CN106943209B - Bionical femur head model and design method - Google Patents

Bionical femur head model and design method Download PDF

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CN106943209B
CN106943209B CN201710238356.2A CN201710238356A CN106943209B CN 106943209 B CN106943209 B CN 106943209B CN 201710238356 A CN201710238356 A CN 201710238356A CN 106943209 B CN106943209 B CN 106943209B
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阎军
郭金陇
王靖元
范少迪
尹权�
尹权一
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Dalian University of Technology
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    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61F—FILTERS IMPLANTABLE INTO BLOOD VESSELS; PROSTHESES; DEVICES PROVIDING PATENCY TO, OR PREVENTING COLLAPSING OF, TUBULAR STRUCTURES OF THE BODY, e.g. STENTS; ORTHOPAEDIC, NURSING OR CONTRACEPTIVE DEVICES; FOMENTATION; TREATMENT OR PROTECTION OF EYES OR EARS; BANDAGES, DRESSINGS OR ABSORBENT PADS; FIRST-AID KITS
    • A61F2/00—Filters implantable into blood vessels; Prostheses, i.e. artificial substitutes or replacements for parts of the body; Appliances for connecting them with the body; Devices providing patency to, or preventing collapsing of, tubular structures of the body, e.g. stents
    • A61F2/02—Prostheses implantable into the body
    • A61F2/28—Bones
    • A—HUMAN NECESSITIES
    • A61—MEDICAL OR VETERINARY SCIENCE; HYGIENE
    • A61F—FILTERS IMPLANTABLE INTO BLOOD VESSELS; PROSTHESES; DEVICES PROVIDING PATENCY TO, OR PREVENTING COLLAPSING OF, TUBULAR STRUCTURES OF THE BODY, e.g. STENTS; ORTHOPAEDIC, NURSING OR CONTRACEPTIVE DEVICES; FOMENTATION; TREATMENT OR PROTECTION OF EYES OR EARS; BANDAGES, DRESSINGS OR ABSORBENT PADS; FIRST-AID KITS
    • A61F2/00—Filters implantable into blood vessels; Prostheses, i.e. artificial substitutes or replacements for parts of the body; Appliances for connecting them with the body; Devices providing patency to, or preventing collapsing of, tubular structures of the body, e.g. stents
    • A61F2/02—Prostheses implantable into the body
    • A61F2/28—Bones
    • A61F2002/2825—Femur
    • A61F2002/2828—Femoral head

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  • Orthopedic Medicine & Surgery (AREA)
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Abstract

本发明属于先进制造技术领域，提供了一种仿生股骨头模型及设计方法。仿生股骨头模型的特点：半球型，因为人类的自然股骨头近似半球形；点阵夹层结构，模型全部是由杆件连接组成；半球模型从球心由内向外层层扩展，理论上可以形成n层夹层结构；可以通过控制不同层间的杆件的横截面积来改变整个模型的梯度分布；同样可以通过控制不同层间杆件的杆长来改变整个模型的梯度分布；另，梯度分布种类大致可以分为：外疏内密梯度，均匀梯度，外密内疏梯度。本发明的有益效果：借鉴人类真实股骨头的特性，设计出未来可以替代坏死或损坏的骨头的模型，同时利用3D打印技术及时进行骨移植替代物制造，从而改善人们的生活。

The invention belongs to the technical field of advanced manufacturing, and provides a bionic femoral head model and a design method. Features of the bionic femoral head model: hemispherical shape, because the natural human femoral head is approximately hemispherical; lattice sandwich structure, the model is all connected by rods; the hemispherical model extends from the center of the sphere from the inside to the outside layer by layer, which can theoretically be formed n-layer sandwich structure; the gradient distribution of the entire model can be changed by controlling the cross-sectional area of the rods between different layers; the gradient distribution of the entire model can also be changed by controlling the rod lengths of the rods between different layers; in addition, the gradient distribution The types can be roughly divided into: outer sparse inner dense gradient, uniform gradient, outer dense inner sparse gradient. Beneficial effects of the invention: Drawing on the characteristics of real human femoral heads, a model that can replace necrotic or damaged bones in the future is designed, and 3D printing technology is used to manufacture bone graft substitutes in time, thereby improving people's lives.

Description

Bionical femur head model and design method
Technical field
The invention belongs to advanced manufacturing technology field, it is related to a kind of bionical femur head model based on 3D printing technique and sets Meter method.
Background technique
3D printing (3DP) i.e. one kind of rapid shaping technique, it is one kind based on digital model file, with powder The adhesive materials such as shape metal or plastics construct the technology of object by layer-by-layer printing.At present at home: Beijing is big It learns research team and successfully implants 3D printing vertebra for the first time in the whole world for 12 years old boy.BJ University of Aeronautics & Astronautics is in gold Belong to direct manufacture view and achieve progress, breaches the laser forming technique of the difficult-to-machine materials such as titanium alloy, unimach.West The big solidification technology National Key Laboratory of northern polytechnical university has been set up series laser cladding forming and repaired item, can expire The heavy parts and hardly possible of sufficient big machinery equipment are dismantled the in-situ immobilization of part and are remanufactured.In foreign countries: the developed countries such as Great Britain and America For structure lightened, technical standard, common technology, business model development etc. expands research, and is applied to military affairs, space flight, Medicine and other fields.
The needs of bone collection are continuously increased, and 400,000 bone collection operation are performed for more than in Europe every year now, in beauty State then requires more than 600,000 times.In the U.S., people in 2010 have costed up to 1,300,000,000 dollars on bone graft substitute, and And it is predicted to increase with annual 7.4% ratio.China comes into aging society, and in China, this technology, which but compares, falls Afterwards.Bone graft substitute how to be designed and produced into the task of top priority.Natural bone is observed, bone is cut in can see The microstructure in face is the discovery that the similar lattice structure with gradient.Observe the plant structure of other natures, it may have this The variation of kind gradient.Manufacturing difficulty once limited the design of microstructure, but with the development of 3D printing technique, it is already possible to Overcome this difficulty.The present invention is bionical femur head model, it is intended to which future can be as novel bone graft substitute.
Summary of the invention
Of the invention be according to the structure organization characteristic of mankind's nature femoral head, design can substitute necrosis or damage The model of femoral head, to achieve the purpose that as bone graft substitute.
Technical solution of the present invention:
Bionical femur head model be by rod piece and node population at hemisphere dot matrix supporting structure, be divided into interior hemisphere dot matrix branch Frame structure and outer hemisphere dot matrix supporting structure two parts；
Outer hemisphere dot matrix supporting structure is divided into top layer, four part of the second layer, third layer and bottom from top to bottom, from upper It is successively increased to the spacing between lower adjacent two layers；Top layer is divided into vertex and lower surface, i.e. vertex and lower circle, and lower circle uses node 16 Equal part；The second layer, third layer and bottom, every layer is divided into upper and lower surfaces, i.e., upper round and lower circle；Upper circle and lower circle are all made of 16 equal part of node, each interlayer node location correspond to each other；
Interior hemisphere dot matrix supporting structure is divided into top layer and bottom two parts from top to bottom, from top to bottom between adjacent two layers Spacing successively increase；Top layer is divided into vertex and lower surface, i.e. vertex and lower circle, and lower circle uses 8 equal part of node；Bottom is divided into Surface and lower surface, i.e., upper round and lower circle；Upper circle and lower circle are all made of 8 equal part of node, and each interlayer node location corresponds to each other；
In outer hemisphere dot matrix supporting structure and interior hemisphere dot matrix supporting structure, the upper circle of the second layer, third layer and bottom Adjacent two nodes, the adjacent two nodes of lower circle, upper circle two node corresponding with lower circle, the node of upper circle and lower circle are corresponding adjacent Two nodes between be connected；Each node of lower circle of top layer is connected with vertex；It is connected between each node using connecting rod；
Node under bottom in interior hemisphere dot matrix supporting structure on circle is respectively and under outer hemisphere dot matrix supporting structure bottom Corresponding node, corresponding adjacent two nodes connection on circle；Node under bottom in interior hemisphere dot matrix supporting structure on circle It is connected respectively with the corresponding node on outer hemisphere dot matrix supporting structure bottom on circle, corresponding two adjacent nodes；
Under top layer in interior hemisphere dot matrix supporting structure circle on node respectively with the outer hemisphere dot matrix supporting structure second layer Corresponding node, corresponding two adjacent nodes on lower circle are attached；Under top layer in interior hemisphere dot matrix supporting structure on circle Node respectively on the outer hemisphere dot matrix supporting structure second layer on circle corresponding node, corresponding two adjacent nodes are connected It connects；
Under top layer in interior hemisphere dot matrix supporting structure circle on node respectively with outer hemisphere dot matrix supporting structure third layer Corresponding node, corresponding two adjacent nodes on lower circle are attached；
The center of circle under node and bottom under bottom in interior hemisphere dot matrix supporting structure on circle on circle is attached；
The vertex of interior hemisphere dot matrix supporting structure and the vertex of outer hemisphere dot matrix supporting structure are attached.
The characteristics of bionical femur head model:
1. dome-type.Because of the natural femoral head near hemispherical of the mankind；
2. dot matrix sandwich, model is entirely to be made of rod piece connection；
3. half spherical model extends layer by layer from inside to outside from the centre of sphere, n-layer sandwich can be theoretically formed；
4. the gradient distribution of entire model can be changed by the cross-sectional area of the rod piece of the different interlayers of control；
5. again may be by controlling the long gradient distribution to change entire model of bar of different interlayer rod pieces；In addition, gradient Topological classes can substantially be divided into: outer to dredge interior close gradient, uniform gradient, outer close interior thin gradient.
Beneficial effects of the present invention: use for reference the true femoral head of the mankind characteristic, design future can substitute necrosis or damage The model of bad bone, while bone graft substitute manufacture is carried out in time using 3D printing technique, so as to improve people's lives.
Detailed description of the invention
Fig. 1 is bionical femoral head dot matrix stent model main view.
Fig. 2 is bionical femoral head dot matrix stent model bottom view.
Fig. 3 is bionical femoral head dot matrix stent model top view.
Specific embodiment
Below in conjunction with attached drawing and technical solution, a specific embodiment of the invention is further illustrated.
A kind of design method of bionical femur head model, steps are as follows:
(1) spherical coordinate system is established, that is, the coordinate put isR represents radius, and θ is represented and put in x-y plane with origin Line and x-axis angle,The point in spherical coordinate system is represented with the line of origin and the angle of x-y plane；
(2) connecting rod between internal layer bar and internal layer and outer layer is established
(2.1) the first plane S1 plane for selecting origin first, is initially set to x-y plane；Radius is established in S1 plane For three points adjacent on the circle of a, each point is 45 degree at central angle with center of circle institute, if first point is p1；It resettles with half Diameter is three points adjacent on the circle of 2a, and each point is 22.5 degree at central angle with center of circle institute；
(2.2) change againIt is 22.5 degree, establishes three points adjacent on the circle that radius is 2a, Mei Gedian on that plane With the center of circle institute at central angle be 22.5 degree；Change againIt is 45 degree, establishes adjacent on the circle that radius is a three on that plane Point, each point and center of circle institute are 45 degree at central angle；It resettles flat about S1 with six points not in S1 plane in this step Six corresponding points of face mirror symmetry, by p1, remaining all the points is attached with step (2)；If the θ value of p1 is 180 degree, The θ value of remaining point is also that the point of 180 degree is not connect with p1；
The θ value of p1 is increased by 45 degree, then this point is p2, repeatedly step (2) establishment process, and so on, altogether repeatedly 2 It is secondary；
(3) the remaining rod piece of circle under hemisphere bottom is connected
(3.1) in S1 plane, a point is established on the circle that radius is a, is set as p4；It is also built on the circle that radius is 2a Found a point；
(3.2) rotary work plane makesIt is 22.5 degree, establishes a point on the circle that radius is 2a；Rotary work is flat Face makesIt is 45 degree, establishes a point on the circle that radius is a；It resettles and two points not in S1 plane in this step About symmetrical two corresponding points of S1 plane mirror image；By p4 and step (3), remaining point is attached；Above-mentioned established each point θ value is all 0；
(3.3) equally, in S1 plane, a point is established on the circle that radius is a, is set as p5；In the circle that radius is 2a On also establish a point；
(3.4) rotary work plane makesIt is 22.5 degree, establishes a point on the circle that radius is 2a；Rotary work is flat Face makesIt is 45 degree, establishes a point on the circle that radius is a；It resettles and two points not in S1 plane in this step About symmetrical two points of S1 plane mirror image；P5 and remaining point are attached；The θ value of above-mentioned established each point is all 180；
(4) connection is in the rod piece in hemisphere sideline
(4.1) in S1 plane, the point that a θ value is 0 is established on the circle that radius is a, is set as p6；It is 2a's in radius The point that a θ value is 22.5 is established on circle；
(4.2) then rotary work plane, makeIt is 22.5 degree, it is 22.5 that a θ value is established on the circle that radius is 2a Point；Equally, it then does about the symmetrical point of S1 plane mirror image；By p6 and step (4), remaining point is attached；
(5) connection is in the rod piece in hemisphere sideline
(5.1) in S1 plane, the point that a θ value is 180 is established on the circle that radius is a, is set as p7；It is 2a in radius Circle on establish a θ value be 167.5 point；
(5.2) then rotary work plane, makeIt is 22.5 degree, it is 167.5 that a θ value is established on the circle that radius is 2a Point；Equally, it then does about the symmetrical point of S1 plane mirror image；By p7 and step (5), remaining point is attached；
(6) S1 plane being rotated 45 degree around z-axis, this plane is S2 plane, step (2)-(5) are repeated, and so on, altogether It is repeated 3 times；
(7) rod piece of outermost layer hemisphere is connected
The first plane S5 plane for selecting origin first, is initially set to x-y plane；It is on 2a that radius is established in S5 plane Three adjacent points, each point and center of circle institute are 22.5 degree at central angle, if first point is n1；Change againMakeFor 22.5 degree, it is three points adjacent on 2a that radius is established in S5 plane, and same each point is 22.5 at central angle with center of circle institute Degree；Equally, identical and symmetrical three points of S1 plane mirror image are resettled；By n1 and step (7), remaining point is attached；If The θ value of p6 is 180 degree, then θ value is also that the point of 180 degree is not connect with n1 in remaining point；
Then the θ value of n1 being increased by 22.5 degree, then this point is n2, step (2) establishment process is repeated, and so on, weight It is 6 times multiple；
(8) the remaining rod piece in hemisphere bottom surface is connected
(8.1) in S5 plane, a point is established on the circle that radius is 2a, is set as n8；
(8.2) then rotary work plane, makeIt is 22.5 degree, establishes a point on the circle that radius is 2a；Equally, then It does about the symmetrical point of S5 plane mirror image；By n8 and step (8), remaining point is attached；The θ of above-mentioned established each point Value is all 0 degree；
(8.3) equally, working face is re-rotated into S5 plane, establishes a point on the circle that radius is 2a, is set as n9；
(8.4) then rotary work plane, makeIt is 22.5 degree, establishes a point on the circle that radius is 2a；Equally, then It does about the symmetrical point of S5 plane mirror image；By n9 and step (8), remaining point is attached；The θ of above-mentioned established each point Value is all 180 degree；
(9) S5 plane is rotated 22.5 degree around z-axis, sets S6 plane for this plane, (7)-(8) step is repeated, with this Analogize, is repeated 7 times altogether；
(10) rod piece of innermost layer is connected
Working face is initialized as x-y plane, establishing radius is a, each two points that θ value is 22.5 degree and 45 degree；So Afterwards around z-axis rotary work plane, makeIt is 45, repeats the foundation of above-mentioned point, be repeated 8 times altogether, by all the points and origin of foundation It is attached；Finally by x-y plane, θ value is that 90 degree of point is attached with origin.

Claims (2)

1.一种仿生股骨头模型，其特征在于，所述的仿生股骨头模型为由杆件和节点组装成的半球点阵支架结构，分为内半球点阵支架结构和外半球点阵支架结构两部分；1. a bionic femoral head model, is characterized in that, described bionic femoral head model is the hemisphere lattice support structure assembled by rod and node, is divided into inner hemisphere lattice support structure and outer hemisphere lattice support structure two parts; 外半球点阵支架结构从上到下依次分为顶层、第二层、第三层和底层四部分，从上到下相邻两层间的间距依次增加；顶层分为顶点和下表面，即顶点和下圆，下圆采用节点16等分；第二层、第三层和底层，每层分为上表面和下表面，即上圆和下圆；上圆和下圆均采用节点16等分，各层间节点位置相互对应；The outer hemisphere lattice scaffold structure is divided into four parts: top layer, second layer, third layer and bottom layer from top to bottom, and the distance between the two adjacent layers increases from top to bottom; The vertex and the lower circle, the lower circle is equally divided by node 16; the second layer, the third layer and the bottom layer, each layer is divided into the upper surface and the lower surface, that is, the upper circle and the lower circle; the upper circle and the lower circle are divided by node 16, etc. The positions of nodes in each layer correspond to each other; 内半球点阵支架结构从上到下依次分为顶层和底层两部分，从上到下相邻两层间的间距依次增加；顶层分为顶点和下表面，即顶点和下圆，下圆采用节点8等分；底层分为上表面和下表面，即上圆和下圆；上圆和下圆均采用节点8等分，各层间节点位置相互对应；The inner hemisphere lattice scaffold structure is divided into two parts from top to bottom, the top layer and the bottom layer, and the distance between the two adjacent layers increases from top to bottom. The nodes are divided into 8 equal parts; the bottom layer is divided into the upper surface and the lower surface, namely the upper circle and the lower circle; the upper circle and the lower circle are divided into 8 equal parts by nodes, and the positions of the nodes between the layers correspond to each other; 外半球点阵支架结构和内半球点阵支架结构中，第二层、第三层和底层的上圆的相邻两节点、下圆的相邻两节点、上圆与下圆相对应的两节点、上圆的节点与下圆对应相邻的两节点间相连；顶层的下圆各节点均与顶点相连；各节点间采用连杆连接；In the outer hemisphere lattice support structure and the inner hemisphere lattice support structure, the second layer, the third layer and the bottom layer have two adjacent nodes in the upper circle, two adjacent nodes in the lower circle, and two nodes corresponding to the upper circle and the lower circle. The nodes and the nodes of the upper circle are connected with the two adjacent nodes of the lower circle; each node of the lower circle on the top layer is connected with the vertex; each node is connected by a connecting rod; 内半球点阵支架结构中的底层下圆上的节点分别与外半球点阵支架结构底层下圆上的对应节点、对应相邻的两个节点连接；内半球点阵支架结构中的底层下圆上的节点分别与外半球点阵支架结构底层上圆上的对应节点、对应相邻的两个节点连接；The nodes on the bottom circle of the inner hemisphere lattice support structure are respectively connected with the corresponding nodes on the bottom circle of the outer hemisphere lattice support structure and the corresponding two adjacent nodes; the bottom circle in the inner hemisphere lattice support structure The nodes on the outer hemisphere lattice support structure are respectively connected with the corresponding nodes on the bottom circle of the outer hemisphere lattice support structure, and the corresponding two adjacent nodes; 内半球点阵支架结构中的顶层下圆上的节点分别与外半球点阵支架结构第二层下圆上的对应节点、对应相邻的两个节点进行连接；内半球点阵支架结构中的顶层下圆上的节点分别与外半球点阵支架结构第二层上圆上的对应节点、对应相邻的两个节点进行连接；The nodes on the lower circle of the top layer in the inner hemisphere lattice support structure are respectively connected with the corresponding nodes on the lower circle of the second layer of the outer hemisphere lattice support structure and the corresponding two adjacent nodes; The nodes on the lower circle of the top layer are respectively connected with the corresponding nodes on the upper circle of the second layer of the outer hemisphere lattice support structure and the corresponding two adjacent nodes; 内半球点阵支架结构中的顶层下圆上的节点分别与外半球点阵支架结构第三层下圆上的对应节点、对应相邻的两个节点进行连接；The nodes on the lower circle of the top layer in the inner hemisphere lattice support structure are respectively connected with the corresponding nodes on the lower circle of the third layer of the outer hemisphere lattice support structure and the corresponding two adjacent nodes; 内半球点阵支架结构中的底层下圆上的节点与底层下圆上的圆心进行连接；The nodes on the bottom circle in the inner hemisphere lattice support structure are connected with the center of the circle on the bottom circle; 内半球点阵支架结构的顶点与外半球点阵支架结构的顶点进行连接。The apex of the inner hemisphere lattice support structure is connected with the apex of the outer hemisphere lattice support structure. 2.一种仿生股骨头模型的设计方法，其特征在于，步骤如下：2. a design method of bionic femoral head model, is characterized in that, step is as follows: (1)建立球坐标系，即点的坐标为r代表半径，θ代表x-y平面上点跟原点的连线与x轴的夹角，代表球坐标系中的点跟原点的连线与x-y平面的夹角；(1) Establish a spherical coordinate system, that is, the coordinates of the point are r represents the radius, θ represents the angle between the line connecting the point and the origin on the xy plane and the x-axis, Represents the angle between the line connecting the point in the spherical coordinate system and the origin and the xy plane; (2)建立内层杆和内层与外层之间的连杆(2) Establish the inner layer rod and the connecting rod between the inner layer and the outer layer (2.1)首先选定过原点的初平面S1平面，最初设为x-y平面；在S1平面上建立半径为a的圆上相邻的三个点，每个点与圆心所成圆心角为45度，设第一个点为p1；再建立以半径为2a的圆上相邻的三个点，每个点与圆心所成圆心角为22.5度；(2.1) First select the initial plane S1 plane that passes through the origin, and initially set it as the x-y plane; establish three adjacent points on the circle with radius a on the S1 plane, and the central angle formed by each point and the center of the circle is 45 degrees , set the first point as p1; then establish three adjacent points on a circle with a radius of 2a, and the central angle formed by each point and the center of the circle is 22.5 degrees; (2.2)再改变为22.5度，在此平面上建立半径为2a的圆上相邻的三个点，每个点与圆心所成圆心角为22.5度；再改变为45度，在此平面上建立半径为a的圆上相邻的三个点，每个点与圆心所成圆心角为45度；再建立与不在S1平面上本步骤中的六个点关于S1平面镜像对称的六个对应点，将p1与步骤(2)其余所有点进行连接；如果p1的θ值为180度，则其余点的θ值也为180度的点不与p1连接；(2.2) Change again is 22.5 degrees, establish three adjacent points on a circle with a radius of 2a on this plane, and the central angle formed by each point and the center of the circle is 22.5 degrees; then change is 45 degrees, establish three adjacent points on the circle with radius a on this plane, and the central angle formed by each point and the center of the circle is 45 degrees; For the six corresponding points of mirror symmetry on the S1 plane, connect p1 to all the remaining points in step (2); if the θ value of p1 is 180 degrees, the points whose θ values are also 180 degrees are not connected to p1; 将p1的θ值增加45度，则此点即为p2，重复步骤(2)建立过程，以此类推，共重复2次；Increase the θ value of p1 by 45 degrees, then this point is p2, repeat the establishment process of step (2), and so on, repeat 2 times in total; (3)连接半球底层下圆剩余的杆件(3) Connect the remaining rods in the lower circle of the bottom layer of the hemisphere (3.1)在S1平面上，在半径为a的圆上建立一个点，设为p4；在半径为2a的圆上也建立一个点；(3.1) On the S1 plane, establish a point on the circle with radius a, set it as p4; also establish a point on the circle with radius 2a; (3.2)旋转工作平面，使为22.5度，在半径为2a的圆上建立一个点；旋转工作平面，使为45度，在半径为a的圆上建立一个点；再建立与不在S1平面上本步骤中的两个点关于S1平面镜像对称的两个对应点；将p4与步骤(3)其余点进行连接；上述所建立的各点的θ值都为0；(3.2) Rotate the work plane so that is 22.5 degrees, establish a point on a circle of radius 2a; rotate the work plane so that For 45 degrees, establish a point on the circle with radius a; then establish two corresponding points that are mirror-symmetrical about the S1 plane with the two points in this step that are not on the S1 plane; connection; the θ value of each point established above is 0; (3.3)同样，在S1平面上，在半径为a的圆上建立一个点，设为p5；在半径为2a的圆上也建立一个点；(3.3) Similarly, on the S1 plane, establish a point on the circle with radius a, set it as p5; also establish a point on the circle with radius 2a; (3.4)旋转工作平面，使为22.5度，在半径为2a的圆上建立一个点；旋转工作平面，使为45度，在半径为a的圆上建立一个点；再建立与不在S1平面上本步骤中的两个点关于S1平面镜像对称的两个点；将p5与其余点进行连接；上述所建立的各点的θ值都为180；(3.4) Rotate the work plane so that is 22.5 degrees, establish a point on a circle of radius 2a; rotate the work plane so that is 45 degrees, establish a point on the circle with radius a; then establish two points that are mirror-symmetrical about the S1 plane with the two points in this step that are not on the S1 plane; connect p5 with the other points; the above established The θ value of each point is 180; (4)连接处于半球边线的杆件(4) Connect the rods on the edge of the hemisphere (4.1)在S1平面上，在半径为a的圆上建立一个θ值为0的点，设为p6；在半径为2a的圆上建立一个θ值为22.5的点；(4.1) On the S1 plane, establish a point with a θ value of 0 on a circle with a radius of a, and set it as p6; establish a point with a θ value of 22.5 on a circle with a radius of 2a; (4.2)然后旋转工作平面，使为22.5度，在半径为2a的圆上建立一个θ值为22.5的点；同样，再做关于S1平面镜像对称的一个点；将p6与步骤(4)其余点进行连接；(4.2) Then rotate the work plane so that is 22.5 degrees, establish a point with a θ value of 22.5 on a circle with a radius of 2a; similarly, make a mirror-symmetric point about the S1 plane; connect p6 with the rest of the points in step (4); (5)连接处于半球边线的杆件(5) Connect the rods on the edge of the hemisphere (5.1)在S1平面上，在半径为a的圆上建立一个θ值为180的点，设为p7；在半径为2a的圆上建立一个θ值为167.5的点；(5.1) On the S1 plane, establish a point with a θ value of 180 on a circle with a radius of a, and set it as p7; establish a point with a θ value of 167.5 on a circle with a radius of 2a; (5.2)然后旋转工作平面，使为22.5度，在半径为2a的圆上建立一个θ值为167.5的点；同样，再做关于S1平面镜像对称的一个点；将p7与步骤(5)其余点进行连接；(5.2) Then rotate the work plane so that is 22.5 degrees, establish a point with a θ value of 167.5 on a circle with a radius of 2a; similarly, make a mirror-symmetric point about the S1 plane; connect p7 with the rest of the points in step (5); (6)将S1平面绕z轴旋转45度，此平面即为S2平面，重复步骤(2)-(5)，以此类推，共重复3次；(6) Rotate the S1 plane by 45 degrees around the z-axis, this plane is the S2 plane, repeat steps (2)-(5), and so on, repeat 3 times in total; (7)连接最外层半球的杆件(7) The rod connecting the outermost hemisphere 首先选定过原点的初平面S5平面，最初设为x-y平面；在S5平面上建立半径为2a上相邻的三个点，每个点与圆心所成圆心角为22.5度，设第一个点为n1；再改变使为22.5度，在S5平面上建立半径为2a上相邻的三个点，同样每个点与圆心所成圆心角为22.5度；同样，再建立相同的与S1平面镜像对称的三个点；将n1与步骤(7)其余点进行连接；如果p6的θ值为180度，则其余点中θ值也为180度的点不与n1连接；First select the initial plane S5 plane that has passed the origin, and initially set it as the xy plane; establish three adjacent points on the S5 plane with a radius of 2a, and the central angle formed by each point and the center of the circle is 22.5 degrees. point is n1; then change Make is 22.5 degrees, establish three adjacent points on the S5 plane with a radius of 2a, and the central angle formed by each point and the center of the circle is 22.5 degrees; also, establish the same three points that are mirror-symmetrical with the S1 plane; Connect n1 to the rest of the points in step (7); if the θ value of p6 is 180 degrees, the points with the θ value of 180 degrees in the remaining points are not connected to n1; 然后将n1的θ值增加22.5度，则此点即为n2，重复步骤(2)建立过程，以此类推，重复6次；Then increase the θ value of n1 by 22.5 degrees, then this point is n2, repeat the establishment process of step (2), and so on, repeat 6 times; (8)连接半球底面剩余的杆件(8) Connect the remaining rods on the bottom surface of the hemisphere (8.1)在S5平面上，在半径为2a的圆上建立一个点，设为n8；(8.1) On the S5 plane, establish a point on a circle with a radius of 2a and set it as n8; (8.2)然后旋转工作平面，使为22.5度，在半径为2a的圆上建立一个点；同样，再做关于S5平面镜像对称的一个点；将n8与步骤(8)其余点进行连接；上述所建立的各点的θ值都为0度；(8.2) Then rotate the work plane so that is 22.5 degrees, establish a point on a circle with a radius of 2a; similarly, make a mirror-symmetric point about the S5 plane; connect n8 to the rest of the points in step (8); the θ values of the above-established points are all is 0 degrees; (8.3)同样，将工作平面重新旋转到S5平面，在半径为2a的圆上建立一个点，设为n9；(8.3) Similarly, re-rotate the working plane to the S5 plane, and establish a point on the circle with a radius of 2a, set it as n9; (8.4)然后旋转工作平面，使为22.5度，在半径为2a的圆上建立一个点；同样，再做关于S5平面镜像对称的一个点；将n9与步骤(8)其余点进行连接；上述所建立的各点的θ值都为180度；(8.4) Then rotate the work plane so that is 22.5 degrees, establish a point on a circle with a radius of 2a; similarly, make a mirror-symmetric point about the S5 plane; connect n9 to the rest of the points in step (8); the θ values of the above-established points are all is 180 degrees; (9)将S5平面绕z轴旋转22.5度，将此平面设置为S6平面，重复(7)-(8)步骤，以此类推，共重复7次；(9) Rotate the S5 plane by 22.5 degrees around the z-axis, set this plane as the S6 plane, repeat steps (7)-(8), and so on, repeat 7 times in total; (10)连接最内层的杆件(10) Connect the innermost member 将工作平面初始化为x-y平面，建立半径为a，θ值为22.5度和45度的各两个点；然后绕z轴旋转工作平面，使为45，重复上述点的建立，共重复8次，将建立的所有点与原点进行连接；最后将x-y平面上，θ值为90度的点与原点进行连接。Initialize the work plane as the xy plane, establish two points with radius a and θ values of 22.5 degrees and 45 degrees; then rotate the work plane around the z axis to make For 45, repeat the establishment of the above-mentioned points for a total of 8 times, and connect all the established points to the origin; finally, connect the points with a θ value of 90 degrees on the xy plane to the origin.
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