool can be obtained through the one-dimensional Boolean operation of the Dexcel line segment between the workpiece and the Tri-dexel model of the tool. Through verification, the Tri-dexel efficiency of the complex three-dimensional model in machining simulation is improved, and the calculation amount and the storage space of machining simulation can be effectively reduced when a complex three-dimensional scene in numerical control machining simulation is coped with.

Description

Technical Field

[0001] The present invention relates to the technical field of intelligent numerical control machining, and in particular to an efficient cutting simulation method of a Tri-dexel model based on octree acceleration. Background Art

[0002] Traditional 3D model representation methods, such as polygonal mesh models, can intuitively describe the surface shape of objects, but they have many drawbacks when dealing with large-scale and complex 3D scenes.

[0003] Traditional 3D model representation methods suffer from inefficient spatial representation when dealing with large-scale, complex scenes. In terms of computational complexity, Boolean operations, collision detection, and other operations on 3D models require traversing and intersecting a large number of polygonal facets. For example, in CNC simulation, the cutting simulation of the tool and workpiece model requires constant determination of the intersection of the tool path and the polygons on the workpiece surface. This results in an exponential increase in computational complexity, extremely slow processing speeds, and difficulty meeting real-time requirements. In terms of storage space, the number of polygons in large-scale, complex 3D scenes can reach millions or even more, and each polygon requires storing information such as its vertex coordinates and normals, which consumes a significant amount of storage space. Furthermore, as model complexity increases, data redundancy becomes more severe, further exacerbating storage pressure.

[0004] In the field of CNC simulation, it is necessary to simulate the cutting process between the tool and the workpiece in real time to accurately predict the machining results. Existing model representation methods are computationally inefficient and cannot meet the real-time requirements of high-speed machining, which affects the accuracy and practicality of CNC simulation. Summary of the Invention

[0005] The purpose of the present invention is to provide an efficient cutting simulation method for a Tri-dexel model based on octree acceleration, aiming to improve the Tri-dexel efficiency of complex three-dimensional models in mechanical processing simulation, and to effectively reduce the computational complexity and storage space of the processing simulation when dealing with complex three-dimensional scenes in CNC processing simulation.

[0006] To achieve the above object, the present invention provides an efficient cutting simulation method of a Tri-dexel model based on octree acceleration, comprising the following steps:

[0007] Step 1: Select the workpiece to be cut and generate a Tri-dexel model;

[0008] Step 2: Mesh the selected 3D model to obtain multiple triangular mesh units, and generate Dexel rays based on the 3D model;

[0009] Step 3: Store the triangles in the triangular mesh model in the form of an octree;

[0010] Step 4: Group the Dexel rays according to the depth of the octree, use the Dexel rays to find the triangles that may intersect with them in the octree and calculate the intersection points, find the duplicate intersection points and perform deduplication processing;

[0011] Step 5: Connect the intersection points of the same Dexel ray to form Dexel line segments that can describe the 3D model, and finally generate the Tri-dexel model of the workpiece. The generation method of the tool Tri-dexel model is the same as that of the workpiece.

[0012] Step 6: After converting the mesh model into a Tri_dexel model, the three-dimensional Boolean operation between the tool and the workpiece is converted into a one-dimensional Boolean operation between line segments. By comparing the Dexel line segments of the tool and the workpiece, the Tri-dexel model of the workpiece after cutting can be obtained.

[0013] Optionally, the Tri-dexel model is a model that uses discrete line segments to represent entities, using a one-dimensional linear model to represent a three-dimensional entity;

[0014] The three-dimensional model of the workpiece to be cut in step 1 is a three-dimensional model drawn or imported in CAD / CAM software, and its surface must be a closed surface.

[0015] Optionally, the generation process of the Dexel ray is specifically to obtain its bounding box according to the three-dimensional model, obtain the projection rectangle of the bounding box on the xoy plane, discretize the rectangle into a rectangular grid with a certain accuracy, take the sub-rectangle vertex of the rectangular grid as the starting point, and make a ray with the positive direction of the z-axis as the direction. This ray is the Dexel ray.

[0016] Optionally, the execution process of step 3 includes the following steps:

[0017] Step 3.1: Arrange the triangles in descending order of area, starting with the triangle with the largest area, and insert them into the octree in sequence;

[0018] Step 3.2: Initialize the octree, create a root node, pass in the boundaries of the 3D model, i.e. the maximum and minimum points of the bounding box, and set the upper limit T for the number of nodes to be stored. max , when the number of triangles stored in the node exceeds the upper limit T max After that, the node begins to split into eight equal-sized child nodes;

[0019] Step 3.3: First determine whether the triangle intersects with the current node. If not, traverse the remaining child nodes until an intersecting node is found. If so, query the number of triangles n stored in the intersecting node.

[0020] If n>T max , split the current node into eight equal-sized child nodes, and record the depth of the octree at this time; if n <T max , insert the triangle into the current node;

[0021] Traverse all current child nodes and repeat this step until the triangles are inserted into all nodes that intersect with them;

[0022] Step 3.4: Traverse all triangles and repeat step 3.3 until all triangles are inserted into the octree.

[0023] Optionally, the execution process of step 4 includes the following steps:

[0024] Step 4.1: Calculate the number of groups p based on the depth obtained in step 3 num , the calculation formula is as follows:

[0025] p num =2 depth 2 depth ;

[0026] Step 4.2: Divide the projection rectangle of the 3D model bounding box into p num sub-rectangles, and divide the Dexel rays in each sub-rectangle into a group;

[0027] Step 4.3: Select the first ray in each group of rays and check if it intersects with the current node; if not, proceed to the next node; if so, store the triangles in the node into an array;

[0028] At the same time, duplicate triangles are removed. That is, when storing, first check whether there are identical triangles in the array. If there are identical triangles, the current triangle does not need to be stored; if there are no identical triangles, the current triangle is stored in the array.

[0029] Until all nodes are traversed;

[0030] Step 4.4: Starting from the first ray in the current ray group, traverse all triangles in the array and calculate the intersection point P. Then calculate the area of the triangle according to the following formula and determine the position of the intersection point in triangle ABC. Then perform the corresponding three operations:

[0031]

[0032] If U>0, the intersection point is outside triangle ABC and no processing is done;

[0033] If U<0, the intersection point is inside the triangle and the point is stored in the intersection array;

[0034] If U = 0, the intersection point is on the edge of the triangle. This is a singular case. You need to offset the current ray into the triangle by a sufficiently small distance, then recalculate the intersection point P and determine its location.

[0035] Step 4.5: After traversing all triangles in the triangle array, all intersection points between the current ray and the triangle model are obtained. The points in the intersection array are arranged in ascending order of z coordinates, and the intersection array is stored in a two-dimensional array. Then, the intersection check for the next ray in the current ray group begins, and steps 4.4 and 4.5 are repeated until all rays in the current ray group are traversed.

[0036] Step 4.6: Repeat steps 4.3 to 4.5 until all ray groups are traversed.

[0037] The present invention provides a tri-dexel model efficient cutting simulation method based on octree acceleration, which first selects a three-dimensional model that needs to be tri-dexelized; meshes the three-dimensional model to obtain multiple triangular mesh units, and generates Dexel rays according to the three-dimensional model; stores the triangular facets in the triangular mesh in the form of an octree; groups the Dexel rays according to the depth of the octree, and uses the Dexel rays to find facets that may intersect with them in the octree for intersection; de-duplicates the repeated intersections; connects the intersections of the same Dexel ray in pairs, and finally generates a tri-dexel model; then, through a one-dimensional Boolean operation of the Dexel line segments between the workpiece and the tool tri-dexel model, the tri-dexel model of the workpiece after tool cutting can be obtained. It has been verified that the present invention improves the tri-dexel efficiency of complex three-dimensional models in mechanical processing simulation, and can also effectively reduce the computational complexity and storage space of the processing simulation when dealing with complex three-dimensional scenes in numerical control processing simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0038] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying any creative work.

[0039] Figure 1 This is a schematic flow chart of the steps of an efficient cutting simulation method of a Tri-dexel model based on octree acceleration of the present invention.

[0040] Figure 2 It is a schematic diagram of the three-dimensional model and its triangular mesh model used in the specific embodiment of the present invention.

[0041] Figure 3 Schematic diagram of Dexel rays generated according to a specific embodiment of the present invention.

[0042] Figure 4 It is a schematic diagram of the process of storing triangular facets in the form of an octree in a specific embodiment of the present invention.

[0043] Figure 5 It is the octree space structure used in the specific embodiment of the present invention.

[0044] Figure 6 4 is a schematic diagram of a specific execution flow of step 4 of the present invention.

[0045] Figure 7 It is a schematic diagram of the minimum node and grouping in a specific embodiment of the present invention.

[0046] Figure 8 Schematic diagram of intersection connection in a specific embodiment of the present invention.

[0047] Figure 9 It is a Tri-dexel model obtained by superimposing dexel models in three directions in a specific embodiment of the present invention.

[0048] Figure 10 It is a one-dimensional Boolean operation on the Dexel line segment between the tool and the workpiece Tri-dexel model in the specific embodiment of the present invention.

[0049] Figure 11 It is a Tri-dexel model of the spatial position of the tool and the workpiece and the cutting result in a specific embodiment of the present invention. DETAILED DESCRIPTION

[0050] The following describes embodiments of the present invention in detail, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, and are not to be construed as limiting the present invention.

[0051] The present invention provides a Tri-dexel model efficient cutting simulation method based on octree acceleration, comprising the following steps:

[0052] S1: Select the workpiece to be cut to generate a Tri-dexel model;

[0053] S2: Mesh the selected 3D model to obtain multiple triangular mesh units, and generate Dexel rays based on the 3D model;

[0054] S3: Store the triangles in the triangular mesh model in the form of an octree;

[0055] S4: Group the Dexel rays according to the depth of the octree, use the Dexel rays to find the triangles that may intersect with them in the octree and calculate the intersection points, find the repeated intersection points and perform deduplication processing;

[0056] S5: Connect the intersection points of the same Dexel ray in pairs to form Dexel line segments that can describe the three-dimensional model, and finally generate the Tri-dexel model of the workpiece. The generation method of the tool Tri-dexel model is the same as that of the workpiece;

[0057] S6: After the mesh model is converted into a Tri_dexel model, the three-dimensional Boolean operation between the tool and the workpiece is converted into a one-dimensional Boolean operation between line segments. By comparing the Dexel line segments of the tool and the workpiece, the Tri-dexel model of the workpiece after cutting can be obtained.

[0058] Figure 1The present invention provides a step flow diagram of an efficient cutting simulation method of a Tri-dexel model based on octree acceleration, and the corresponding execution process is as follows: first, a workpiece model with a closed surface is drawn or imported into the CAD / CAM software for Tri-dexelization; then the three-dimensional model is meshed to obtain multiple triangular mesh units, and its bounding box is obtained according to the model, and the bounding box is projected on the xoy plane into a rectangular mesh with a certain accuracy, and a Dexel ray is made with the small rectangle vertex as the starting point and the positive direction of the z-axis as the direction; then the triangles in the triangular mesh are arranged from large to small according to the area, the octree is initialized and the upper limit of the node storage number is set, and it is judged whether the triangle intersects with the node, and whether the node is split according to the number of triangles in the node, and the triangle is inserted into all the triangles that intersect with it. The octree is a set of nodes. The projected rectangle of the 3D model's bounding box is then divided into corresponding smaller rectangles. The Dexel rays within each smaller rectangle are grouped together. The first ray in each group is selected for intersection with the node, and the intersecting triangles are selected and stored in an array. The intersection points between the rays and the triangles are calculated, and the intersection locations are determined using the triangle area formula. Singular cases and duplicate intersections are handled, and the intersection points are sorted by z-coordinate and stored in a two-dimensional array. Finally, the intersection points of the same Dexel ray in the two-dimensional array are connected to form Dexel segments that describe the 3D model. This set of Dexel segments is the Dexel model in the z-direction. The same method is used to obtain the Dexel models in the x- and y-directions. The Dexel models in the three directions are superimposed to form a tri-dexel model. The same method is used to obtain the tri-dexel model of the tool. Then, a one-dimensional Boolean operation on the Dexel segments between the workpiece and the tool's tri-dexel model is used to obtain the tri-dexel model of the workpiece after cutting.

[0059] See also Figures 2 to 11 , the following is further explained with reference to specific embodiments and execution steps:

[0060] S1: Select the workpiece to be cut to generate a Tri-dexel model;

[0061] In step S1 , the Tri-dexel model is a model that uses discrete line segments to represent entities, and uses a one-dimensional model to represent a three-dimensional entity.

[0062] In this embodiment, a three-dimensional model is first drawn or imported, and its surface must be a closed surface for subsequent generation of a Tri-dexel model.

[0063] S2: Mesh the selected 3D model to obtain multiple triangular mesh units, and generate Dexel rays based on the 3D model;

[0064] like Figure 2 As shown, in this embodiment, the bounding box of the 3D model is obtained, and the projection rectangle of the bounding box on the xoy plane is obtained. The rectangle is discretized into a rectangular grid with a certain accuracy. The vertices of the sub-rectangles of the rectangular grid are used as the starting point, and the ray is drawn in the positive direction of the z-axis. The ray is the Dexel ray. The Dexel ray is shown as follows Figure 3 shown.

[0065] S3: Store the triangles in the triangle mesh model in the form of an octree, such as Figure 4 The specific operations are as follows:

[0066] S301. Arrange the triangles in descending order of area, starting with the triangle with the largest area, and insert them into the octree in sequence.

[0067] S302, initialize the octree, its spatial structure is as follows Figure 5 As shown, create a root node, pass in the boundary of the 3D model, that is, the maximum and minimum points of the bounding box, and set the upper limit T of the node storage quantity max ,When the number of triangles stored in a node exceeds the upper limit, the node begins to split into eight child nodes of equal size.

[0068] S303, first determine whether the triangle intersects with the current node. If not, traverse the remaining child nodes until an intersecting node is found; if intersecting, query the number of triangles n stored in the intersecting node. Further, if n>T max , split the current node into eight equal-sized child nodes, and record the depth of the octree at this time; if n <T max , insert the triangle into the current node. Then traverse all the current child nodes and repeat this step until the triangle is inserted into all the nodes that intersect with it, because a triangle may intersect with more than one node, that is, a triangle may be stored in several nodes.

[0069] S304: Traverse all triangles and repeat step 303 until all triangles are inserted into the octree.

[0070] For further information, see Figure 6 :

[0071] S4: Group the Dexel rays according to the depth of the octree, use the Dexel rays to find the triangles that may intersect with them in the octree and calculate the intersection points, find the repeated intersection points and perform deduplication processing; the specific operations are as follows:

[0072] S401, calculate the number of groups p according to the depth depth obtained in step S3 num , the calculation formula is as follows:

[0073] p num =2 depth 2 depth

[0074] S402, such as Figure 7 As shown, the projection rectangle of the 3D model bounding box is divided into p num Sub-rectangles are formed, and the Dexel rays in each sub-rectangle are divided into a group. At this time, the intersection of the rays in each group and the octree nodes is the same, and it is only necessary to determine whether a ray in the same group intersects with the current node.

[0075] S403. Select the first ray in each group of rays and perform an intersection check with the current node. If they do not intersect, proceed to the next node; if they do intersect, store the triangles in the node into an array and deduplicate the triangles. That is, when storing, first check whether there are identical triangles in the array. If there are identical triangles, the current triangle does not need to be stored; if there are no identical triangles, store the current triangle in the array. This continues until all nodes are traversed. At this point, all triangles that may intersect with all rays in the current ray group are stored in the array.

[0076] S404. Starting from the first ray in the current ray group, traverse all triangles in the array, calculate the intersection point P, then calculate the area of the triangle according to the following formula and determine the position of the intersection point in triangle ABC, and then perform the corresponding three processing.

[0077]

[0078] (1) If U>0, the intersection point is outside triangle ABC and no processing is performed.

[0079] (2) If U < 0, the intersection point is inside the triangle and the point is stored in the intersection array.

[0080] (3) If U = 0, the intersection point is on the edge of the triangle, which is a singular case. When a ray intersects multiple triangles, repeated intersection points will appear. Therefore, it is necessary to offset the current ray a small enough distance into the triangle, and then recalculate the intersection point P and determine its location.

[0081] S405: After traversing all triangles in the triangle array, all intersection points between the current ray and the triangle model are obtained. The points in the intersection array are sorted in ascending order of z coordinates, and the intersection array is stored in a two-dimensional array. Intersection determination is then started for the next ray in the current ray group, and steps S404 and S405 are repeated until all rays in the current ray group have been traversed.

[0082] S406: Repeat steps S403, S404, and S405 until all ray groups are traversed. At this point, the intersection points of all rays are stored in a two-dimensional intersection point array.

[0083] Finally, if Figure 8 As shown, S5: connect the intersection points of the same Dexel ray in pairs to form Dexel line segments that can describe the three-dimensional model, and finally generate a Tri-dexel model.

[0084] In this embodiment, the Tri-dexel model includes dexel models in the x, y, and z directions. The above steps take the z-direction dexel model as an example, and finally the intersection points of the same Dexel ray in the two-dimensional array are connected to form a dexel line segment that can describe the three-dimensional model. The collection of the dexel line segments is the z-direction dexel model. Figure 9 As shown in the figure, the same method can be used to obtain the Dexel models in the x and y directions, and finally the Tri-dexel model can be obtained by superimposing the Dexel models in the three directions. Then the same method is used to obtain the Tri-dexel model of the tool.

[0085] The one-dimensional Boolean operation of the Dexel line segment between the tool and the workpiece Tri-dexel model is as follows: Figure 10 As shown in the figure, the Tri-Dexel model of the tool is abbreviated as DT, and the Tri-Dexel model of the workpiece is abbreviated as DW. Taking the xoy plane as an example, assuming that a Dexel line segment of the workpiece is (DW_max, DW_min), and a Dexel line segment of the tool is (DT_max, DT_min). To verify this method, the spatial position of the tool and the workpiece and the Tri-Dexel model of the cutting result are shown as follows: Figure 11 As shown,

[0086] As shown in Table 1, as the division accuracy increases (that is, the resolution of the grid increases), the advantage of octree acceleration becomes increasingly apparent. This is because as the three-dimensional model becomes more refined, the number of grid cells increases, and the efficiency of directly calculating the intersection decreases, while the octree optimizes this process by spatial partitioning, reducing the number of triangles that need to be checked. Therefore, in practical applications, for the generation of high-precision models, this embodiment can greatly improve performance and is a very effective optimization method. As the division accuracy increases (that is, the resolution of the grid increases), the advantage of octree acceleration becomes increasingly apparent. This is because as the three-dimensional model becomes more refined, the number of grid cells increases, and the efficiency of directly calculating the intersection decreases, while the octree optimizes this process by spatial partitioning, reducing the number of triangles that need to be checked.

[0087] Table 1

[0088]

[0089] In summary, compared with the existing technology, the beneficial effects of the present invention are:

[0090] In terms of computational efficiency, by storing triangles in the form of an octree and grouping Dexel rays, the amount of computation required to intersect rays with triangles is reduced, while optimizing the handling of singular cases to ensure accurate and efficient calculations. In terms of storage space, the octree storage structure can store triangle information more compactly, saving a lot of space. In terms of model representation accuracy, duplicate intersections are deduplicated so that the generated Tri-dexel model can more accurately describe the three-dimensional model. In addition, this method improves the spatial representation efficiency of the three-dimensional model and has broad application prospects in the field of numerical control simulation, making up for the shortcomings of existing technologies in computational efficiency, storage, and application scope.

[0091] The above disclosure is merely one or more preferred embodiments of the present invention, and certainly cannot be used to limit the scope of the present invention. A person skilled in the art can understand that all or part of the processes of the above embodiments and equivalent changes made in accordance with the claims of the present invention still fall within the scope of the invention.

Claims

1. An efficient cutting simulation method of Tri-dexel model based on octree acceleration, characterized in that: The following steps are involved: Step 1: Select the workpiece to be cut and generate a Tri-dexel model; Step 2: Mesh the selected 3D model to obtain multiple triangular mesh units, and generate Dexel rays based on the 3D model; Step 3: Store the triangles in the triangular mesh model in the form of an octree; Step 4: Group the Dexel rays according to the depth of the octree, use the Dexel rays to find the triangles that may intersect with them in the octree and calculate the intersection points, find the duplicate intersection points and perform deduplication processing; Step 5: Connect the intersection points of the same Dexel ray to form Dexel line segments that can describe the 3D model, and finally generate the Tri-dexel model of the workpiece. The generation method of the tool Tri-dexel model is the same as that of the workpiece. Step 6: After converting the mesh model into a Tri_dexel model, the three-dimensional Boolean operation between the tool and the workpiece is converted into a one-dimensional Boolean operation between line segments. By comparing the Dexel line segments of the tool and the workpiece, the Tri-dexel model of the workpiece after cutting can be obtained.

2. The efficient cutting simulation method of Tri-dexel model based on octree acceleration according to claim 1, characterized in that: The Tri-dexel model is a model that uses discrete line segments to represent entities, and uses a one-dimensional linear model to represent three-dimensional entities; The three-dimensional mesh model that needs to be cut in step 1 is a three-dimensional model drawn or imported in CAD / CAM software, and its surface must be a closed surface.

3. The efficient cutting simulation method of Tri-dexel model based on octree acceleration according to claim 2, characterized in that: The generation process of the Dexel ray is specifically to obtain its bounding box according to the 3D model, obtain the projection rectangle of the bounding box on the xoy plane, discretize the rectangle into a rectangular grid with a certain accuracy, use the sub-rectangle vertex of the rectangular grid as the starting point, and draw a ray in the positive direction of the z-axis. This ray is the Dexel ray.

4. The efficient cutting simulation method of the Tri-dexel model based on octree acceleration according to claim 3, characterized in that: The execution process of step 3 includes the following steps: Step 3.1: Arrange the triangles in descending order of area, starting with the triangle with the largest area, and insert them into the octree in sequence; Step 3.2: Initialize the octree, create a root node, pass in the boundaries of the 3D model, i.e. the maximum and minimum points of the bounding box, and set the upper limit T for the number of nodes to be stored. max , when the number of triangles stored in the node exceeds the upper limit T max After that, the node begins to split into eight equal-sized child nodes; Step 3.3: First determine whether the triangle intersects with the current node. If not, traverse the remaining child nodes until an intersecting node is found; If they intersect, query the number of triangles n stored in the intersection node; If n>T max , split the current node into eight equal-sized child nodes, and record the depth of the octree at this time; if n <T max , insert the triangle into the current node; Traverse all current child nodes and repeat this step until the triangles are inserted into all nodes that intersect with them; Step 3.4: Traverse all triangles and repeat step 3.3 until all triangles are inserted into the octree.

5. The efficient cutting simulation method of Tri-dexel model based on octree acceleration according to claim 4, characterized in that: The execution process of step 4 includes the following steps: Step 4.1: Calculate the number of groups p based on the depth obtained in step 3 num , the calculation formula is as follows: p num ＝2 depth ·2 depth ； Step 4.2: Divide the projection rectangle of the 3D model bounding box into p num sub-rectangles, and divide the Dexel rays in each sub-rectangle into a group; Step 4.3: Select the first ray in each group of rays and check if it intersects with the current node; if not, proceed to the next node; if so, store the triangles in the node into an array; At the same time, duplicate triangles are removed. That is, when storing, first check whether there are identical triangles in the array. If there are identical triangles, the current triangle does not need to be stored; if there are no identical triangles, the current triangle is stored in the array. Until all nodes are traversed; Step 4.4: Starting from the first ray in the current ray group, traverse all triangles in the array and calculate the intersection point P. Then calculate the area of the triangle according to the following formula and determine the position of the intersection point in triangle ABC. Then perform the corresponding three operations: If U>0, the intersection point is outside triangle ABC and no processing is done; If U<0, the intersection point is inside the triangle and the point is stored in the intersection array; If U = 0, the intersection point is on the edge of the triangle. This is a singular case. You need to offset the current ray into the triangle by a sufficiently small distance, then recalculate the intersection point P and determine its location. Step 4.5: After traversing all triangles in the triangle array, all intersection points between the current ray and the triangle model are obtained. The points in the intersection array are arranged in ascending order of z coordinates, and the intersection array is stored in a two-dimensional array. Then, the intersection check for the next ray in the current ray group begins, and steps 4.4 and 4.5 are repeated until all rays in the current ray group are traversed. Step 4.6: Repeat steps 4.3 to 4.5 until all ray groups are traversed.

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