Display method and device of three-dimensional illustration and storage medium

By saving the depth information of the three-dimensional model in the drawing buffer area and performing the back extrinsing drawing, the problem of high computational cost, high cost and poor real-time interaction in the prior art is solved, and efficient three-dimensional illustration generation and display are achieved.

CN120236018AActive Publication Date: 2025-07-01BEIJING YUANHUI TECH CO LTD

Patent Information

Application Number
CN202510714732.5
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-05-30
Publication Date
2025-07-01
Estimated Expiration
2045-05-30

AI Technical Summary

Technical Problem

In the prior art, the calculation of edge line completion of three-dimensional illustrations is high, costly and poor real-time interactivity, especially when processing large models, the generation time may be as long as several hours.

Method used

The generation of the initial 3D illustration is achieved by saving only the grid depth information of the 3D model in the drawing buffer and superimposing the depth, color and template information of the outline in the drawn grid. Then, set the color information of the grid to the color information of the outline, and perform the back extubation drawing to form the final three-dimensional illustration.

Benefits of technology

This method greatly reduces the calculation amount, reduces the geometric data calculation of edge fill lines, improves rendering performance, makes the generation and display of three-dimensional illustrations more efficient, and the real-time interactivity is significantly improved.

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Abstract

The invention discloses a three-dimensional illustration display method and device and a storage medium, and relates to the technical field of image processing. According to the three-dimensional illustration display method, firstly, objects which do not need to be displayed are removed, a grid of a target object is drawn, and then a contour line of the target object is overlaid and drawn in the drawn grid. And then, setting the color information of all the grids as the color information of the contour line, and performing back expansion drawing on all the grids in the initial three-dimensional illustration based on the data in the drawing cache region and the set color information to obtain a final three-dimensional illustration. According to the three-dimensional illustration, actual geometric data calculation does not need to be carried out, a real edge line does not need to be displayed, the edge line repairing function is achieved through simple rendering skills, and the calculation amount is greatly reduced.
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Description

Technical Field

[0001] This application relates to the field of image processing technology, and in particular, to a method, device, and storage medium for displaying three-dimensional illustrations. Background Art

[0002] In industrial manufacturing fields such as aerospace, automotive, and shipbuilding, as well as in the fields of 3D modeling and game development, the detailed forms of objects are accurately stored in the form of 3D models. Although high-precision 3D models are the core data carriers, they still need to be converted into 3D illustrations in scenarios such as technical documents and assembly guides. By simplifying the structure, hiding irrelevant components, adding annotations or hatching, 3D illustrations can more clearly highlight the key parts. For example, in the design of automotive engines in the industrial manufacturing field, it is often necessary to convert complex 3D assemblies into 3D illustrations such as exploded views or sectional schematic diagrams to show the spatial relationships between parts.

[0003] When converting a 3D model into a 3D illustration, the drawing of the contour lines of the 3D model depends on predefined geometric parameters (such as edge coordinates). When the model rotates, these preset edges will change with the coordinate system transformation and may disappear due to projection degradation. In order to ensure that the outer contour in the current view is always displayed regardless of how the object rotates, edge compensation line processing is required. Through the edge compensation line function, the visual contour missing due to projection degradation is reconstructed to maintain the integrity of the shape expression.

[0004] In the prior art, edge compensation lines are mostly based on geometric projection algorithms (such as orthographic projection / axonometric projection), and vector lines are generated by calculating the visible edges and contours on the surface of the model. That is, every time the viewing angle changes, all the patches need to be traversed again, the visibility of the edges of each patch is calculated, and then geometric projection calculations are performed on the missing contour edges to determine the coordinates and lengths of the edges to be filled. After that, the missing contour edges are drawn and displayed. It can be seen that the disadvantages of the edge compensation line scheme in the prior art are: it is necessary to process visibility judgment and geometric calculations patch by patch. When dealing with large models with millions of patches (such as the entire ship assembly), the generation time may be as long as several hours, with a large amount of calculation, high cost, and poor real-time interactivity. Summary of the Invention

[0005] Embodiments of the present disclosure provide a method, device, and storage medium for displaying three-dimensional illustrations to at least solve the technical problems of large calculation amount, high cost, and poor real-time interactivity in edge line completion of three-dimensional illustrations in the prior art.

[0006] According to one aspect of the embodiments of the present disclosure, an illustration display method is provided. The three-dimensional illustration is a stereoscopic image generated by rendering a three-dimensional model onto a screen plane. The display method includes: drawing a mesh of a target object in the three-dimensional model and only saving the depth information of the mesh in a drawing buffer; wherein the target object is an object to be displayed in the three-dimensional model; overlaying and drawing a contour line of the target object on the drawn mesh and saving the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object and does not show the mesh surface of the target object, and there are missing parts in the contour line shown in some visual areas of the screen display; setting the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and performing back-face expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back-face drawing in the expanded meshes form continuous complementary lines visually with the missing contour lines.

[0007] According to another aspect of the embodiments of the present disclosure, a storage medium is further provided. The storage medium includes a stored program, wherein, when the program runs, the method described in any one of the above is executed by a processor.

[0008] According to another aspect of the embodiments of the present disclosure, an illustration display device is further provided. The three-dimensional illustration is a three-dimensional model rendered onto a screen plane to generate a stereoscopic image. The display device includes: a contour occlusion module configured to draw a mesh of a target object in the three-dimensional model and only save the depth information of the mesh in a drawing buffer; wherein the target object is an object to be displayed in the three-dimensional model; superimpose and draw the contour line of the target object on the drawn mesh, and save the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object and does not show the mesh surface of the target object, and there are missing parts in some visual areas of the contour line displayed on the screen; an edge filling module configured to set the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and perform back-face expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back-face drawing in each expanded mesh form continuous complementary lines visually with the missing contour lines.

[0009] According to another aspect of the embodiments of the present disclosure, an illustration display device is provided, including: a processor; and a memory connected to the processor for providing instructions for the processor to perform the following processing steps: draw a mesh of a target object in the three-dimensional model and only save the depth information of the mesh in a drawing buffer; wherein the target object is an object to be displayed in the three-dimensional model; superimpose and draw the contour line of the target object on the drawn mesh, and save the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object and does not show the mesh surface of the target object, and there are missing parts in some visual areas of the contour line displayed on the screen; set the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and perform back-face expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back-face drawing in each expanded mesh form continuous complementary lines visually with the missing contour lines; the three-dimensional illustration is a three-dimensional model rendered onto a screen plane to generate a stereoscopic image.

[0010] In the embodiments of the present disclosure, instead of performing actual geometric data calculations and displaying a real edge line, simple rendering techniques are used to implement the edge complement line function. The present invention uses rendering techniques to achieve the effect of illustrations, which can meet the implementation of illustration effects for large-scale scene models. In addition, the present invention uses rendering to replace the geometric data calculations of traditional edge complement lines, greatly reducing the amount of calculation. To ensure that the rendering performance of this solution remains in the millisecond level, this solution uses a culling algorithm to reduce the number of objects to be displayed before the first drawing, without causing a decline in rendering performance. Brief Description of the Drawings

[0011] The drawings described herein are used to provide a further understanding of the present disclosure and form a part of this application. The illustrative embodiments of the present disclosure and their descriptions are used to explain the present disclosure and do not constitute an improper limitation to the present disclosure. In the drawings: Figure 1 is a hardware structure block diagram of a computing device for implementing the method according to Embodiment 1 of the present disclosure; Figure 2 is a schematic flowchart of a method for displaying a three-dimensional illustration according to the first aspect of Embodiment 1 of the present disclosure; Figure 3 is a schematic flowchart of performing backside outward expansion drawing on each grid according to Embodiment 1 of the present disclosure; Figure 4 is a schematic diagram of a first illustration without edge complement line; Figure 5 is a schematic diagram of the final effect of the three-dimensional illustration display of the present invention; Figure 6 is a schematic diagram of a display device for the three-dimensional illustration of the present invention; and Figure 7 is a schematic diagram of another display device for the three-dimensional illustration of the present invention. Detailed Embodiments

[0012] In order to enable those skilled in the art of the present technology to better understand the technical solutions of the present disclosure, the technical solutions in the embodiments of the present disclosure will be clearly and completely described below in conjunction with the drawings in the embodiments of the present disclosure. Obviously, the described embodiments are only a part of the embodiments of the present disclosure, rather than all of the embodiments. Based on the embodiments in the present disclosure, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present disclosure.

[0013] It should be noted that the terms "first", "second", etc. in the specification, claims and above-mentioned drawings of the present disclosure are used to distinguish similar objects, and do not necessarily need to describe a specific order or sequence. It should be understood that the data used in this way can be interchanged under appropriate circumstances, so that the embodiments of the present disclosure described herein can be implemented in an order other than those illustrated or described herein. In addition, the terms "comprising" and "having" and any variations thereof are intended to cover non-exclusive inclusion. For example, a process, method, system, product or device comprising a series of steps or units does not necessarily have to be limited to those steps or units clearly listed, but may include other steps or units not clearly listed or inherent to these processes, methods, products or devices.

[0014] View Space / Camera Space: A coordinate system with the camera as the origin, and the coordinates of all objects are relative to the position and orientation of the camera. In this space, the camera is located at the origin, and the orientation is usually the -Z axis, which is convenient for subsequent projection calculations (such as perspective projection).

[0015] NDC Space (Normalized Device Coordinates): That is, the normalized device coordinate space, which is an intermediate space used to standardize vertex coordinates in 3D rendering, and its coordinate range is [-1, 1] (the x, y, and z axes are all in this range). The core function of this space is to unify the vertex coordinates of different projection types into the standardized range, which is convenient for subsequent conversion to screen space coordinates.

[0016] Screen Space: The coordinates in the NDC space are transformed to the screen space through the viewport transformation matrix (Viewport Transformation), and the coordinates in the screen space correspond to the pixel coordinates finally displayed on the screen.

[0017] Embodiment 1 According to this embodiment, a display embodiment of a three-dimensional illustration is provided. It should be noted that the steps shown in the flowchart of the drawings can be executed in a computer system such as a set of computer-executable instructions, and although the logical order is shown in the flowchart, in some cases, the steps shown or described herein can be executed in a different order than here.

[0018] The method embodiment provided in this embodiment can be executed on a mobile terminal, a computer terminal, a server or a similar computing device. Figure 1 A hardware structure block diagram of a computing device for implementing the display of a three-dimensional illustration is shown. As Figure 1As shown, the computing device may include one or more processors (the processors may include, but are not limited to, processing devices such as microprocessor MCUs or programmable logic devices FPGAs), a memory for storing data, a transmission device for communication functions, and an input / output interface. The memory, the transmission device, and the input / output interface are connected to the processor via a bus. In addition, it may further include: a display, a keyboard, and a cursor control device connected to the input / output interface. Those of ordinary skill in the art can understand that Figure 1 the structure shown is only schematic and does not limit the structure of the above-mentioned electronic device. For example, the computing device may further include more or fewer components than those Figure 1 shown in, or have a different configuration from that Figure 1 shown.

[0019] It should be noted that the above one or more processors and / or other data processing circuits are generally referred to as "data processing circuits" herein. The data processing circuit may be embodied in software, hardware, firmware, or any combination thereof, in whole or in part. In addition, the data processing circuit may be a single independent processing module, or be incorporated in whole or in part into any one of the other elements in the computing device. As involved in the embodiments of the present disclosure, the data processing circuit is a kind of processor control (such as the selection of a variable resistance terminal path connected to an interface).

[0020] The memory can be used to store software programs and modules of application software, such as the program instructions / data storage device corresponding to the display of three-dimensional illustrations in the embodiments of the present disclosure. The processor executes various functional applications and data processing by running the software programs and modules stored in the memory, that is, implements the method for displaying three-dimensional illustrations of the above application program. The memory may include high-speed random access memory, and may also include non-volatile memory, such as one or more magnetic storage devices, flash memory, or other non-volatile solid-state memories. In some instances, the memory may further include a memory remotely located relative to the processor, and these remote memories can be connected to the computing device through a network. Examples of the above network include, but are not limited to, the Internet, an enterprise intranet, a local area network, a mobile communication network, and combinations thereof.

[0021] The transmission device is used to receive or send data via a network. Specific examples of the above network may include a wireless network provided by a communication provider of the computing device. In one instance, the transmission device includes a network adapter (Network Interface Controller, NIC), which can be connected to other network devices through a base station and thus communicate with the Internet. In one instance, the transmission device may be a radio frequency (RF) module, which is used to communicate with the Internet wirelessly.

[0022] The display may be, for example, a touch screen liquid crystal display (LCD), which enables a user to interact with the user interface of the computing device.

[0023] It should be noted here that, in some alternative embodiments, the above Figure 1 illustrated computing device may include hardware elements (including circuits), software elements (including computer code stored on a computer-readable medium), or a combination of both hardware elements and software elements. It should be pointed out that Figure 1 is only an example of a specific specific instance and is intended to illustrate the types of components that may exist in the above computing device.

[0024] Under the above operating environment, according to the first aspect of this embodiment, a method for displaying a three-dimensional illustration is provided. Figure 2 The flowchart of the method is shown. Referring to Figure 2 as shown, the method includes: S202. Draw the mesh of the target object in the three-dimensional model and only save the depth information of the mesh in the drawing buffer.

[0025] Wherein, the target object is the object to be displayed in the three-dimensional model.

[0026] In the present invention, a three-dimensional illustration is a stereoscopic image generated by rendering a three-dimensional model onto a screen plane.

[0027] A mesh is a geometric structure composed of vertices, edges, and faces, and is used to describe the shape of an object. In the present invention, the method for drawing the mesh of the target object may include, but is not limited to, the following methods: Polygon modeling, using basic geometric bodies (such as cubes, spheres, cylinders, etc.) as a starting point, and gradually refining the model by adding, deleting, moving vertices, edges, and faces.

[0028] Curve modeling, using curves (such as Bezier curves, NURBS curves) to define the contour of an object. Generate a mesh by lofting, revolving lofting, etc.

[0029] Scan modeling, using a 3D scanning device (such as a laser scanner, structured light scanner, etc.) to obtain the surface data of an object, and then converting it into a mesh.

[0030] Image-based modeling, reconstructing a three-dimensional mesh through multiple photos of an object. Use two or more photos taken from different angles, and reconstruct the depth information by calculating the disparity. Or use multiple photos taken under different lighting conditions, and reconstruct the surface normal of the object by analyzing the lighting changes, and then reconstruct the mesh.

[0031] Subdivision surface modeling starts from a low-resolution mesh and gradually generates a high-resolution smooth mesh through subdivision algorithms.

[0032] Physics-based modeling generates a mesh by simulating physical processes (such as hydrodynamics, elasticity, etc.). The flow of a liquid is simulated using hydrodynamics algorithms to generate a mesh on the liquid surface. Or the elastic deformation of an object is simulated using the finite element method to generate a mesh after deformation.

[0033] In the present invention, the existing high-resolution mesh can also be optimized to reduce the number of vertices and faces while maintaining the visual effect of the model.

[0034] S204. Superimpose and draw the contour line of the target object on the drawn mesh, and save the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration.

[0035] Among them, the initial three-dimensional illustration only shows the contour line of the target object and does not show the mesh surface of the target object, and there are missing parts in the visual area of the part of the contour line displayed on the screen.

[0036] In the present invention, the drawing buffer may include: The depth buffer is used to store the depth information of the drawing; The color buffer is used to store the color information of the drawing; The stencil buffer is used to store the module information of the drawing.

[0037] S206. Set the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and perform back face outward expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration.

[0038] Among them, all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back face drawing in each expanded mesh form continuous complementary lines visually with the missing contour lines.

[0039] Optionally, before the operation of drawing the mesh of the target object in the three-dimensional model in 202, it further includes: Using a preset culling algorithm to cull the objects that do not need to be displayed in the three-dimensional model and retain the objects that need to be displayed in the three-dimensional model.

[0040] It should be noted that in the present invention, the preset culling algorithm can be set in advance before implementing the present invention. The culling algorithm is used to reduce the rendering calculation amount and improve the rendering efficiency. For example, in the present invention, the culling algorithm may include, but is not limited to: 1. Frustum culling algorithm, which determines whether a 3D model or object is within the camera's frustum. If an object is completely outside the frustum, it will not be rendered and will be culled. This method decomposes the frustum into six planes (left - right, up - down, front - back), and determines whether an object is inside the frustum by calculating the intersections of the object's bounding box with these planes. If the bounding box of the object intersects all six planes of the frustum, the object is inside the frustum; otherwise, the object is outside the frustum.

[0041] 2. Back - face culling algorithm, which determines the orientation of a polygon. If the normal direction of a polygon points away from the camera, it is considered a back - face polygon and can be culled. This method calculates the normal direction of the polygon, and then determines the orientation of the polygon based on the position and direction of the camera. Usually, it is determined by calculating the dot product of the normal vector of the polygon vertices and the camera direction vector. If the dot product is negative, the polygon is a back - face.

[0042] 3. Occlusion culling algorithm, which determines whether an object is occluded by other objects. If it is completely occluded, it will not be rendered. During the rendering process, this method uses the depth buffer to record the depth value of each pixel, and determines whether it is occluded by comparing the depth value of the newly rendered object. The objects in the scene are represented by hierarchical bounding boxes, and the occluded objects are culled by determining the occlusion relationship between the bounding boxes.

[0043] 4. Level - of - detail culling algorithm, which uses models with different levels of detail for rendering according to the distance between the object and the camera. Objects at a greater distance use low - detail models, and objects at a closer distance use high - detail models. This method prepares multiple models with different levels of detail for each object in advance. During rendering, a suitable model is dynamically selected for rendering according to the distance between the object and the camera.

[0044] 5. Screen - space culling algorithm, in screen space, determines whether an object needs to be rendered based on the projection size and position of the object on the screen. This method calculates the projection bounding box of the object on the screen. If the bounding box exceeds the screen range, the object is culled. For objects that exceed the screen range but are partially visible, it can be further determined whether they intersect with the screen.

[0045] 6. Spatial - partitioning culling algorithm, which divides the scene into multiple sub - spaces (such as octrees, quadtrees, etc.), and culls the objects in the entire sub - space by determining the relationship between the sub - space and the frustum or the camera.

[0046] Optionally, in S204, the step of superimposing and drawing the contour line of the target object on the drawn grid includes: Comparing the depth value of each line segment in the currently drawn contour line with the existing depth values in the depth buffer; wherein, the existing depth values include the depth information of all the drawn grids and the depth information of the drawn contour lines; Determining the line segments to be rendered onto the screen according to the comparison result.

[0047] In this step, the contour lines of the objects to be displayed can also be superimposed and drawn on the drawn grid. At this time, because the depth data of the grid surface drawing still exists and depth testing is performed with the currently drawn contour lines, the effect that the lines are blocked by the grid surface is produced.

[0048] In the present invention, depth testing refers to: by comparing the depth value of each line segment in the currently drawn contour line with the existing depth values in the depth buffer (including the drawn grids and contour lines, and the depth, color, and module information of each completed contour line are automatically saved to the buffer), determining which line segments should be rendered onto the screen. Optionally, the rule can be: if the line segment depth is less than the buffer value, it is visible (i.e., drawn); if the line segment depth is greater than or equal to the buffer value, it is blocked (i.e., not drawn), thereby ensuring the correct occlusion relationship.

[0049] After this step, an initial three-dimensional illustration is obtained. In the present invention, the initial three-dimensional illustration is referred to as the first illustration. For example, as Figure 4 shown, the display effect of this illustration is to only display the border lines of the object and not display the grid surface of the object, but the displayed border lines can also be blocked by the grid surfaces that are not displayed (i.e., the occlusion relationship is still retained). It should be noted that the first illustration has not been supplemented with edge lines.

[0050] Optionally, in S206, the operation of only saving the depth information of the grid in the drawing buffer includes: After drawing the grid of the target object in the three-dimensional model, saving the depth information, color information, and template information of the grid to the drawing buffer; and clearing the color information and template information of all the grids in the drawing buffer.

[0051] In this step, the color and template of the grids in the buffer data are cleared, and only the data in the depth buffer is retained. That is, only the depth information is retained, and the color information and module information are not retained, so that the display effect of the subsequent illustration only has border lines (contour lines) and the grids are not displayed.

[0052] In this step, clearing the color and template of the grids in the buffer data and only retaining the data in the depth buffer can achieve the following technical effects: 1. Improve rendering efficiency. For example, reduce unnecessary fragment shading calculations. In the rendering process of the prior art, each fragment passes through the fragment shader to calculate the color value. If a fragment is finally occluded by other closer fragments, then the color calculation for it is redundant. By only retaining the depth information, it is possible to determine at an early stage which fragments are invisible, thereby skipping the color calculation for these fragments. This can significantly reduce the computational burden on the fragment shader and improve the rendering efficiency. Another example is to reduce bandwidth occupancy. During the rendering process, the read and write operations of color and stencil information occupy the video memory bandwidth. If only the depth information is retained, these unnecessary read and write operations can be reduced, thereby reducing the use of the video memory bandwidth and further improving the rendering performance.

[0053] 2. Optimize the rendering of complex scenes. In complex scenes, there may be a large number of occlusion relationships between objects. By only retaining the depth information, these occlusion relationships can be processed more efficiently, avoiding unnecessary rendering of occluded objects. For example, in a scene containing a large number of transparent or semi-transparent objects, depth pre-testing can help quickly determine which objects do not need to perform transparency blending calculations, thereby optimizing the rendering process.

[0054] 3. Reduce resource consumption. Clearing the color and stencil buffer areas can reduce the occupancy of the video memory. On some devices with limited resources, this can significantly improve the rendering performance. In addition, only retaining the depth information can reduce the fragmentation of the buffer and improve the utilization rate of the buffer. This helps to improve the data access efficiency during the rendering process and further enhance the performance.

[0055] 4. Improve the performance of shadow rendering. In shadow mapping, usually only the depth information is required to calculate the shadows. By only retaining the depth information, shadow maps can be generated more efficiently, thereby improving the performance of shadow rendering. In addition, ambient occlusion calculations usually rely on the depth information. By only retaining the depth information, the ambient occlusion effect can be calculated more efficiently, thereby improving the realism of the scene.

[0056] 5. Reduce rendering errors. When dealing with transparent objects, the rendering methods in the prior art may cause sorting errors, resulting in incorrect rendering results. By only retaining the depth information and using deferred shading technology, the rendering of transparent objects can be processed more accurately, avoiding these errors. In addition, by reducing unnecessary calculations and data processing, the stability of the rendering process can be improved, reducing rendering errors and anomalies.

[0057] Optionally, in S206, perform backside extrusion drawing on each mesh in the initial three-dimensional illustration, as Figure 3 shown, including: S302. Calculate the first position information of the vertex in the view space based on the position information of the vertex in each grid under the local coordinate system of the 3D model. In this step, the purpose of calculating the position of the vertex in the view space is to transform the vertex coordinates into the coordinate system with the camera as the origin, which is convenient for subsequent projection calculations.

[0058] Optionally, the coordinate calculation relationship can be: the position of the vertex in the view space = the position of the model vertex * the model view matrix.

[0059] Optionally, the model view matrix is calculated by the following formula: ; Where, is the model matrix, is the view matrix, is the model view matrix.

[0060] Optionally, the model matrix represents the transformation of the object from the local coordinate system to the world coordinate system, which is composed of the following basic transformations: Translation: Placing the object at a position in the world coordinate system. Rotation: Adjusting the orientation of the object. Scale: Adjusting the size of the object.

[0061] Assume that the model needs to be scaled (S) first, then rotated (R), and finally translated (T), then the model matrix is: ; Optionally, the view matrix converts the world coordinate system to the camera coordinate system (with the camera as the origin and facing the -z axis). Its core is to calculate the three orthogonal axes of the camera: the right vector (R), the up vector (U), and the forward vector (F), and construct a rotation matrix with them. The specific process includes: 1. Input parameters: Camera position: , representing the x-axis coordinate of the camera , the y-axis coordinate and the z-axis coordinate ; Observation target point: , representing the x-axis coordinate of the observation point , the y-axis coordinate and the z-axis coordinate ; World space up direction: (usually the Y axis).

[0062] 2. Calculate the three orthogonal vectors Forward vector (F): The direction pointing to the observation target, which needs to be normalized.

[0063] ; F represents the x-axis coordinate of the forward vector , the y-axis coordinate and the z-axis coordinate , indicating the normalization operation.

[0064] Right vector (R): Obtained by taking the cross product of the forward vector F and the "up direction" in world space, and needs to be normalized.

[0065] ; R represents the x-axis coordinate of the right vector , the y-axis coordinate and the z-axis coordinate .

[0066] Up vector (U): Obtained by taking the cross product of the right vector and the forward vector to ensure orthogonality of the three vectors.

[0067] ; U represents the x-axis coordinate of the up vector , the y-axis coordinate and the z-axis coordinate .

[0068] 3. Construct the view matrix. The view matrix consists of a rotation part and a translation part: Rotation matrix : Align the world coordinate system to the camera coordinate system. The camera right vector (R) corresponds to the +x axis of the view coordinate system, and the camera up vector (U) corresponds to the +y axis of the view coordinate system. The camera forward vector (-F) corresponds to the -z axis of the view coordinate system (due to the default orientation being -z).

[0069] ; is the component of the right vector R corresponding to the x axis of the view coordinate system.

[0070] is the component of the up vector U corresponding to the y axis of the view coordinate system.

[0071] is the opposite direction of the forward vector F corresponding to the -z axis of the view coordinate system.

[0072] Translation matrix : Move the camera position to the origin.

[0073] ; Finally, the view matrix is: ; S304. Calculate the outward expansion direction of the vertex in the normalized device coordinate space based on the normal of the three-dimensional model; Optionally, this step includes the following steps: 1. Calculate the normal in world space and set the z-axis value to -0.5 (to prevent the vertices after back face expansion from blocking the front face patches); where the world space is the coordinate system of the entire scene, and in this space, the positions of all objects are relative to the origin of the scene.

[0074] Normal in world space = Model normal Normal matrix; 2. Calculate the normal in NDC space; Normal in NDC space = Normal in world space Projection matrix 3. Normalize the normal in NDC space to ensure that the outward expansion direction is a unit vector. Here, the outward expansion direction of the vertex is obtained, that is, the vertex will expand along this normal direction subsequently.

[0075] Among them, the normal matrix is used to transform the normal vector in model space to world space, and its mathematical essence is the inverse transpose matrix of the model matrix.

[0076] The model matrix is: ; T is the translation matrix and does not affect the direction of the normal; R is the rotation matrix and is an orthogonal matrix, that is, it satisfies ; S is the scaling matrix, and non-uniform scaling will cause the normal direction to be incorrect.

[0077] Since the normal is a direction vector, the influence of translation needs to be eliminated and the direction deviation caused by scaling needs to be corrected.

[0078] ; Therefore, the normal matrix can be expressed as: ; When the model only includes rotation and translation and does not include scaling, the normal matrix is simplified to .

[0079] The projection matrix P converts the coordinates within the view frustum to clip space coordinates. The core is to define the geometric parameters of the view frustum. Taking perspective projection as an example: The parameters of perspective projection are: near plane distance n, far plane distance f, field of view fov (vertical direction), and aspect ratio aspect.

[0080] The perspective projection matrix P is as follows: ; The formula for normalizing the normal vector in the NDC space is as follows:

[0081] In the formula, n normalized is the normalized plane normal vector, that is, the normal vector in the normalized NDC space; n ndc_xy is the normal vector in the NDC space, is the modulus of the normal vector in the NDC space.

[0082] S306. Calculate the second position information of the vertex in the normalized device coordinate space based on the first position information; Optionally, the calculation method is: The position of the vertex in the NDC space = the position of the vertex in the view space Projection matrix.

[0083] S308. Calculate the third position information of the vertex in the normalized device coordinate space after the vertex is expanded by a preset width along the expansion direction; Calculate the position of the vertex after the model vertex is expanded by a preset width along the normalized plane normal vector. As a preferred example, the preset width is the width of the contour line.

[0084] The position of the vertex after expansion in the NDC space = the position of the vertex in the NDC space + the normalized plane normal vector Preset width.

[0085] S310. Calculate the fourth position information of the vertex in the screen space based on the third position information; Optionally, the calculation method is: The position of the vertex in the screen space = the position of the vertex after expansion in the NDC space Viewport transformation matrix; The viewport transformation matrix (Viewport Transformation Matrix) is a matrix in computer graphics used to convert normalized device coordinates (Normalized Device Coordinates, abbreviated as NDC) to screen coordinates (Screen Coordinates). The NDC coordinate system is a cube with a coordinate range of -1 to 1 in the x, y, and z directions. The viewport transformation maps these coordinates to a rectangular area on the actual display device, that is, the viewport.

[0086] The viewport transformation matrix is a 4x4 homogeneous coordinate matrix. Assume the bottom-left coordinates of the viewport on the screen are with width and height being respectively W and H , and the depth range is (usually n = 0, f = 1). Then the viewport transformation matrix M viewport can be expressed as: .

[0087] Among them, the first row of the viewport transformation matrix scales the x coordinate and translates , mapping [-1, 1] to .

[0088] The second row of the viewport transformation matrix scales the y coordinate and translates , mapping [-1, 1] to , where the positive direction of the screen y-axis is downward.

[0089] The third row of the viewport transformation matrix scales the z coordinate and translates , mapping [0, 1] to n, f .

[0090] The transformation process is: Given the normalized device coordinates , the screen coordinates are obtained through matrix multiplication. That is, the x-axis coordinate of the normalized device coordinates is , the y-axis coordinate is , the z-axis coordinate is , the x-axis coordinate of the screen coordinates is , the y-axis coordinate is , and the z-axis coordinate is .

[0091] The calculation process is: .

[0092] For example, assume the viewport parameters are , and the normalized coordinates are (0.5, -0.5, 0.8, 1). Then the transformed screen coordinates are: ; ; ; The final screen coordinates are (700, 550, 0.9).

[0093] S312. Only perform back drawing on each grid in the initial three-dimensional illustration based on the data in the drawing buffer, the set color information, and the fourth position information.

[0094] After the above steps, the final three-dimensional illustration is obtained. All grids in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back drawing in each expanded grid and the missing contour lines visually form continuous complementary lines. The final three-dimensional illustration, also referred to as the second illustration in the present invention, is as Figure 5 shown. Since the depth buffer is retained when implementing the contour line occlusion effect, when drawing the back of the first illustration, the part of the grid that coincides with the object drawn in the first illustration will not be displayed, and only the expanded part will be displayed. For the position of the missing contour line, the expanded part displayed resembles a contour line, thus indirectly implementing the line complement function.

[0095] Using the three-dimensional illustration display method of the present invention, there is no need to perform actual geometric data calculation and display a real edge line, but use simple rendering techniques to implement the edge line complement function. That is to say, the present invention uses rendering techniques to achieve the illustration effect, which can meet the implementation of the illustration effect of large-scale scene models. In addition, since rendering is used instead of the geometric data calculation of traditional edge line complement, the calculation amount is greatly reduced. To ensure that the rendering performance of this solution remains in the millisecond level, this solution uses a culling algorithm to reduce the number of objects to be displayed before the first drawing, so it will not cause a decline in rendering performance.

[0096] In addition, according to the second aspect of this embodiment, a storage medium is provided. The storage medium includes a stored program, wherein when the program runs, the three-dimensional illustration display method described in any one of the above is executed by a processor.

[0097] It should be noted that for the foregoing method embodiments, for the sake of simple description, they are all expressed as a series of action combinations. However, those skilled in the art should know that the present invention is not limited by the described action sequence, because according to the present invention, certain steps can be performed in other sequences or simultaneously. Secondly, those skilled in the art should also know that the embodiments described in the specification are all preferred embodiments, and the actions and modules involved are not necessarily essential to the present invention.

[0098] Through the description of the above embodiments, those skilled in the art can clearly understand that the method according to the above embodiments can be implemented by means of software plus a necessary general hardware platform. Of course, it can also be implemented by hardware, but in many cases, the former is a better implementation method. Based on such an understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art can be embodied in the form of a software product. This computer software product is stored in a storage medium (such as ROM / RAM, magnetic disk, optical disk), and includes several instructions for causing a terminal device (which can be a mobile phone, computer, server, or network device, etc.) to execute the methods described in various embodiments of the present invention.

[0099] Embodiment 2 Figure 6 An illustration display device according to the present embodiment is shown, and this device corresponds to the method described in the first aspect of Embodiment 1. Refer to Figure 6 As shown, the device includes: A contour occlusion module 610, configured to draw a mesh of a target object in the three-dimensional model and only save the depth information of the mesh in a drawing buffer; wherein, the target object is an object to be displayed in the three-dimensional model; superimpose and draw the contour line of the target object on the drawn mesh, and save the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object, does not show the mesh surface of the target object, and there are missing parts in the visual area of the part of the contour line displayed on the screen. An edge filling module 620, configured to set the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and perform backside expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein, all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric areas displayed due to backside drawing in the expanded meshes form continuous filled lines visually with the missing contour lines.

[0100] Wherein, the three-dimensional illustration is a three-dimensional image generated by rendering a three-dimensional model onto a screen plane.

[0101] Optionally, the contour occlusion module 610 is further configured to, before the operation of drawing the mesh of the target object in the three-dimensional model, use a preset culling algorithm to cull the objects that do not need to be displayed in the three-dimensional model and retain the objects that need to be displayed in the three-dimensional model.

[0102] Optionally, the contour occlusion module 610 is further configured to save only the depth information of the mesh in the rendering buffer, specifically including: After rendering the mesh of the target object in the three-dimensional model, saving the depth information, color information, and stencil information of the mesh to the rendering buffer; and Clearing the color information and stencil information of all meshes in the rendering buffer.

[0103] Optionally, the contour occlusion module 610 is further configured to superimpose and render the contour lines of the target object on the rendered mesh, specifically including: Comparing the depth value of each line segment in the currently rendered contour line with the existing depth values in the depth buffer; wherein, the existing depth values include the depth information of all rendered meshes and the depth information of the rendered contour lines; Determining the line segments to be rendered onto the screen according to the comparison result.

[0104] Optionally, the edge filling module 620 is further configured to perform backside expansion rendering on each mesh in the initial three-dimensional illustration through the following steps: Based on the position information of the vertices in each mesh in the local coordinate system of the three-dimensional model, calculating the first position information of the vertices in the view space; Based on the normal of the three-dimensional model, calculating the expansion direction of the vertices in the normalized device coordinate space; Based on the first position information, calculating the second position information of the vertices in the normalized device coordinate space; Based on the second position information, calculating the third position information of the vertices in the normalized device coordinate space after the vertices are expanded by a preset width along the expansion direction; Based on the third position information, calculating the fourth position information of the vertices in the screen space; Based on the data in the rendering buffer, the set color information, and the fourth position information, performing only backside rendering on each mesh in the initial three-dimensional illustration.

[0105] Optionally, the preset width is the width of the contour line.

[0106] Optionally, the depth information is saved in the depth buffer, the color information is saved in the color buffer, and the stencil information is saved in the stencil buffer.

[0107] According to this embodiment, instead of performing actual geometric data calculations and displaying a real edge line, simple rendering techniques are used to implement the edge line filling function. The present invention uses rendering techniques to achieve the effect of illustrations, which can meet the implementation of illustration effects for large-scale scene models. In addition, the present invention uses rendering to replace the geometric data calculations of traditional edge line filling, greatly reducing the amount of calculation. To ensure that the rendering performance of this solution remains in the millisecond level, this solution uses a culling algorithm to reduce the number of objects to be displayed before the first drawing, so as not to cause a decline in rendering performance.

[0108] Embodiment 3 Figure 7 An illustration display device according to this embodiment is shown, and this device corresponds to the method described in the first aspect of Embodiment 1. Referring Figure 7 as shown, this device includes: a processor 710; and a memory 720, connected to the processor 710, for providing instructions for the processor 710 to process the following processing steps: Drawing the mesh of the target object in the three-dimensional model and only saving the depth information of the mesh in the drawing buffer; wherein, the target object is the object to be displayed in the three-dimensional model; Overlaying and drawing the contour line of the target object on the drawn mesh, and saving the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object and does not show the mesh surface of the target object, and there are missing parts in the visual area of the displayed contour line on the screen; Setting the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and performing back face outward expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein, all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric areas displayed due to back face drawing in the expanded meshes form continuous complementary lines visually with the missing contour lines; The three-dimensional illustration is a three-dimensional image generated by rendering a three-dimensional model onto a screen plane.

[0109] Optionally, the memory 720 is further used to provide instructions for the processor 710 to process the following processing steps: Before the operation of drawing the mesh of the target object in the three-dimensional model, it further includes: Using a preset culling algorithm to cull the objects that do not need to be displayed in the three-dimensional model and retain the objects that need to be displayed in the three-dimensional model.

[0110] Optionally, the memory 720 is further configured to provide instructions for the processor 710 to process the following processing steps: Only save the depth information of the mesh in the rendering buffer, specifically including: After rendering the mesh of the target object in the three-dimensional model, save the depth information, color information, and stencil information of the mesh to the rendering buffer; and Clear the color information and stencil information of all meshes in the rendering buffer.

[0111] Optionally, the memory 720 is further configured to provide instructions for the processor 710 to process the following processing steps: Overlay and render the contour lines of the target object on the rendered meshes, specifically including: Compare the depth value of each line segment in the currently rendered contour line with the existing depth values in the depth buffer; wherein, the existing depth values include the depth information of all rendered meshes and the depth information of the rendered contour lines; Determine the line segments to be rendered onto the screen based on the comparison results.

[0112] Optionally, the memory 720 is further configured to provide instructions for the processor 710 to process the following processing steps: Perform backside extrusion rendering on each mesh in the initial three-dimensional illustration through the following steps: Based on the position information of the vertices in each mesh in the local coordinate system of the three-dimensional model, calculate the first position information of the vertices in the view space; Based on the normal of the three-dimensional model, calculate the extrusion direction of the vertices in the normalized device coordinate space; Based on the first position information, calculate the second position information of the vertices in the normalized device coordinate space; Based on the second position information, calculate the third position information of the vertices in the normalized device coordinate space after the vertices are extruded by a preset width along the extrusion direction; Based on the third position information, calculate the fourth position information of the vertices in the screen space; Based on the data in the rendering buffer, the set color information, and the fourth position information, perform only backside rendering on each mesh in the initial three-dimensional illustration.

[0113] Optionally, the preset width is the width of the contour line.

[0114] Optionally, the depth information is saved in the depth buffer, the color information is saved in the color buffer, and the stencil information is saved in the stencil buffer.

[0115] According to this embodiment, instead of performing actual geometric data calculations and displaying a real edge line, a simple rendering technique is used to implement the edge filling function. The present invention uses rendering techniques to achieve the effect of illustrations, which can meet the realization of illustration effects for large-scale scene models. In addition, the present invention uses rendering to replace the geometric data calculations of traditional edge filling, greatly reducing the amount of calculation. To ensure that the rendering performance of this solution remains in the millisecond level, the culling algorithm is used to reduce the number of objects to be displayed before the first drawing, so as not to cause a decline in rendering performance.

[0116] The serial numbers of the above embodiments of the present invention are only for description and do not represent the advantages or disadvantages of the embodiments.

[0117] In the above embodiments of the present invention, the descriptions of each embodiment have their own focuses. For the parts not detailed in a certain embodiment, reference can be made to the relevant descriptions of other embodiments.

[0118] In several embodiments provided by the present application, it should be understood that the disclosed technical content can be implemented in other ways. Among them, the device embodiments described above are only illustrative. For example, the division of the units is only a logical function division. In actual implementation, there may be other division methods. For example, multiple units or components can be combined or integrated into another system, or some features can be ignored or not executed. Another point is that the displayed or discussed couplings or direct couplings or communication connections to each other can be through some interfaces. The indirect couplings or communication connections of the units or modules can be in electrical or other forms.

[0119] The units described as separate components may or may not be physically separated. The components displayed as units may or may not be physical units, that is, they can be located in one place or distributed to multiple network units. Some or all of the units can be selected according to actual needs to achieve the purpose of the solution of this embodiment.

[0120] In addition, the functional units in each embodiment of the present invention can be integrated into one processing unit, or each unit can exist physically alone, or two or more units can be integrated into one unit. The above integrated units can be implemented in the form of hardware or in the form of software functional units.

[0121] When the integrated unit is implemented in the form of a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of this technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions for causing a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes: various media that can store program codes, such as USB flash drives, read-only memories (ROMs), random access memories (RAMs), mobile hard disks, magnetic disks, or optical discs.

[0122] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention.

Claims

1. A method for displaying a three-dimensional illustration, characterized in that, The three-dimensional illustration is a stereoscopic image generated by rendering a three-dimensional model onto a screen plane. The display method includes: Drawing a mesh of a target object in the three-dimensional model and only saving the depth information of the mesh in a drawing buffer; wherein, the target object is an object to be displayed in the three-dimensional model; Overlaying and drawing the contour lines of the target object on the drawn mesh, and saving the depth information, color information, and stencil information of the contour lines in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour lines of the target object and does not show the mesh surface of the target object, and there are missing parts in the contour lines shown in some visual areas of the screen display; Setting the color information of all meshes in the initial three-dimensional illustration to the color information of the contour lines, and performing back-face expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein, all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions shown due to back-face drawing in each expanded mesh form visually continuous complementary lines with the missing contour lines.

2. The method according to claim 1, wherein, Before the operation of drawing the mesh of the target object in the three-dimensional model, it further includes: Using a preset culling algorithm to cull the objects that do not need to be displayed in the three-dimensional model and retaining the objects that need to be displayed in the three-dimensional model.

3. According to the method described in claim 1, only saving the depth information of the mesh in the drawing buffer includes: After drawing the mesh of the target object in the three-dimensional model, saving the depth information, color information, and stencil information of the mesh to the drawing buffer; and Clearing the color information and stencil information of all meshes in the drawing buffer.

4. The method according to claim 1, wherein The overlaying and drawing the contour lines of the target object on the drawn mesh includes: Comparing the depth value of each line segment in the currently drawn contour line with the existing depth values in the depth buffer; wherein, the existing depth values include the depth information of all drawn meshes and the depth information of the drawn contour lines; Determining the line segments to be rendered onto the screen according to the comparison result.

5. The method according to claim 1, wherein Performing back-face expansion drawing on each mesh in the initial three-dimensional illustration through the following steps: Based on the position information of the vertices in each mesh in the local coordinate system of the three-dimensional model, calculating the first position information of the vertices in the view space; Based on the normal of the three-dimensional model, calculating the expansion direction of the vertices in the normalized device coordinate space; Based on the first position information, calculating the second position information of the vertices in the normalized device coordinate space; Based on the second position information, calculating the third position information of the vertices in the normalized device coordinate space after the vertices are expanded by a preset width along the expansion direction; Based on the third position information, calculating the fourth position information of the vertices in the screen space; Based on the data in the drawing buffer, the set color information, and the fourth position information, only back drawing is performed on each grid in the initial three-dimensional illustration.

6. The method according to claim 5, wherein The preset width is the width of the contour line.

7. The method according to claim 1, wherein The depth information is stored in the depth buffer, the color information is stored in the color buffer, and the stencil information is stored in the stencil buffer.

8. A storage medium, characterized in that, The storage medium includes a stored program, wherein, when the program runs, the method according to any one of claims 1 to 7 is executed by a processor.

9. A display device for three-dimensional illustrations, characterized in that, The three-dimensional illustration is a three-dimensional image generated by rendering a three-dimensional model onto a screen plane, and the display device includes: A contour occlusion module configured to draw the grids of the target object in the three-dimensional model and only save the depth information of the grids in the drawing buffer; wherein, the target object is the object to be displayed in the three-dimensional model; overlay and draw the contour line of the target object on the drawn grids, and save the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object and does not show the grid surface of the target object, and there are missing parts in the visual area of the displayed contour line on the screen. An edge filling line module configured to set the color information of all grids in the initial three-dimensional illustration to the color information of the contour line, and perform back expansion drawing on all grids in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein, all grids in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back drawing in the expanded grids form continuous filling lines visually with the missing contour lines.

10. An illustration display device, characterized in that, Comprising: A processor; And A memory connected to the processor for providing instructions for the processor to perform the following processing steps: Draw the grids of the target object in the three-dimensional model and only save the depth information of the grids in the drawing buffer; wherein, the target object is the object to be displayed in the three-dimensional model; Overlay and draw the contour line of the target object on the drawn grids, and save the depth information, color information, and stencil information of the contour line in the drawing buffer to obtain an initial three-dimensional illustration; wherein the initial three-dimensional illustration only shows the contour line of the target object and does not show the grid surface of the target object, and there are missing parts in the visual area of the displayed contour line on the screen. Set the color information of all meshes in the initial three-dimensional illustration to the color information of the contour line, and perform back expansion drawing on all meshes in the initial three-dimensional illustration based on the data in the drawing buffer and the set color information to obtain a final three-dimensional illustration; wherein, all meshes in the final three-dimensional illustration are expanded by a preset width along the normal direction in the normalized device coordinate space, and the geometric regions displayed due to back drawing in each expanded mesh form visually continuous complementary lines with the missing contour lines. The three-dimensional illustration is a three-dimensional image generated by rendering a three-dimensional model onto a screen plane with a three-dimensional sense.

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