Modeling Method for Fitting Generalized Cylinder Surfaces Based on the Derived Structure of Water-Bomb Origami

Abstract

The present invention proposes a modeling method for fitting a generalized cylindrical surface based on the derivative structure of water bomb origami. First, the optimized origami grid model is mapped from a three-dimensional state to a two-dimensional plane, and linear regression processing is performed using the vertex information to achieve the purpose of model precision. A three-dimensional grid model is constructed from the processed data to maximize the satisfaction of the unfoldable constraint and the flat-foldable constraint of the origami model. The present invention has three types of derivative structures based on water bomb origami, which can simulate the target surface more diversely. Secondly, by controlling a single variable, the rigid origami movement process of the origami is simulated. On the basis of studying the modeling method of the derivative structure of water bomb origami, considering the degree of freedom problem of each vertex, the number of connecting rods around the vertex is reduced, thereby reducing the degree of freedom of the vertex. This adds favorable conditions for the research on the fitting of the target surface.

Description

Technical Field

[0001] The present invention relates to the field of computer graphics for fitting surfaces with generalized cylinder characteristics by computer origami, and specifically relates to a derivative structure obtained by adding rectangular blocks based on Waterbomb to form a basic origami unit, and simulating its rigid folding process. Background Art

[0002] When a rigid object is subjected to an external force, its shape does not change, specifically, it does not undergo stretching, deformation, etc. From small brackets to the deployment of large solar sails to the braking and propulsion of robots, the application of rigid deformation in people's lives is becoming more and more extensive, and the origami structure can provide a template for deformation. Origami is an ancient origami art because it has unique properties, such as deployability, flat foldability, etc. The application of the origami structure in terms of deformation is mainly to simulate the target surface by tiling the origami structure according to certain structural characteristics. For example, in the invention patent: A driving model of a dielectric elastomer-driven rigid origami-like flexible finger joint (CN113199485A), a driving model between the position, driving force, and field strength of a flexible finger joint based on dielectric elastomer driving and rigid origami is established, which is used to calculate the driving force of the flexible finger joint and achieve precise position control of the flexible finger joint when loaded or unloaded, ensuring the motion accuracy of the flexible finger joint. The invention patent A rigid origami folding method based on an optimized path planning algorithm (CN112734079A) combines rigid origami folding path planning and folding path smoothing to solve the problem of lack of folding stability. Summary of the Invention

[0003] The present invention proposes a modeling method for fitting a generalized cylinder surface based on a derivative structure of water bomb origami. First, the optimized origami grid model is mapped from a three-dimensional state to a two-dimensional plane, and linear regression processing is performed using the vertex information to achieve the purpose of model precision. A three-dimensional grid model is constructed from the processed data to maximize the satisfaction of the deployability constraint and flat foldability constraint of the origami model. The present invention has three types of derivative variant origami unit structures based on Waterbomb (water bomb), which can simulate the target surface more diversely. Secondly, by controlling a single variable, the rigid origami movement process of the origami is simulated.

[0004] To achieve the above object, the technical solutions provided by the present invention are as follows:

[0005] A modeling method for fitting a generalized cylinder surface based on a derivative structure of water bomb origami, comprising the following steps:

[0006] S1: Obtain a two-dimensional mapping and additional information from a three-dimensional Waterbomb basic strip

[0007] ​S2: Obtain the left side of the 3D Waterbomb origami strip that satisfies the unfoldable constraint and the flat-foldable constraint from the two-dimensional mapping and vertex information.

[0008] S3: Based on Construct the target origami structure based on the Waterbomb-derived origami structure.

[0009] S4: Simulate the rigid folding motion process of the origami structure.

[0010] Furthermore, the specific steps of the above S1 are as follows:

[0011] S11: Determine the two-dimensional p1 point coordinates, that is, the origin (note: uppercase P represents the 3D vertex, and lowercase p represents its corresponding 2D vertex).

[0012] S12: Determine the two-dimensional p3 point coordinates: Obtain the distance L between P1 and P3 through the 3D Waterbomb base strip where L 1，3 represents the distance between vertices P i，j and P i and P j , and determine the p3 coordinates as (0, L 1，3 ).

[0013] S13: Determine the two-dimensional p2 point coordinates: Obtain the distances L and L 1，2 and L 3，2 through the 3D Waterbomb base strip. According to formula (1), obtain two candidate points; let the coordinates of the first candidate point be (c 2，x1 , c 2，y1 ), and the coordinates of the second candidate point be (c 2，x2 , c 2，y2 ), and their calculation formulas are as follows:

[0014]

[0015] where X represents the x coordinate of p3 minus the x coordinate of p1, Y represents the y coordinate of p3 minus the y coordinate of p1, r1 = L 1，2 , r2 = L 3，2 ;

[0016] If c 2，x1 is less than 0, then take (c 2，x2 , c 2，y2 ) as the p2 point, otherwise take (c 2，x1 , c 2，y1 ) as the p2 point; after the coordinate selection is successful, perform vector addition with p2 and p1, and add the local coordinate point p2 to the global coordinate system based on p1.

[0017] S14. Obtain other two-dimensional vertex p i (i = 4, 5,...) Coordinates: Through the three-dimensional Waterbomb basic strip Obtain P i-2 , P i-1 to P i distance L i-2，i , L i-1，i , take the distance L i-2，i to replace r1 in formula (1), take the distance L i-1，i to replace r2 in formula (1), take X as the x-axis coordinate of p i-1 minus the x-axis coordinate of p i-2 , Y is the y-axis coordinate of p i-1 minus the y-axis coordinate of p i-2 , use formula (1) to find two candidate points of p i (c i，x1 , c i，y1 ), (c i，x2 , c i，y2 ), use the first candidate point (c i，x1 , c i，y1 ) and p i-2 , p i-1 to form a triangular face face1, use the second candidate point (c i，x2 , c i，y2 ) and p i-2 , p i-1 to form a triangular face face2. After the vertex coordinate p i is successfully selected, perform vector addition with p i and p i-2 , and add the local coordinate point p i to the global coordinate system where p i-2 is located to obtain the mapping of all three-dimensional vertices to the two-dimensional plane;

[0018] S15. Using the two-dimensional vertex configuration information obtained in S14, process the vertex data by linear regression to obtain the regression lines l c , l1, and obtain the projection point p i ' of the two-dimensional vertex p i ;

[0019] S16. Using the projection point p i ' to form a new p i point, and use the newly generated p i vertex data as the basic data for the two-dimensional to three-dimensional mapping;

[0020] S17. For the line l c, with p1 as the rotation center, rotate counterclockwise by an angle α in the x-y plane, where the angle α:

[0021]

[0022] S18. Obtain the data of the three-dimensional mesh model mapped to the two-dimensional plane: The data D represents Φ when establishing the initial strip L and Φ C distance, where Φ C represents the plane x-z, that is, the plane where the vertices with odd subscripts are located, and Φ L represents the plane y = D, that is, the plane where the vertices with even subscripts are located; the data Repeats represents the number of times the strip is replicated; store the angle Beta, the total number of control points, and the coordinates of the first point in the original three-dimensional space.

[0023] Furthermore, the above step S2 specifically includes the following contents:

[0024] S21. Determine the three-dimensional P1 vertex: According to the two-dimensional configuration file, first determine the P1 point in the three-dimensional space, and take the plane Φ C , that is, the x-z plane;

[0025] S22. Determine the three-dimensional P2 vertex: Determine the plane Φ by the D value L , the plane Φ L is formed by translating the plane Φ C along the positive y-axis by a distance D; with P1 as the center of the sphere and a distance l 1，2 as the radius to make a sphere, the sphere intersects with the plane Φ L , and a circle C with the point P′1 obtained by projecting the point P1 onto the plane Φ L as the center is obtained on the plane Φ L . Take a point P on the circle C such that the angle between the vector and the positive Z-axis is Beta, then take this point as P2, where l i，j represents the distance between p i and p j ;

[0026] S23. Determine the three-dimensional P3 vertex: Respectively take P1 and P2 as the centers of the spheres and l 1，3 , l 2，3 as the radii to make spheres, and two intersecting circles are obtained in the plane Φ C , that is, there are two intersection points, and both of these points are candidate points for P3;

[0027] S24. Determine the three-dimensional P i , i = 4, 5, 6…: Respectively take P i-2 , P i-1 as the centers of the spheres and l i-2，i , li-1，i Make spheres with a radius: When i is odd, the above two spheres have two intersecting circles in the plane Φ C and there are two intersection points. Select the point that meets the geometric structure characteristics among the intersection points as P i ; When i is even, then the above two spheres have two intersecting circles in the plane Φ L and there are two intersection points. Select the point that meets the geometric structure characteristics among the intersection points as P i ;

[0028] S25. Generate the left side of the three-dimensional Waterbomb folding strip from two-dimensional configuration data. The grid structure satisfies the deployable constraint and the flat-foldable constraint, and is named

[0029] Furthermore, the above p i The selection of the two candidate points (c i，x1 , c i，y1 ), (c i，x2 , c i，y2 ) is determined based on the following conditions:

[0030] 1) When i is odd and the z-axis component of the face1 normal vector is less than 0, take the first candidate point as p i , otherwise take the second candidate point as p i ;

[0031] 2) When i is even and the z-axis component of the face1 normal vector is greater than 0, take the first candidate point as p i , otherwise take the second candidate point as p i .

[0032] Furthermore, the processing of vertex data using linear regression in the above S15 includes the following content:

[0033] Perform linear regression processing on the data with odd point subscripts:

[0034] 1) Obtain the x coordinates and y coordinates of all two-dimensional vertices and denote them as x i , y i (i = 1, 2,..., n odd ), where n odd is the number of odd points;

[0035] 2) Use the linear regression equation to obtain the regression line: where The method is as follows:

[0036]

[0037] where Let the regression line be l c ;

[0038] 3) Take each p i point's projection onto the line l c and denote this projection point as p i ';

[0039] For the points with even subscripts, take the x and y coordinates of each vertex and perform linear regression using formula (2) to obtain the line l l and take each p i point's projection onto the line l l and denote this projection point as p i '.

[0040] Furthermore, there are the following three cases in constructing the Waterbomb-derived origami structure in step S3 above:

[0041] S31. The Waterbomb-derived origami structure of Type L fits a surface with the geometric characteristics of a generalized cylinder:

[0042] Based on for the vertices with even subscripts in this structure, extrude them along the positive y-axis by a distance d l , where the distance d l is specified by the user. The corresponding point after extrusion for each vertex P i is A i , then the quadrilateral P i A i A i+2 P i+2 is obtained, where i = 2, 4, …, and the obtained quadrilateral is on the left side of . And through mirror symmetry (i.e., about the x-z plane) of

[0043] and the quadrilateral structure, a complete Waterbomb Type L base origami strip is obtained. According to the number of copies input by the user, replicate and tile it along the y-axis to generate the target mesh model;

[0044] Based on for the vertices with odd subscripts in this structure, extrude them along the negative y-axis by a distance d r , where the distance d r is specified by the user. The corresponding point after extrusion for each vertex P i is B i , then the quadrilateral B i P i P i+2 B i+2 is obtained, where i = 1, 3, …, and the obtained quadrilateral is on the On the right side, at this time, the entire grid model needs to be translated a distance d in the positive y-axis direction. r , the and the quadrilateral structure are mirror-symmetric (i.e., about the x-z plane) to obtain a complete Waterbomb Type R basic origami strip. According to the number of copies input by the user, it is replicated and tiled along the y-axis to generate the target grid model.

[0045] S33. The Type B Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder:

[0046] Based on For the vertices with even subscripts in this structure, extrude them a distance d in the negative y-axis direction. l , the distance d l is specified by the user. For each vertex P i , the corresponding point after extrusion is A i , then the quadrilateral P i A i A i+2 P i+2 is obtained, where i = 2, 4, …, and the obtained quadrilateral is on the left side; for the vertices with odd subscripts in this structure, extrude them a distance d in the negative y-axis direction. r , the distance d r is specified by the user. For each vertex P i extruded by d r , the corresponding point is B i , then the quadrilateral B i P i P i+2 B i+2 is obtained, where i = 1, 3, …, and the obtained quadrilateral is on the right side. At this time, the entire grid model needs to be translated a distance d in the positive y-axis direction. r , and then the whole is mirror-symmetric (i.e., about the x-z plane) to obtain a complete Waterbomb Type B basic origami strip. According to the number of copies input by the user, it is replicated and tiled along the y-axis to generate the target grid model.

[0047] Furthermore, the above step S4 specifically includes the following content:

[0048] S41. By changing the value of D′, simulate the rigid folding motion process of the origami structure:

[0049] The rigid folding process is the process from the target origami grid state to the completely flat folding. Each time, the grid model is reconstructed by the value of D′ until the value of D′ is 0. At this time, it represents that the grid model state is the flat folding state. The above sequence of grid models constitutes the rigid folding process of the origami structure.

[0050] The rigid folding motion of the target grid model is along the y-axis. Define the folding rate Ψ, and its formula is defined as:

[0051]

[0052] where the value range of D′ is 0 to D, and D represents Φ when establishing the initial strip. L and Φ C distance, where Φ C represents the plane x-z, that is, the plane where the vertices with odd subscripts are located, and Φ L represents the plane y = D, that is, the plane where the vertices with even subscripts are located;

[0053] S42. The rigid constraint of the left half of the strip. According to the corresponding D′ value, reconstruct the corresponding origami grid model:

[0054] For its original triangular structure of the Waterbomb pattern, during its folding motion, the side length of the triangle remains unchanged, which can ensure its rigid characteristics;

[0055] For the added left quadrilateral part, side B i P i (i = 2, 4,...) is perpendicular to the plane Φ L , that is, perpendicular to the plane Φ L on the line segment with P i as the vertex, then the right-angle characteristic of the quadrilateral can be ensured, that is, the angle remains unchanged; at the same time, according to the generation characteristics of the target grid, that is, the extrusion length of the vertex remains unchanged, then side B i P i length remains unchanged, and the quadrilateral can ensure that the angle and length remain unchanged during the motion, that is, it can meet the rigid characteristics;

[0056] For the added right quadrilateral part, side A i P i (i = 1, 3,...) is perpendicular to the plane Φ C , that is, perpendicular to the plane Φ C on the line segment with P i as the vertex, then the right-angle characteristic of the quadrilateral can be ensured, that is, the angle remains unchanged; at the same time, according to the generation characteristics of the target grid, that is, the extrusion length of the vertex remains unchanged, then side A i P i length remains unchanged, and the quadrilateral can ensure that the angle and length remain unchanged during the motion, meeting the rigid characteristics; the left side of the origami strip can ensure the rigid characteristics, and the complete strip is generated by mirror symmetry of the left strip about the plane x-z, which can meet the rigid characteristics of the strip. The target grid model is replicated and tiled along the y-axis by the strip, which can meet the rigid characteristics.

[0057] The present invention has the following beneficial effects:

[0058] The present invention uses a waterbomb origami derivative structure to fit a target surface with the characteristics of a generalized cylinder. Waterbomb is a type of origami pattern. The internal vertices of its pattern unit have six adjacent vertices, forming six edges, among which the distribution is four valley folds and two mountain folds. The origami pattern used in the present invention is based on the waterbomb derivative structure, which is a derivative origami structure invented based on the consideration of the internal vertex degrees of freedom of origami. A new strip of origami for fitting the target surface is obtained by deriving and adjusting on the left side of the original waterbomb strip. The present invention constructs three different types of waterbomb derivative structures respectively, all of which can well fit the target surface and can meet the deployable constraint. And the present invention can ensure the diversification of fitting the target surface to a certain extent while meeting the flat-foldable characteristic of origami. In addition, the present invention also simulates the rigid motion process of the origami structure. Brief Description of the Drawings

[0059] Figure 1 is a flowchart of the method of the present invention;

[0060] Figure 2 is an example of the optimized waterbomb basic strip;

[0061] Figure 3 are grid models of three types of waterbomb derivative structures. Detailed Embodiments

[0062] The present invention will be described in detail below in conjunction with the embodiments shown in the drawings. However, these embodiments do not limit the present invention, and any structural, method, or functional transformation made by those of ordinary skill in the art based on these embodiments is included in the protection scope of the present invention.

[0063] As Figure 1 shown, the present invention is a modeling method for fitting a generalized cylinder surface based on a waterbomb origami derivative structure, including the following steps:

[0064] S1: Obtain a two-dimensional mapping and additional information from a three-dimensional waterbomb basic strip ;

[0065] S2: Obtain the left side of the three-dimensional waterbomb origami strip that meets the deployable constraint and the flat-foldable constraint from the two-dimensional mapping and vertex information An example of the optimized waterbomb basic strip is as Figure 2 shown.

[0066] S3: Based on Construct a target origami structure based on a waterbomb-derived origami structure;

[0067] S4: Simulate the rigid folding motion process of the origami structure.

[0068] As a preferred embodiment of the present invention, the specific content of step S1 of the present invention includes:

[0069] S11. Determine the two-dimensional p1 point coordinates, that is, the origin. Note: Uppercase P represents a three-dimensional vertex, and lowercase p represents its corresponding two-dimensional vertex;

[0070] S12. Determine the two-dimensional p3 point coordinates: Obtain the distance L between P1 and P3 through the three-dimensional Waterbomb base strip where L 1，3 , and L i，j represents the distance between vertices P i and P j . Determine the p3 coordinates as (0, L 1，3 );

[0071] S13. Determine the two-dimensional p2 point coordinates: Obtain the distances L and L 1，2 as well as L 3，2 through the three-dimensional Waterbomb base strip. According to formula (1), obtain two candidate points; Let the coordinates of the first candidate point be (c 2，x1 , c 2，y1 ), and the coordinates of the second candidate point be (c 2，x2 , c 2，y2 ). Their calculation formulas are as follows:

[0072]

[0073]

[0074] where X represents the x coordinate of p3 minus the x coordinate of p1, Y represents the y coordinate of p3 minus the y coordinate of p1, r1 = L 1，2 , r2 = L 3，2 ;

[0075] If c 2，x1 is less than 0, then take (c 2，x2 , c 2，y2 ) as the p2 point, otherwise take (c 2，x1 , c 2，y1 ) as the p2 point; After the coordinate selection is successful, perform vector addition with p2 and p1, and add the local coordinate point p2 to the global coordinate system based on p1;

[0076] S14. Obtain other two-dimensional vertices p i(i = 4, 5, …) Coordinates: Through the three-dimensional Waterbomb base strip Obtain P i-2 , P i-1 to P i distance L i-2，i , L i-1，i , take the distance L i-2，i to replace r1 in formula (1), take the distance L i-1，i to replace r2 in formula (1), take X as the x-axis coordinate of p i-1 minus the x-axis coordinate of p i-2 , Y is the y-axis coordinate of p i-1 minus the y-axis coordinate of p i-2 , use formula (1) to find two candidate points of p i (c i，x1 , c i，y1 ), use the first candidate point (c i，x1 , c i，y1 ) and p i-2 , p i-1 to form a triangular face face1, use the second candidate point (c i，x2 , c i，y2 ) and p i-2 , p i-1 to form a triangular face face2. After the vertex coordinate p i is successfully selected, perform vector addition on p i and p i-2 , add the local coordinate point p i to the global coordinate system where p i-2 is located to obtain the mapping of all three-dimensional vertices to the two-dimensional plane;

[0077] As a preferred embodiment of the present invention, for the two candidate points of p i (c i，x1 , c i，y1 ), (c i，x2 , c i，y2 ), the selection is determined based on the following conditions:

[0078] 1) When i is odd and the z-axis component of the normal vector of face1 is less than 0, take the first candidate point as p i , otherwise take the second candidate point as p i ;

[0079] 2) When i is even and the z-axis component of the normal vector of face1 is greater than 0, take the first candidate point as p i , otherwise take the second candidate point as p i .

[0080] S15. Using the two-dimensional vertex configuration information obtained in S14, process the vertex data by linear regression to obtain the regression line l c 、l l , and obtain the projection point p i ' of the two-dimensional vertex p i ';

[0081] As a preferred embodiment of the present invention, the processing of vertex data by linear regression in the above S15 includes the following content:

[0082] Perform linear regression processing on the data with odd subscripts:

[0083] 1) Obtain the x coordinates and y coordinates of all two-dimensional vertices, denoted as x i , y i (i = 1, 2,..., n odd ), where n odd is the number of points with odd subscripts;

[0084] 2) Use the linear regression equation to obtain the regression line: where The calculation method is as follows:

[0085]

[0086] where Let this regression line be l c ;

[0087] 3) Take the projection of each p i point onto the line l c , and let this projection point be p i ';

[0088] For the points with even subscripts, take the x and y coordinates of each vertex, perform linear regression using formula (2) to obtain the line l l , take the projection of each p i point onto the line l l , and let this projection point be p i '.

[0089] S16. Use the projection points p i ' to form new p i points, and use the newly generated p i vertex data as the basic data for two-dimensional to three-dimensional mapping;

[0090] S17. For the line l c , with p1 as the rotation center, rotate counterclockwise by an angle α in the x-y plane, where the angle α:

[0091]

[0092] The purpose is to rotate the straight line l c to coincide with the x-axis, which can ensure that there will be no splitting when the left side of the strip is mirror-symmetric about the x-z plane, and ensure the effectiveness of generating the origami model;

[0093] S18. Obtain the data of the three-dimensional mesh model mapped to the two-dimensional plane: The data D represents Φ when establishing the initial strip L and Φ C distance, where Φ C represents the plane x-z, that is, the plane where the vertices with odd subscripts are located, and Φ L represents the plane y = D, that is, the plane where the vertices with even subscripts are located; The data Repeats represents the number of times the strip is replicated; Store the angle Beta, the total number of control points, and the coordinates of the first point in the original three-dimensional space.

[0094] As a preferred embodiment of the present invention, the above step S2 specifically includes the following contents:

[0095] S21. Determine the three-dimensional P1 vertex: According to the two-dimensional configuration file, first determine the P1 point in the three-dimensional space, and take the plane Φ C passing through this point, that is, the x-z plane;

[0096] S22. Determine the three-dimensional P2 vertex: Determine the plane Φ by the D value L , and the plane Φ L is formed by translating the plane Φ C along the positive y-axis by a distance D; Taking P1 as the center of the sphere and the distance l 1，2 (l i，j represents the distance between p i and p j ) as the radius to make a sphere. At this time, the sphere intersects with the plane Φ L , and a circle C with the projection point P′1 of the point P1 on the plane Φ L as the center is obtained on the plane Φ L . Take a point P on the circle C such that the angle between the vector and the positive direction of the Z-axis is Beta, then take this point as P2;

[0097] S23. Determine the three-dimensional P3 vertex: Take P1 and P2 as the centers of the spheres respectively, and take l 1，3 , l 2，3 as the radii to make spheres, and two intersecting circles are obtained in the plane Φ C , that is, there are two intersection points, and both of these two points are candidate points for P3;

[0098] S24. Determine the three-dimensional P i , i = 4, 5, 6…: Take P i-2 and P i-1Taking [the point] as the center of the sphere and with l i-2，i and l i-1，i as the radii to form spheres: When i is odd, the above two spheres have two intersecting circles in the plane Φ C , then there are two intersection points, and the point that conforms to the geometric structure characteristics among the intersection points is taken as P i ; When i is even, then the above two spheres have two intersecting circles in the plane Φ L , then there are two intersection points, and the point that conforms to the geometric structure characteristics among the intersection points is taken as P i ;

[0099] S25. Generate the left side of the three-dimensional Waterbomb folded strip from two-dimensional configuration data. The grid structure satisfies the deployable constraint and the flat-foldable constraint, and is named

[0100] As a preferred embodiment of the present invention, as shown in Figure 3 , (a) is the grid model of the Type L Waterbomb-derived structure. (b) is the grid model of the Type R Waterbomb-derived structure. (c) is the grid model of the Type B Waterbomb-derived structure. There are the following three cases for constructing the Waterbomb-derived origami structure,

[0101] S31. The Type L Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder:

[0102] Based on , for the vertices with even subscripts in this structure, extrude them along the positive y-axis by a distance d l , the distance d l is specified by the user. The corresponding point after extrusion for each vertex P i is A i , then the quadrilateral P i A i A i+2 P i+2 is obtained, where i = 2, 4, …, and the obtained quadrilateral is on the left side of . By mirror symmetry (i.e., about the x-z plane) of and the quadrilateral structure, a complete Waterbomb Type L basic folded strip is obtained, and according to the number of copies input by the user, it is tiled and replicated along the y-axis to generate the target grid model;

[0103] S32. The Type R Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder:

[0104] Based on , for the vertices with odd subscripts in this structure, extrude them along the negative y-axis by a distance dr , the distance d r is specified by the user, and for each vertex P i the corresponding point after extrusion is B i , then the quadrilateral B i P i P i+2 B i+2 is obtained, where i = 1, 3, …, and the obtained quadrilateral is on the right side of, and at this time the entire mesh model needs to be translated in the positive y-axis direction by a distance d r , and and the quadrilateral structure are mirror-symmetric (i.e., about the x-z plane) to obtain a complete Waterbomb Type R basic origami strip. According to the number of copies input by the user, it is replicated and tiled along the y-axis to generate the target mesh model;

[0105] S33. The Type B Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder:

[0106] Based on for the vertices with even subscripts in this structure, they are extruded along the negative y-axis direction by a distance d l , the distance d l is specified by the user, and for each vertex P i the corresponding point after extrusion is A i , then the quadrilateral P i A i A i+2 P i+2 is obtained, where i = 2, 4, …, and the obtained quadrilateral is on the left side of; for the vertices with odd subscripts in this structure, they are extruded along the negative y-axis direction by a distance d r , the distance d r is specified by the user, and for each vertex P i extruded by d r the corresponding point after extrusion is B i , then the quadrilateral B i P i P i+2 B i+2 is obtained, where i = 1, 3, …, and the obtained quadrilateral is on the right side of. At this time, the entire mesh model needs to be translated in the positive y-axis direction by a distance d r , and then the whole is mirror-symmetric (i.e., about the x-z plane) to obtain a complete Waterbomb Type B basic origami strip. According to the number of copies input by the user, it is replicated and tiled along the y-axis to generate the target mesh model.

[0107] On the deployable constraints of the Waterbomb-derived structure. Here we verify the deployable constraints of the Waterbomb Type B origami structure. For the left side of the Type B Waterbomb origami strip formed above, for each vertex inside it, the vertex has five angles, and the angles follow the following rules:

[0108] a. When the subscript i of vertex P i is even:

[0109] For its quadrilateral addition module, extrude the side P i A i perpendicular to the vertical plane Φ L , then P i A i must be perpendicular to the side inside the plane Φ L . For A i P i perpendicular to the line P i P i-2 , that is, in the quadrilateral P i-2 A i-2 A i P i , ∠A i P i P i-2 = π / 2. In the quadrilateral P i A i A i+ 2P i+2 , for A i P i perpendicular to the line P i P i+2 , ∠P i+2 P i A i = π / 2. It can be obtained that ∠P i+2 P i A i + ∠A i P i P i-2 = π. And for the origami structure Waterbomb pattern module, since this three-dimensional structure has been optimized and refined, it can be known that ∠P i-2 P i P i-1 + ∠P i-1 P i P i+1 + ∠P i+1 P i P i+2 = π, then for the internal vertex P i , the sum of the angles around this point is ∠P i+2 P i Ai + ∠A i P i P i-2 + ∠P i-1 P i P i-1 + ∠P i-1 P i P i+1 + ∠P i+1 P i P i+2 = 2π, that is, the internal vertex P i satisfies the developable constraint;

[0110] b. When the vertex P i and the subscript i is odd:

[0111] For its quadrilateral addition module, the extrusion edge B i P i is perpendicular to the plane Φ C , then B i P i must be perpendicular to the edges in the plane Φ C . For B i P i being perpendicular to the edge P i P i-2 , that is, in the quadrilateral B i-2 P i-2 P i B i , ∠P i-2 P i B i = π / 2. For B i P i being perpendicular to the edge P i P i+2 , in the quadrilateral B i P i P i+2 B i+2 , ∠B i P i P i+2 = π / 2. It can be obtained that ∠P i-2 P i B i + ∠B i P i P i+2 = π. And for the origami structure Waterbomb origami module, since this three-dimensional structure has been optimized and refined, it can be known that ∠P i+2 P i P i+1 + ∠P i+1 P i P i-1 + ∠P i-1P i P i-2 = π, then for the internal vertex P i , the sum of the angles around this point is ∠P i-2 P i B i + ∠B i P i P i+2 + ∠P i+2 P i P i+1 + ∠P i+1 P i P i-1 + ∠P i-1 P i P i-2 = 2π, that is, it satisfies the deployable constraint;

[0112] c. In summary, the left side of the Type B Waterbomb crepe strip satisfies the deployable constraint. Since the complete crepe strip is symmetric about the x - y plane mirror from the left half, it can be ensured that the structure of a single crepe strip can satisfy the deployable constraint. For the entire origami structure, by translating and replicating a single crepe strip along the y - axis, for the newly formed internal vertices, the deployable constraint of the vertices can still be satisfied. In summary, this origami structure can satisfy the deployable constraint.

[0113] d. Since Type L and Type R are special cases where Type B is about d r and d l equal to zero, so that Type B can satisfy the deployable constraint indicates that Type L and Type R can also satisfy the deployable constraint;

[0114] The rigid folding motion process means that the origami structure changes from the initial folded origami state that fits the target surface to its folded structure becoming a completely flat - folded state, that is, finally the origami mechanism can be contracted into a planar structure. For the origami structure of the present invention, in its flat - folded state, except for the derived rectangular added blocks, the remaining part can still satisfy the flat - folding characteristics, and during this process, this origami structure can ensure the rigid constraint conditions of its faces and edges.

[0115] As a preferred embodiment of the present invention, step S4 specifically includes the following content:

[0116] S41. By changing the value of D′, simulate the rigid folding motion process of the origami structure:

[0117] The rigid folding process is the process from the target origami grid state to the fully flat fold. Each time, the grid model is reconstructed based on the value of D′ until the value of D′ is 0, which represents that the grid model state is in the flat fold state. The grid models in the above sequence constitute the rigid folding process of the origami structure;

[0118] The rigid folding motion of the target grid model is along the y-axis. Define the folding rate Ψ, and its formula is defined as:

[0119]

[0120] where the value range of D′ is 0 to D, and D represents Φ when establishing the initial strip L and Φ C distance, where Φ C represents the plane x-z, that is, the plane where the vertices with odd subscripts are located, and Φ L represents the plane y = D, that is, the plane where the vertices with even subscripts are located;

[0121] S42. The rigid constraint of the left half of the strip. According to the corresponding value of D′, reconstruct the corresponding origami grid model:

[0122] For the original triangular structure with the Waterbomb pattern, during its folding motion, the invariance of the triangle side length can ensure its rigid characteristics;

[0123] For the added part of the left quadrilateral, side B i P i (i = 2, 4,...) is perpendicular to the plane Φ L , that is, perpendicular to the plane Φ L on the line segment with P i as the vertex, which can ensure the right-angle characteristic of the quadrilateral, that is, the angle remains unchanged; at the same time, according to the generation characteristics of the target grid, that is, the extrusion length of the vertex remains unchanged, then side B i P i has a constant length. The quadrilateral can ensure the invariance of the angle and length during the motion, that is, it can meet the rigid characteristics;

[0124] For the added part of the right quadrilateral, side A i P i (i = 1, 3,...) is perpendicular to the plane Φ C , that is, perpendicular to the plane Φ C on the line segment with P i as the vertex, which can ensure the right-angle characteristic of the quadrilateral, that is, the angle remains unchanged; at the same time, according to the generation characteristics of the target grid, that is, the extrusion length of the vertex remains unchanged, then side A i P iWith the length remaining unchanged, the quadrilateral can ensure that the angles and lengths remain unchanged during the movement, meeting the rigid characteristics; the left side of the folded paper strip can ensure the rigid characteristics, and the complete strip is generated by mirror symmetry of the left strip with respect to the x-z plane, which can meet the rigid characteristics of the strip. The target mesh model is replicated and tiled along the y-axis by the strip, which can meet the rigid characteristics.

Claims

1. A modeling method for fitting the surface of a generalized cylinder based on the derivative structure of water bomb origami, characterized in that, The method includes the following steps: S1: Obtain a two-dimensional mapping and additional information from a three-dimensional Waterbomb base strip ; S2: Obtain the left side of the 3D Waterbomb folding strip that satisfies the unfoldable constraint and the flat-foldable constraint from the two-dimensional mapping and vertex information S3: Based on Construct a target origami structure based on the Waterbomb-derived origami structure; S4: Simulate the rigid folding motion process of the origami structure; The specific content of step S1 is as follows: S11. Determine the coordinates of the two-dimensional p1 point, which is the origin. Here, the capital P represents the three-dimensional vertex, and the lowercase p represents the two-dimensional vertex corresponding to the three-dimensional vertex; S12. Determine the two-dimensional coordinates of point p3: Through the three-dimensional Waterbomb basic strip Obtain the distance L between P1 and P3 1，3 , where L i，j represents the distance between vertex P i and P j . Determine the coordinates of p3 as (0, L 1，3 ); S13. Determine the coordinates of the two-dimensional p2 point: Through the three-dimensional Waterbomb basic strip Obtain the distance L 1，2 and L 3，2 , according to formula (1), obtain two candidate points; Let the coordinates of the first candidate point be (c 2，x1 , c 2，y1 ), and the coordinates of the second candidate point be (c 2，x2 , c 2，y2 ), and their calculation formulas are as follows: Where X represents the x - coordinate of p3 minus the x - coordinate of p1, Y represents the y - coordinate of p3 minus the y - coordinate of p1, r1 = L 1，2 , r2 = L 3，2 ; If c 2，x1 is less than 0, then take (c 2,x2 , c 2,y2 ) as point p2, otherwise take (c 2,x1 , c 2,y1 ) as point p2; after the coordinate selection is successful, perform vector addition on p2 and p1, and add the local coordinate point p2 to the global coordinate system based on p1; S14. Obtain other two-dimensional vertex p i , i = 4, 5, …, Coordinates: Through the three-dimensional Waterbomb basic strip Obtain P i-2 , P i-1 to P i distance L i-2，i , L i-1，i , Take the distance L i-2,i to replace r1 in formula (1), take the distance L i-1,i to replace r2 in formula (1), take X as the x-axis coordinate of p i-1 minus the x-axis coordinate of p i-2 , Y is the y-axis coordinate of p i-1 minus the y-axis coordinate of p i-2 , Use formula (1) to find two candidate points of p i , (c i，x1 , c i，y1 ), (c i，x2 , c i，y2 ), Use the first candidate point (c i，x1 , c i，y1 ) and p i-2 , p i-1 to form triangle face face1, use the second candidate point (c i，x2 , c i，y2 ) and p i-2 , p i-1 to form triangle face face2. After the vertex coordinate p i is successfully selected, use p i and p i-2 for vector addition, and add the local coordinate point p i to the global coordinate system where p i-2 is located to obtain the mapping of all three-dimensional vertices to the two-dimensional plane; S15. Using the two-dimensional vertex configuration information obtained in S14, process the vertex data by linear regression to obtain the regression line l c and l l , and obtain the projection point p i ' of the two-dimensional vertex p i '; S16. Use the projection point p i ' to form a new p i point. The newly generated p i vertex is the model vertex of the two-dimensional to three-dimensional mapping; S17. For the straight line l c , taking p1 as the rotation center, rotate counterclockwise by an angle α in the x-y plane, where the angle α: is the regression line l c coefficient; S18. Obtain the data mapped from the three-dimensional mesh model to the two-dimensional plane: The data D represents Φ when establishing the initial strip L and Φ C distance, where Φ C represents the plane x-z, that is, the plane where the vertices with odd subscripts are located, and Φ L represents the plane y = D, that is, the plane where the vertices with even subscripts are located; The data Repeats represents the number of times the strip is repeatedly copied; Store the angle Beta, the total number of control points, and the coordinates of the first point in the original three-dimensional space; The specific content of step S2 includes the following: S21. Determine the three-dimensional P1 vertex: According to the two-dimensional configuration file, first determine the P1 point in three-dimensional space and take the plane Φ passing through this point C , that is, the x-z plane; S22. Determine the three-dimensional P2 vertex: Determine the plane Φ from the D value L , the plane Φ L is formed by translating the plane Φ C along the positive y-axis by a distance D; taking P1 as the center of the sphere and a distance l 1，2 as the radius to make a sphere, the sphere intersects the plane Φ L , and a circle C with the point P′1, which is the projection of the point P1 onto the plane Φ L , as the center is obtained on the plane Φ L . Take a point P on the circle C such that the angle between the vector and the positive Z-axis is Beta, then take this point as P2, where l i，j represents the distance between p i and p j ; S23. Determine the three-dimensional P3 vertex: Respectively take P1 and P2 as the centers of the spheres, and take l 1，3 , l 2，3 as the radii to make spheres. Two intersecting circles are obtained within the plane Φ c . That is, there are two intersection points, and both of these two points are candidate points for P3; S24. Determine the three-dimensional P i , where i = 4, 5, 6...: Respectively, with P i-2 , P i-1 as the centers of the spheres and l i-2，i , l i-1，i as the radii to form spheres: When i is odd, the above two spheres have two intersecting circles in the plane Φ C , and there are two intersection points. Select the points that meet the geometric structure characteristics among the intersection points as P i ; When i is even, the above two spheres have two intersecting circles in the plane Φ L , and there are two intersection points. Select the points that meet the geometric structure characteristics among the intersection points as P i ; S25. The left side of the three-dimensional Waterbomb strip paper generated from the two-dimensional configuration data satisfies the deployable constraint and the flat-foldable constraint, and is named There are three cases for constructing the Waterbomb-derived origami structure in step S3: S31. The Type L Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder: Based on For the vertices with even subscripts in this structure, extrude them along the positive y-axis by a distance d l , the distance d l is specified by the user. For each vertex P i , the corresponding point after extrusion is A i , then the quadrilateral P i A i A i+2 P i+2 is obtained, where i = 2, 4, …, and the obtained quadrilateral is on the left of . Mirror-symmetrize and the quadrilateral structure to obtain a complete Waterbomb Type L-shaped base folded paper strip. According to the number of copies input by the user, tile and generate the target grid model by replicating along the y-axis; S32. The Type R Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder: Based on For the vertices with odd subscripts in this structure, extrude them along the negative y-axis by a distance d r , the distance d r is specified by the user. For each vertex P i , the corresponding point after extrusion is B i , then the quadrilateral B i P i P i+2 B i+2 is obtained, where i = 1, 3, …, and the obtained quadrilateral is on the right side of . At this time, the entire mesh model needs to be translated along the positive y-axis by a distance d r . Then and the quadrilateral structure are mirror-symmetric to obtain a complete Waterbomb Type R-based folded paper strip. According to the number of copies input by the user, it is replicated and tiled along the y-axis to generate the target mesh model; S33. The Type B Waterbomb-derived origami structure fits a surface with the geometric characteristics of a generalized cylinder: Based on For the vertices with even subscripts in this structure, extrude them along the negative y-axis by a distance d l , the distance d l is specified by the user. For each vertex P i , the corresponding point after extrusion is A i , then the quadrilateral P i A i A i+2 P i+2 is obtained, where i = 2, 4, …, and the obtained quadrilateral is on the left side of; For the vertices with odd subscripts in this structure, extrude them along the negative y-axis by a distance d r , the distance d r is specified by the user. For each vertex P i extruded by d r , the corresponding point is B i , then the quadrilateral B i P i P i+2 B i+2 is obtained, where i = 1, 3, …, and the obtained quadrilateral is on the right side of; At this time, the entire mesh model needs to be translated along the positive y-axis by a distance d r , and then the whole is mirror-symmetrical, that is, about the x-z plane, to obtain a complete Waterbomb Type B basic folding strip. According to the number of copies input by the user, copy and tile it along the y-axis to generate the target mesh model; The specific content of step S4 includes the following: S41. Simulate the rigid folding motion process of the origami structure by changing the value of D': The rigid folding process is the process from the target origami grid state to the fully flat folding. Each time, the grid model is reconstructed with the value of D', until the value of D' is 0. At this time, it represents that the grid model state is the flat folding state. The grid models during the decreasing process of D' constitute the rigid folding process of the origami structure; The rigid folding motion of the target grid model is along the y-axis. Define the folding rate Ψ, and its formula is defined as: Where the value range of D′ is 0 to D, and D represents Φ when establishing the initial strip L and Φ C distance, where Φ C represents the plane x-z, that is, the plane where the vertices with odd subscripts are located, and Φ L represents the plane y = D, that is, the plane where the vertices with even subscripts are located; S42. The rigid constraint of the left half of the strip. Reconstruct the corresponding origami grid model according to the corresponding value of D'; For the original triangular structure of the Waterbomb pattern, during its folding motion process, the side length of the triangle remains unchanged, which can ensure its rigid characteristics; For the left quadrilateral addition part, side B i P i , i = 2, 4,... is perpendicular to plane Φ L , that is, perpendicular to plane Φ L on the line segment with P i as the vertex, the right-angle property of the quadrilateral can be guaranteed, that is, the angle remains unchanged; at the same time, according to the generation characteristics of the target grid, that is, the extrusion length of the vertex remains unchanged, then side B i P i has a constant length, and the quadrilateral can guarantee the invariance of the angle and length during the movement, that is, it can meet the rigid characteristics; For the added part of the quadrilateral on its right side, side A i P i , i = 1, 3, … is perpendicular to plane Φ C , that is, perpendicular to plane Φ C on the line segment with P i as the vertex, the right-angle property of the quadrilateral can be guaranteed, that is, the angle remains unchanged; at the same time, according to the generation characteristics of the target grid, that is, the extrusion length of the vertex remains unchanged, then side A i P i has a constant length. The quadrilateral can ensure that the angle and length remain unchanged during the movement, meeting the rigid characteristics; the left side of the folded paper strip can ensure the rigid characteristics. The complete strip is generated by mirror symmetry of the left strip about the x-z plane, which can meet the rigid characteristics of the strip. The target grid model is replicated and tiled along the y-axis by the strip, which can meet the rigid characteristics.

2. The modeling method for fitting a generalized cylindrical surface based on the water bullet origami-derived structure according to claim 1, characterized in that The said p i Two candidate points (c i，x1 , c i，y1 ), (c i，x2 , c i，y2 ) are selected based on the following conditional determination: 1) When i is odd and the z-axis component of the normal vector of face1 is less than 0, take the first candidate point as p i , otherwise take the second candidate point as p i ; 2) When i is even and the z-axis component of the normal vector of face1 is greater than 0, take the first candidate point as p i , otherwise take the second candidate point as p i .

3. The modeling method for fitting a generalized cylindrical surface based on the water bullet origami-derived structure according to claim 1, wherein The content of using the linear regression method to process the vertex data in S15 includes the following: Perform linear regression processing on the data with odd subscripts: 1) Obtain the x - coordinates and y - coordinates of all two - dimensional vertices, denoted as x i , y i , i = 1, 2, …, n odd , where n odd is the number of odd points; 2) Use the linear regression equation to obtain the regression line: where is the regression line coefficient, and the calculation method is as follows: Among them Let the regression line be l c ; 3) Take each p i point's projection onto the straight line l c and let the projection point be p i '; For points with even subscripts, take the x and y coordinates of each vertex, perform linear regression using formula (2), and obtain line l i , take each p i point's projection onto line l l , and let this projection point be p i '.

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