A curvature compensation-based optimized directional slice convex hull volume calculation method

Abstract

The application discloses a curvature compensation-based optimization direction slice convex hull volume calculation method, belongs to the technical field of point cloud data processing, and is used for convex hull volume calculation and comprises the following steps: automatically analyzing a main extension direction based on point cloud distribution characteristics to adaptively perform layered slicing, simultaneously calculating local density distribution and surface concave-convex features for each layer of point cloud, correcting sparse area errors through density compensation, combining curvature energy analysis to dynamically adjust volume calculation of a concave surface area, adopting a parallel accelerated trapezoidal integration method to fuse correction results of each layer, outputting high-precision earthwork volume data, and feeding back to a cockpit terminal in real time to guide operation adjustment, and realizing full-process automatic processing from data acquisition to volume calculation. The application satisfies the hardware limitation of low computing power of a current engineering machinery controller, increases the accuracy of point cloud volume calculation, and balances the relationship between accuracy and cost.

Description

TECHNICAL FIELD

[0001] The application discloses a curvature compensation-based optimized direction slice convex hull volume calculation method and belongs to the technical field of point cloud data processing. BACKGROUND

[0002] With the maturity of laser radar technology, point cloud-based volume measurement methods have gradually become mainstream, but in the field of engineering machinery, the following challenges still exist: dynamic operation interference, vibration generated by excavator operation, and dust environment leading to significant point cloud noise, making it difficult for traditional filtering algorithms to effectively denoise; complex shape modeling, irregular concave surface and loose structure of soil accumulation, high error of standard convex hull method, and inability to meet the engineering settlement accuracy requirement (usually <5%); real-time bottleneck, existing algorithms rely on cloud computing, which cannot achieve real-time feedback in scenarios where network coverage is insufficient in mining areas, affecting operation efficiency optimization. In view of the above problems, the prior art attempts to improve accuracy through multi-sensor fusion (such as laser radar + IMU) or high-density scanning, but has defects such as high cost and high system complexity, and some deep learning-based algorithms have high hardware computing power requirements and low real-time performance. SUMMARY

[0003] The application aims to provide a curvature compensation-based optimized direction slice convex hull volume calculation method to solve the problem of low convex hull volume calculation efficiency in the prior art.

[0004] A curvature compensation-based optimized direction slice convex hull volume calculation method selects four non-coplanar points in point cloud data to form an initial tetrahedral convex hull. For points in the point cloud data other than the four points, it is determined whether a point is outside the initial tetrahedral convex hull. If a point is outside the initial tetrahedral convex hull, the point is considered to be a temporary vertex, the visible faces of the temporary vertex are found, the shared edges between the visible faces are removed, the non-shared edges are retained, the temporary vertex is connected to the two endpoints of the shared edge, and a new triangular face is generated. If a point is located inside the initial tetrahedral convex hull, the point is skipped. When all points are located inside the initial tetrahedral convex hull, the three-dimensional convex hull is constructed, and the volume of the three-dimensional convex hull is calculated.

[0005] Calculating the volume of the three-dimensional convex hull includes selecting the center of gravity of the point cloud data

[0006]

[0007] In the formula, is the number of point cloud data in the grid, represents the i-th grid, represents the j-th point cloud data in the grid.

[0008] ​​​​The calculating the volume of the three-dimensional convex hull comprises constructing a covariance matrix :

[0009] ;

[0010] ;

[0011] wherein, is equivalent to , is a central coordinate, , , is a horizontal, vertical and longitudinal coordinate value of the central coordinate.

[0012] The calculating the volume of the three-dimensional convex hull comprises solving , is an eigenvalue matrix, is an eigenvector matrix;

[0013] After solving, eigenvalues and orthogonal eigenvectors corresponding to the eigenvalues , are obtained.

[0014] An orthogonal projection base is constructed to generate a two-dimensional projection plane perpendicular to the main direction of the point cloud distribution, and an orthogonal base is constructed by Gram-Schmidt orthogonalization , and a unitized main direction is calculated. is normalized to a unit length:

[0015] ;

[0016] The second direction is orthogonalized, and is projected to a direction perpendicular to :

[0017] ;

[0018] The third direction is obtained by cross multiplication, and the base forms a right-handed coordinate system:

[0019] ;

[0020] After the orthogonal base is calculated, the point cloud is projected to plane.

[0021] The coordinates of all points along the main direction are projected:

[0022] ;

[0023] wherein, is the projection value of the point along the main direction;

[0024] determining the range of the slice:

[0025] ;

[0026] ;

[0027] wherein, is the minimum value of the slice, is the maximum value of the slice;

[0028] adaptively calculating the number of slices :

[0029] ;

[0030] ;

[0031] ;

[0032] ;

[0033] wherein, is used to limit the number of slices between 200 and 1000, , is the maximum and minimum value of the horizontal axis of the point cloud data, , is the maximum and minimum value of the vertical axis of the point cloud data, , is the maximum and minimum value of the vertical axis of the point cloud data, , , is the length of the three directions, is the average value of the length of the three directions, is the minimum value of the length of the three directions.

[0034] For the first layer slice, extract the point set in the layer :

[0035] ;

[0036] ;

[0037] wherein, is the intermediate parameter, projecting the point set in the layer to the plane :

[0038] ;

[0039] divide the integral interval into Layers, each layer thickness , calculate three-dimensional convex hull volume :

[0040] ;

[0041] ;

[0042] In the formula, is the vertical axis coordinate value, is the first layer cross-section of the two-dimensional convex hull area.

[0043] For Optimization of the approximate calculation formula, including calculating the K-nearest neighbor of each layer slice Covariance matrix :

[0044] ;

[0045] In the formula, is the K-nearest neighbor set of , denotes the mean value of , denotes the point cloud data;

[0046] For Eigenvalue decomposition, get the eigenvalue , calculate the average curvature of the local surface :

[0047] ;

[0048] In the formula, The positive value corresponds to the convex surface, and the negative value corresponds to the concave surface.

[0049] Optimization of the approximate calculation formula of Including, calculating the second moment of the cross-section, assuming that the first Layer two-dimensional convex hull vertex , calculate the bending amount of the cross-sectional shape :

[0050] ;

[0051] In the formula, is the maximum value of , according to the curvature and Layer second moment of the compensation amount of the cross-sectional area :

[0052] ;

[0053] In the formula, is a parameter for controlling compensation intensity;

[0054] layer-corrected second-moment cross-sectional area is:

[0055] ;

[0056] wherein, is the second-moment cross-sectional area before correction;

[0057] in the concave region .

[0058] to Optimizing the approximate calculation formula of includes the corrected three-dimensional convex hull volume is:

[0059] .

[0060] Compared with the prior art, the present application has the following beneficial effects: the present application meets the current hardware limitation of low computing power of the engineering machinery controller, increases the accuracy of point cloud volume calculation, and balances the relationship between accuracy and cost. BRIEF DESCRIPTION OF DRAWINGS

[0061] Figure 1 is a convex hull in a two-dimensional space;

[0062] Figure 2 is the point cloud data of a cylindrical iron drum;

[0063] Figure 3 is the convex hull construction result of the cylindrical iron drum;

[0064] Figure 4 is the curvature calculation visual view of the cylindrical iron drum;

[0065] Figure 5 is the optimized direction slice schematic diagram of the cylindrical iron drum;

[0066] Figure 6 is the point cloud data of a conical tower model;

[0067] Figure 7 is the convex hull construction result of the conical tower model;

[0068] Figure 8 is the curvature calculation visual view of the conical tower model;

[0069] Figure 9 is the optimized direction slice schematic diagram of the conical tower model;

[0070] Figure 10 is the point cloud data of a concave hull cube;

[0071] Figure 11 is the convex hull construction result of the concave hull cube;

[0072] Figure 12 is the curvature calculation visual view of the concave hull cube;

[0073] Figure 13 is the full-dip soil material point cloud image. DETAILED DESCRIPTION

[0074] In order to make the objects, technical solutions and advantages of the present application clearer, the technical solutions in the present application are described clearly and completely below. Obviously, the described embodiments are part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative labor fall within the protection scope of the present application.

[0075] A curvature compensation-based optimized direction slice convex hull volume calculation method selects four non-coplanar points in point cloud data to form an initial tetrahedral convex hull. For points in the point cloud data other than the four points, it is judged whether a point is outside the initial tetrahedral convex hull. If a point is outside the initial tetrahedral convex hull, the point is considered as a temporary vertex, the visible faces of the temporary vertex are found, the shared edges between the visible faces are removed, the non-shared edges are retained, and the temporary vertex is connected with the two end points of the shared edge to generate a new triangular face. If a point is located in the initial tetrahedral convex hull, the point is skipped. When all the points are located in the initial tetrahedral convex hull, the three-dimensional convex hull is constructed, and the volume of the three-dimensional convex hull is calculated.

[0076] The calculation of the volume of the three-dimensional convex hull includes selecting the center of gravity of the point cloud data :

[0077] ;

[0078] In the formula, is the number of point cloud data in the grid, represents the i-th grid, represents the i-th point cloud data in the grid. The calculation of the volume of the three-dimensional convex hull includes constructing a covariance matrix :

[0079]

[0080] ;

[0081] ;

[0082] In the formula, is equivalent to , is the centralized coordinate,​​ 、 、 are the horizontal, vertical and depth coordinate values of the central coordinate.

[0083] The calculation of the three-dimensional convex hull volume includes solving , is the eigenvalue matrix, is the eigenvector matrix.

[0084] After solving, the eigenvalue and the orthogonal eigenvector corresponding to the eigenvalue , is the principal direction of the point cloud distribution.

[0085] Construct an orthogonal projection basis to generate a two-dimensional projection plane perpendicular to the principal direction of the point cloud distribution, and use the Gram-Schmidt orthogonalization to construct an orthogonal basis , calculate the unitized principal direction , normalize to unit length:

[0086] ;

[0087] Orthogonalize the second direction, project to the direction perpendicular to :

[0088] ;

[0089] The cross product gets the third direction, which ensures that the basis forms a right-handed coordinate system:

[0090] ;

[0091] After calculating the orthogonal basis, project the point cloud to plane.

[0092] Project the coordinates of all points along the principal direction:

[0093] ;

[0094] In the formula, is the projection value of the point along the principal direction;

[0095] Determine the range of the slice:

[0096] ;

[0097] ;

[0098] In the formula, is the minimum value of the slice, is the maximum value of the slice,

[0099] Adaptive calculation of the number of slices

[0100]

[0101]

[0102]

[0103]

[0104] wherein, for limiting the number of slices between 200 and 1000, is the maximum and minimum value of the horizontal axis of the point cloud data, is the maximum and minimum value of the vertical axis of the point cloud data, is the maximum and minimum value of the vertical axis of the point cloud data, is the length of the three directions, is the average value of the length of the three directions, is the minimum value of the length of the three directions.

[0105] For the first layer slice, the in-layer point set is extracted

[0106]

[0107]

[0108] wherein, is the intermediate parameter, projecting the in-layer point set to the plane

[0109]

[0110] The integral interval is divided into layers, each layer has a thickness of , and the three-dimensional convex hull volume is calculated

[0111]

[0112]

[0113] wherein, is the vertical axis coordinate value, is the first​​​​​​​​​​​​​​​​​​ The two-dimensional convex hull area of the layer cross-section.

[0114] The mean curvature of the local surface is calculated as follows: The approximate calculation formula of the mean curvature of the local surface is optimized, including calculating the second moment of each layer cross-section The covariance matrix of the K-nearest neighbors of the point cloud data :

[0115] ;

[0116] In the formula, is the K-nearest neighbor set of the point cloud data, denotes the mean value of the point cloud data, denotes the point cloud data; The eigenvalues of the feature decomposition of the point cloud data are obtained

[0117] The mean curvature of the local surface is calculated as follows: :

[0118] ;

[0119] In the formula, The positive value corresponds to the convex surface, and the negative value corresponds to the concave surface.

[0120] The approximate calculation formula of the mean curvature of the local surface is optimized, including calculating the second moment of each layer cross-section The vertices of the two-dimensional convex hull of the i-th layer are denoted as , and the bending amount of the cross-sectional shape is calculated :

[0121] ;

[0122] In the formula, is the maximum value of the point cloud data, and the compensation amount of the cross-sectional area is adjusted according to the curvature and the second moment of the i-th layer: :

[0123] ;

[0124] In the formula, is a parameter for controlling the compensation strength;

[0125] The second moment of the cross-sectional area of the i-th layer is modified as follows:

[0126] ;

[0127] In the formula, is the second moment of the cross-sectional area before modification; ​​​​​​

[0128] In the concave region .

[0129] For The optimization of the approximate equal formula includes the modified three-dimensional convex hull volume For:

[0130] .

[0131] Convex hull is an important tool in computational geometry, which is used to determine the smallest convex set that encloses a given set of points. In three-dimensional point cloud data processing, convex hull is often used to estimate the volume of an object. In two-dimensional space, the convex hull is the smallest convex polygon that can contain all the points. It can completely enclose a set of discrete points and ensure that all points are located on the boundary or inside the convex polygon. The convex hull is composed of a series of vertices and boundaries, as shown in Figure 1 One of the important properties of convex hull is convexity, that is, the line between any two points inside or on the boundary of the convex hull is also completely inside the convex hull. In three-dimensional space, the convex hull is a polyhedron, and all points in space are inside or on the surface of the polyhedron. The three-dimensional convex hull is composed of multiple faces, boundaries and vertices, where each face is a planar polygon, the boundary is the intersection of the faces, and the vertex is the intersection of the boundaries. Like the properties of two-dimensional convex hull, three-dimensional convex hull also has convexity, which ensures that the convex hull can cover the entire point set with the smallest boundary.

[0132] Convex hull method has high accuracy for convex point cloud data. However, due to its neglect of concave characteristics, it usually overestimates the true volume when facing concave point cloud data, resulting in large errors. To compensate for this error, the point cloud data can be calculated by the slice convex hull method. The principle of slice convex hull method is to decompose the three-dimensional problem into two-dimensional slices along the fixed axis (such as Z axis), calculate the two-dimensional convex hull area layer by layer, and obtain the total volume by integration.

[0133] When slicing the convex polyhedron, the fixed-axis slice convex hull method uses a specified number of slices, which may cause the slice of small volume point cloud to be too dense or redundant. Therefore, the slice parameters need to be calculated autonomously in the optimization direction slice convex hull method.

[0134] The experimental point cloud data includes an open source data set and excavated materials and standard material iron barrels collected by a laser radar. The volume measurement experiment is performed by using a standard convex hull method, a fixed shaft slice convex hull method, an optimized direction slice convex hull method and a slice convex hull method based on curvature compensation to measure the volumes of a plurality of point cloud models, and compared with the true volume to calculate the relative error of the measurement value. The data set is obtained by scanning a cylindrical iron cylinder using a laser radar and acquiring point cloud data of the cylindrical iron cylinder. The volume of the cylindrical iron cylinder point cloud data is calculated by the convex hull method and compared with the theoretical volume. The cylindrical cylinder point cloud data is shown in Figure 2 .

[0135] The volume calculation formula of the cylinder is , wherein is the radius of the bottom surface of the cylinder, is the height of the cylinder, the height of the measured cylindrical barrel is 1.5 m, the outer diameter of the barrel is 1.03 m, and the radius is 0.515 m. The theoretical volume of the cylindrical barrel is calculated to be 1.2490 m³. The volume calculated by the standard convex hull method is 1.3215 m³; the volume calculated by the fixed shaft slice convex hull method is 1.3359 m³; the volume calculated by the optimized direction slice convex hull method is 1.2326 m³; and the volume calculated by the optimized direction convex hull algorithm based on curvature compensation is 1.2172 m³. The calculation can obtain that the error of the standard convex hull method is 5.804%, the error of the fixed shaft slice convex hull method is 6.958%, the error of the optimized direction slice convex hull method is -1.313%, and the error of the optimized direction convex hull algorithm based on curvature compensation is -2.546%. The calculation result of the optimized direction convex hull algorithm based on curvature compensation is within the standard of ±5% which is generally acceptable in engineering.

[0136] Figure 3 is the convex hull construction result of the cylindrical barrel, wherein the grid of the convex hull is expanded by 50% for visualization. Figure 4 is the visualization of the point cloud data points calculated by the curvature compensation algorithm of the present application for each slice of the cylindrical iron barrel. Figure 5 is an optimized direction slice diagram, and it can be seen that the slice direction is from the lower left corner to the upper right corner of the cylinder, which conforms to the maximum distribution direction of the point cloud.

[0137] The data set is a tower model composed of two cones (hereinafter referred to as a cone tower model), as shown in Figure 6 , the height of the three-prism pyramid is 5 m, the bottom surface radius is 2 m, and the volume of the three-prism pyramid is , wherein , wherein is the height of the three-prism pyramid, is the bottom surface radius of the three-prism pyramid, Take 3.14. The true volume of the cone tower model can be calculated as 41.87 m³. The volume calculated by the standard convex hull method is 83.5439 m³; the volume calculated by the fixed axis slice convex hull method is 84.7863 m³; the volume calculated by the optimized direction slice convex hull method is 42.5126 m³; and the volume calculated by the optimized direction convex hull algorithm based on curvature compensation proposed in the application is 42.3988 m³. Through calculation, it can be obtained that the error of the standard convex hull method and the fixed axis slice convex hull method is large, the error of the optimized direction slice convex hull method is 1.535%, and the error of the optimized direction convex hull algorithm based on curvature compensation is 1.263%. The calculation result of the optimized direction convex hull algorithm based on curvature compensation is within the standard of ± 5% generally accepted in engineering.

[0138] Figure 7 The convex hull construction result of the cone tower model. Figure 8 The visualization of the point cloud data points calculated for each layer slice of the cone tower model in the curvature compensation algorithm of the application. Figure 9 The optimized direction slice diagram of the cone tower model. It can be seen that the slice direction is the symmetric normal direction of the two cones, which conforms to the maximum distribution direction of the point cloud.

[0139] The above data do not have a particularly obvious concave hull region. To verify the effect of curvature compensation, this group of data is a cube point cloud, the upper part of which is excavated by a half sphere with the same diameter as the side length of the cube, forming a point cloud data with a concave region. The shape is as shown in Figure 10 .

[0140] As shown in Figure 10 , the height of the cube is 10 m, and the concave pattern is a half sphere with a radius of 5 m. Through the volume formula of the sphere and the cube, it can be obtained that the volume of the concave cube is about 738.33 m³. The volume calculated by the standard convex hull method is 999.9169 m³; the volume calculated by the fixed axis slice convex hull method is 999.9500 m³; the volume calculated by the optimized direction slice convex hull method is 844.3069 m³; and the volume calculated by the optimized direction convex hull algorithm based on curvature compensation proposed in the application is 750.2656 m³. It can be obtained that the error of the standard convex hull method and the fixed axis slice convex hull method is about 26%, the error of the optimized direction slice convex hull method is 15.569%, and the error of the optimized direction convex hull algorithm based on curvature compensation is 1.617%. The calculation result of the optimized direction convex hull algorithm based on curvature compensation is within the standard of ± 5% generally accepted in engineering.

[0141] Figure 11 The convex hull construction result of the concave cube under the standard convex hull method. Figure 12For the visualization of the point cloud data calculated in the curvature compensation algorithm of the present application for each slice, it can be seen that only a small part of the entire point cloud data is occupied, which greatly reduces the calculation amount. And in Figure 12 the slice direction is a body diagonal of the cube. Since the sphere and the cube are symmetrical, this slice direction conforms to the maximum distribution direction of the point cloud.

[0142] This set of data is the bucket point cloud information collected during the process of excavating soil material by the excavator. The original point cloud data only has the point cloud above the bucket. After the registration and fusion of the point cloud, this set of point cloud is obtained, and the point cloud data is as shown in Figure 13 .

[0143] The actual volume of the excavated soil material is about 5.31m³ measured on site. The volume calculated by the standard convex hull method is 7.3219m³; the volume calculated by the fixed axis slice convex hull method is 6.9995m³; the volume calculated by the optimized direction slice convex hull method is 5.8758m³; and the volume calculated by the optimized direction convex hull algorithm based on curvature compensation proposed by the present application is 4.9944m³. Through calculation, it can be obtained that: the error of the standard convex hull method is about 37.89%, the error of the fixed axis slice convex hull method is about 31.817%, the error of the optimized direction slice convex hull method is 10.655%, and the error of the optimized direction convex hull algorithm based on curvature compensation is 5.944%. Since errors are inevitable in the process of actually measuring the excavated soil material, it is considered that the error of the optimized direction convex hull algorithm based on curvature compensation is within an acceptable range.

[0144] Table 1 is a summary of experimental data.

[0145] Table 1 is a summary of experimental data.

[0146] ;

[0147] The above examples are only used to illustrate the technical solutions of the present application, and not to limit it. Although the present application has been described in detail with reference to the foregoing examples, those skilled in the art should understand that they can still modify the technical solutions recorded in the foregoing examples, or make equivalent substitutions for part or all of the technical features, and these modifications or substitutions do not make the essence of the corresponding technical solutions deviate from the scope of the technical solutions of the embodiments of the present application.

Claims

1. A curvature compensation based optimized directional slice convex hull volume calculation method, characterized in that, Select four non-coplanar points in the point cloud data of the excavated material collected by the laser radar of the excavator, to form an initial tetrahedral convex hull, for points in the point cloud data other than the four points, determine whether a point is outside the initial tetrahedral convex hull, if a point is outside the initial tetrahedral convex hull, consider the point as a temporary vertex, find the visible faces of the temporary vertex, remove the shared edges between the visible faces, retain the non-shared edges, connect the temporary vertex with the two end points of the shared edges, and generate new triangular faces; if a point is located inside the initial tetrahedral convex hull, skip the point, and when all points are located inside the initial tetrahedral convex hull, the three-dimensional convex hull is constructed, and the volume of the three-dimensional convex hull is calculated; Computing a three-dimensional convex hull volume includes selecting a center of mass of the point cloud data : ； In the formula, is the number of point cloud data in the grid, represents the th grid, represents the th point cloud data in the grid; Computing a three-dimensional convex hull volume includes constructing a covariance matrix : ； ； wherein is equivalent to , is a central coordinate, , , is a horizontal, vertical, and vertical coordinate value of the central coordinate; Computing the three-dimensional convex hull volume comprises solving , is a matrix of eigenvalues, is a matrix of eigenvectors; solving the equation to obtain eigenvalues and the corresponding orthogonal eigenvectors of the eigenvalues , is the principal direction of the point cloud distribution; An orthogonal projection basis is constructed to generate a two-dimensional projection plane perpendicular to the main direction of the point cloud distribution, and an orthogonal basis is constructed by using Gram-Schmidt orthogonalization , the unitized main direction is calculated , and is normalized to a unit length: ； orthogonalize the second direction, to project onto a direction perpendicular to the first direction: ； The cross product gives the third direction, ensuring the base Constructing the right-handed coordinate system: ； After computing the orthonormal basis, the point cloud is projected to a plane; Project the coordinates of all points along the main direction: ； In the formula, is the projection value of the point along the main direction; Determine the range of the slice: ； ； In the formula, is the minimum value of the slice, is the maximum value of the slice; Adaptive computation of slice number : ； ； ； ； In the formula, Used to limit the number of slices to between 200 and 1000. , These are the maximum and minimum values ​​on the horizontal axis of the point cloud data. , These are the maximum and minimum values ​​on the vertical axis of the point cloud data. , These are the maximum and minimum values ​​on the vertical axis of the point cloud data. , , It is the length in three directions. It is the average of the lengths in three directions. It takes the minimum value of the length in the three directions; For the first slices, extract intra-layer point sets : ； ； wherein is an intermediate parameter that projects the set of points in the layer to the plane : ； The integral interval is divided into layers, each layer thickness , calculate the three-dimensional convex hull volume : ； ； In the formula, is the vertical axis coordinate value, is the two-dimensional convex hull area of the section of the first layer, is the two-dimensional convex hull area of the section of the first layer, Optimizing the approximate calculation formula for , including calculating the covariance matrix of K-nearest neighbors for each slice of :​ ； wherein is a K-neighbors set of denotes a mean value of denotes point cloud data; To perform eigen decomposition to obtain eigenvalue , calculate the average curvature of the local surface : ； In the formula, Positive values correspond to convex surfaces, negative values to concave surfaces. Optimizing the approximate calculation formula of the includes calculating the second moment of the section, assuming that the vertices of the two-dimensional convex hull of the layer are , and calculating the bending amount of the section shape : ； wherein is the maximum value of the curvature and the compensation of the cross-sectional area adjusted by the second moment of area of the layer : ； In the formula, is a parameter that controls the strength of compensation; Layer-corrected second-moment cross-sectional area is: ； wherein is the second moment area before modification; In the concave region ; right The approximate calculation formula is optimized, including the corrected 3D convex hull volume. for: 。

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