A three-dimensional design method for clothes using principles of origami curve geometry

Abstract

The application provides a garment three-dimensional design method using origami curve geometry principles, comprising: folding a two-dimensional origami pattern; scanning the folded origami pattern using a three-dimensional scanner to obtain a three-dimensional model of the origami pattern; performing mathematical deduction on the three-dimensional model of the origami pattern based on a Frenet frame in differential geometry to obtain a mathematical mapping of the developable surface origami model; obtaining body data of a model and a two-dimensional design drawing of a garment; obtaining a three-dimensional design drawing corresponding to the two-dimensional design drawing of the garment according to the mathematical mapping of the developable surface origami model; calculating the surface curve crease curvature and folding angle of the garment structure based on the three-dimensional design drawing; and accurately obtaining a garment prototype piece structure plate type and fabric consumption according to the surface curve crease curvature and folding angle of the garment structure. The method of origami curve geometry is applied to garment design, which reduces the fabric consumption rate of the garment, improves the clothing production efficiency and the three-dimensional effect of the garment.

Description

Technical Field

[0001] This invention relates to the field of clothing design technology, and in particular to a method for three-dimensional clothing design that utilizes the geometric principles of origami curves. Background Technology

[0002] In existing technologies, during the three-dimensional design of clothing, fashion designers typically first design the garment based on experience, then adjust the overall structure to check if it conforms to the human body. If it does not meet wearing requirements, the overall structure is readjusted and redesigned until the design requirements are met. This design process often wastes a lot of fabric, consumes a significant amount of time, and fails to achieve the desired effect.

[0003] Currently, the three-dimensional design techniques used by fashion designers mainly rely on experience and experimentation, constantly trying different design effects with fabrics. This approach has a high cost of trial and error and lacks scientific three-dimensional design methods. Summary of the Invention

[0004] In view of this, the present invention provides a method for three-dimensional clothing design that utilizes the geometric principles of origami curves, thereby improving the efficiency of clothing production and enhancing the three-dimensional effect of clothing.

[0005] Therefore, the present invention provides the following technical solution:

[0006] A method for three-dimensional clothing design utilizing the geometric principles of origami curves, the method comprising:

[0007] Fold a two-dimensional origami graphic to obtain a three-dimensional origami pattern.

[0008] The origami pattern is scanned using a 3D scanner to obtain a 3D model of the pattern;

[0009] Based on the Frenet frame in differential geometry, the three-dimensional model of the paper pattern is mathematically derived to obtain the mathematical mapping of the developable surface origami model. The mathematical mapping of the developable surface origami model represents the mapping relationship between the two-dimensional planar design drawing and the three-dimensional solid design drawing.

[0010] Obtain two-dimensional clothing design drawings; the two-dimensional clothing design drawings include the model's body measurements and styling;

[0011] The three-dimensional design drawing corresponding to the two-dimensional design drawing of the clothing is obtained by mathematical mapping of the developable origami model.

[0012] Calculate the curvature of the surface curves and folding angles of the garment structure based on the three-dimensional design drawing;

[0013] Based on the curvature of the crease and the folding angle of the garment structure, a developable surface is constructed to obtain the garment prototype piece structure pattern and the corresponding fabric usage.

[0014] Furthermore, obtaining two-dimensional design drawings of the garment includes selecting curve creases based on two basic developable surface types.

[0015] Furthermore, obtaining the two-dimensional design drawing of the garment includes: selecting the basic developable surface types on both sides of the determined crease curve type.

[0016] Furthermore, the crease curve is an arc, and the basic developable surface types on both sides of the curve are selected based on the determined crease curve type, including:

[0017] The basic developable surface type on both sides of the arc is a conical surface.

[0018] Furthermore, based on the Frenet frame in differential geometry, the folding angle of the developable surface in the three-dimensional model of the paper pattern is calculated, including:

[0019] Regarding point P, establish two basic developable surfaces ∑ L ,∑ R Unit frame:

[0020]

[0021] Where: e L =r L ×t L ;e R =r R ×t R ;

[0022]

[0023] Where, ∑ L ,∑ R There are two basic developable surfaces connected by a curved crease. On these surfaces are a pair of straight generatrices that intersect at point P. L r R These are the unit direction vectors of a pair of straight generatrices; t L t R Let e ​​be the unit vector of the tangent line to the crease curve drawn through point P; L e R These are the unit normal vectors of the tangent planes of the two expansion surfaces formed through the pair of straight generatrices; θ is e L e R The angle formed.

[0024] Furthermore, based on the Frenet frame in differential geometry, a mathematical derivation is performed on the three-dimensional model of the paper pattern to obtain the mathematical mapping of the developable surface origami model, including:

[0025] Establish a Frenet frame for the folded curve with point P as the base point. After folding, the curve forms a space curve s. Then, the unit tangent vector of the space curve l′0 at point p is represented by T, the principal normal vector by N, and the binormal vector by B. Draw two unit direction vectors r for the straight generatrices. L ,r R Through vector r L ,r R Construct the tangent planes of the two basic developable surfaces respectively, ∑ L ,∑ R The angles between the tangent plane and the closely spaced plane in the Frenet frame are β and β respectively. L ,β R Sectioning along the Frenet frame plane provides a clearer view of the included angle β. L ,β R When the positional relationship is in a two-dimensional planar state, the crease curve is l0, and at this time the curve only has curvature k. 2D Existing without torsion, when a two-dimensional developable origami model is folded to a certain extent, the curve will transform into a spatial curve s. This curve not only has curvature but also undergoes torsion, generating torsion. The curvature at point P is represented by k. 3D Let the torsion be denoted by τ(s), and let θ(s) be the folding angle function of the spatial crease curve s at point P. Then we have the following formula:

[0026] |β L |=|β R |;

[0027] k 3D (s)cosθ(s)=k 2D (s);

[0028]

[0029]

[0030] The formula for expressing curvature under general parameters is:

[0031]

[0032] The formula for expressing deflection under general parameters is:

[0033]

[0034] According to the above relationship, the unit direction vector on the straight generatrix is ​​expressed as:

[0035]

[0036] Construct tangent planes to the basic developable surfaces on both sides of the crease curve. The angle between the normal vectors of the two tangent planes is the fold angle. Let the normal vectors of the two tangent planes be f. L ,f R Then we have:

[0037]

[0038] Based on the above formula, the normal vectors of the two tangent planes are calculated respectively. Then, based on the two normal vectors f... L ,f R Further, the size of the folding angle of the model is determined by... Obtain the fold angle:

[0039] Furthermore, the structural pattern of the garment prototype and the corresponding fabric usage are obtained, including:

[0040] The required fabric area is calculated as follows:

[0041] Where θ is the foldable angle of the surface; r is the radius of the arc, and R2 and R2 are the maximum and minimum values ​​of the arc radius, respectively.

[0042] Advantages and positive effects of the present invention:

[0043] This invention applies the geometric principles of origami curves to three-dimensional clothing design, providing a novel unfolding method. This new method significantly reduces fabric consumption and structural distortion in clothing. By unfolding the fabric into curved surfaces, it allows for precise angle adjustments and improved fit. This method also increases efficiency, combining experience and theory to reduce trial-and-error time and material waste. Attached Figure Description

[0044] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0045] Figure 1 This is a flowchart illustrating a method for three-dimensional clothing design using the geometric principles of origami curves, as described in an embodiment of the present invention.

[0046] Figure 2 This is a plan view of the paper pattern in an embodiment of the present invention;

[0047] Figure 3In this embodiment of the invention, a 3D scanning instrument scans paper samples at different angles;

[0048] Figure 4 This is a spatial model diagram of the folding angle of the paper pattern in a 3D state, as shown in an embodiment of the present invention.

[0049] Figure 5 This is a schematic diagram of the Frenet frame in an embodiment of the present invention;

[0050] Figure 6 The images show the 2D planar model (left), 3D spatial model (middle), and normal plane cross-section (right) of the foldable angle in this embodiment of the invention.

[0051] Figure 7 This is the torsion analysis model in the embodiments of the present invention;

[0052] Figure 8 This is a calculation example in an embodiment of the present invention. Detailed Implementation

[0053] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0054] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0055] like Figure 1 As shown, a flowchart of a three-dimensional clothing design method using origami curve geometry principles is illustrated in an embodiment of the present invention. The method includes the following steps:

[0056] S1. Fold the two-dimensional origami graphic to obtain a three-dimensional origami pattern; scan the origami pattern using a three-dimensional scanner to obtain a three-dimensional model of the pattern.

[0057] like Figure 2 As shown, it illustrates a planar view of the paper pattern in an embodiment of the present invention, that is, a two-dimensional planar origami diagram. A three-dimensional model of the paper pattern is formed by scanning the paper pattern at different angles using a 3D scanning instrument, as shown below. Figure 3 As shown.

[0058] S2. Mathematical derivation of the model structure based on differential geometry; including: calculating the size of the folding angle of the developable surface origami in the three-dimensional model of the paper pattern based on the Frenet frame in differential geometry; and performing mathematical derivation of the three-dimensional model of the paper pattern based on the Frenet frame in differential geometry to obtain the mathematical mapping of the developable surface origami model, wherein the mathematical mapping of the developable surface origami model represents the mapping relationship between the two-dimensional planar design drawing and the three-dimensional solid design drawing.

[0059] The combination and connection of basic developable surfaces is the foundation of the combination of developable surface units. There are two main ways to connect two basic developable surfaces:

[0060] Firstly, when two basic developable surfaces intersect, the resulting intersecting curve is the connecting line between the two basic developable surfaces. There are five types of intersecting curves: 1. hyperbola, 2. parabola, 3. ellipse, 4. circular arc, and 5. sine and cosine curves. However, different curves have certain requirements on the properties of the surfaces on the left and right sides. For example, the developable surfaces derived from the parabola as the basic curve can only be a conical surface and a cylindrical surface.

[0061] Secondly, the two basic developable surfaces are connected by a shared straight generatrix. The advantage of this method is that the connection between the two basic developable surfaces is smoother and the transition is gentler, resulting in a more visually concise and aesthetically pleasing appearance. Therefore, this connection method is widely used in clothing design and some engineering fields. The case of using a shared straight generatrix as the connecting line is relatively simple and will not be considered in this invention.

[0062] When building a paper-folding structure model, you can first select the curve crease based on two basic developable surface types, or you can select the basic developable surface types on both sides of the curve based on the determined crease curve type. For example, if the crease curve is selected as an arc in the example, then the developable surfaces on the left and right sides can only be conical surfaces.

[0063] Next, the size of the surface folding angle is described. The so-called surface folding angle exists for two fundamental developable surfaces ∑ L ,∑ R The two surfaces are connected by a curved crease. On these two developable surfaces, there exists a pair of intersecting straight generatrices at point P. A tangent line is drawn to the crease curve through point P, and the unit vector of the tangent line is represented by t. L (tR The unit direction vector on the two straight generatrices is represented by r. L ,r R Through these two straight generatrices, construct tangent planes to the two developable surfaces. The unit power of the two tangent planes is e. L ,e R The angle θ between the unit normal vectors of two tangent planes is the fold angle of the two developable surfaces. By analyzing the fold angles on the spatial model and establishing a corresponding Frenet frame at point P, various angles are analyzed based on the Frenet frame, such as... Figure 4 As shown.

[0064] Regarding point P, establish two basic developable surfaces ∑ L ,∑ R Unit frame:

[0065]

[0066] Where: e L =r L ×t L ;e R =r R ×t R ,

[0067] The folding angle θ of a developable curved surface in 3D can be calculated using the above formula.

[0068] The next step is to establish a mapping relationship between two-dimensional planar design drawings and three-dimensional solid design drawings, so that the geometric relationship between two-dimensional and three-dimensional can be clearly expressed through mathematical formulas.

[0069] For foldable surfaces, this embodiment of the invention utilizes the Frenet frame from differential geometry. In space, there exists a curve C: r = r(t), vector T(t) is the unit tangent vector of curve C at r(t), the normalized vector N(t) of the curvature vector is the principal normal vector of curve C at r(t), and vector B(t) = T(t) × N(t) is called the secondary normal vector (binormal vector) of curve C at r(t). Then, the unit orthogonal right-handed frame {r(t): T(t), N(t), B(t)} is called the Frenet frame of curve C at r(t). The plane intersecting the unit tangent vector T(t) and the principal normal vector N(t) is called the osculating plane; the plane intersecting the unit tangent vector T(t) and the secondary normal vector B(t) is called the secondary tangent plane; and the plane intersecting the principal normal vector T(t) and the secondary normal vector B(t) is called the normal plane. Figure 5 As shown.

[0070] Through the Frenet frame and related mathematical derivations, the geometric relationships between two-dimensional (2D) and three-dimensional (3D) can be clearly expressed mathematically. The following is a dynamic frame under general parameters:

[0071]

[0072] Folding this two-dimensional origami figure yields its three-dimensional spatial model, as follows: Figure 6 As shown, a Frenet frame for the folded curve is established with point P as the base point. After folding, the curve forms a space curve s. The unit tangent vector of the space curve l′0 at point p is represented by T, the principal normal vector by N, and the binormal vector by B. Two straight generatrices, unit direction vectors r, are then constructed. L ,r R Through vector r L ,r R Construct the tangent planes of the two basic developable surfaces respectively, ∑ L ,∑ R The angles between the tangent plane and the closely spaced plane in the Frenet frame are β and β respectively. L ,β R Sectioning along the Frenet frame plane provides a clearer view of the included angle β. L ,β R When the positional relationship is in a two-dimensional planar state, the crease curve is l0, and at this time the curve only has curvature k. 2D Existing without torsion, when a two-dimensional developable origami model is folded to a certain extent, the curve will transform into a spatial curve s. This curve not only has curvature but also undergoes torsion, generating torsion. The curvature at point P is represented by k. 3D Let the torsion be denoted by τ(s), and let θ(s) be the folding angle function of the spatial crease curve s at point P. Then we have the following formula:

[0073] |β L |=|β R |

[0074] k 3D (s)cosθ(s)=k 2D (s)

[0075]

[0076]

[0077] The formula for expressing curvature under general parameters is:

[0078]

[0079] The formula for expressing deflection under general parameters is:

[0080]

[0081] According to the above relationship, the unit direction vector on the straight generatrix can be expressed as:

[0082]

[0083] Construct tangent planes to the basic developable surfaces on both sides of the crease curve. The angle between the normal vectors of the two tangent planes is the fold angle. Let the normal vectors of the two tangent planes be f. L ,f R Then we have:

[0084]

[0085] Based on the above formula, the normal vectors of the two tangent planes can be calculated separately. Based on the two normal vectors f... L ,f R Furthermore, the size of the folding angle of the model can be determined from: The folding angle can be obtained as follows:

[0086] Therefore, through mathematical modeling, a certain mathematical relationship is established between the two-dimensional planar graphic of the developable origami and the three-dimensional spatial model. Using this relationship, the mathematical relationship in the 3D model can be obtained from the conditions given by the two-dimensional graphic of the developable origami, and vice versa.

[0087] S3. Derive the content based on the described origami structure and apply it to clothing design. Specifically, this includes:

[0088] S31. Obtain a two-dimensional design drawing of the garment; the two-dimensional design drawing of the garment includes the model's body measurements and styling;

[0089] Specifically, MATLAB software was used to set the curves and make initial design settings. Different colors were used to distinguish the lines.

[0090] S32. Obtain the three-dimensional design drawing corresponding to the two-dimensional design drawing of the clothing based on the mathematical mapping of the developable origami model;

[0091] S33. Calculate the curvature of the surface curve crease and the folding angle of the garment structure based on the three-dimensional design drawing;

[0092] S34. Construct a developable surface based on the curvature of the crease and the folding angle of the garment structure to obtain the garment prototype piece structure pattern and the corresponding precise fabric usage.

[0093] In the above embodiments, a spatial foldability analysis was performed on the developable origami model. Using differential geometry, a mathematical connection was established between the two-dimensional planar state and the three-dimensional spatial state of the developable origami model. The mathematical mapping relationship between the two states was analyzed, and the developability conditions of the spatial origami model were given. During the design process, designers can pre-determine the curvature in the software based on the design curve fold requirements and the basic garment design, setting the expression on a two-dimensional plane. Then, based on the garment's fold design, within the range of foldable angles, the deflection is set, which is the spatial torsion of the garment piece structure. This step mainly involves setting the wearer's size. Unlike the cumbersome methods of traditional garment design, this method, when applied to enterprises, not only improves efficiency and reduces resource waste by predicting fabric usage in advance, but is also more convenient for designers.

[0094] To facilitate understanding, a specific example is provided below to illustrate the method for three-dimensional clothing design using the geometric principles of origami curves in this embodiment of the invention.

[0095] In this example, the shape of the crease is determined in advance to be an arc, which also determines that the basic developable surfaces on both sides of the crease are conical surfaces.

[0096] First, a design drawing is provided. Solid lines represent mountain bends, and dashed lines represent valley bends. Figure 2 As shown.

[0097] like Figure 7 As shown, all points P lie on the arc, and their spatial parametric equations can be expressed as: The tangent vector at any point on the arc can be represented as (clockwise): The principal normal vector is From the normal vector Where t is the angle between the line connecting the center of the circle to any point on the circle and the positive x-axis, and R is the radius of the arc. Here, R is the radius of the circle containing point P. The two straight generatrices r on either side of the curve crease... L ,r R The angle between the line and the tangent is represented by α. L ,α R This indicates that all angles are 90°. From the expression for the normal vector, it is easy to see that the circular arc does not leave its oscillating plane, therefore its torsion is 0.

[0098] Using the above mathematical derivation, assume the angle between the osculating plane on the arc crease and the tangent planes of the developable surfaces on both sides of the crease is: |β L |=|β R If |=45°, then the angle between the normal vectors of the two tangent planes is the fold angle. Let the normal vectors of the two tangent planes be f. L ,f R Then we have:

[0099]

[0100] Depend on The folding angle between the two developable surfaces is further obtained as follows:

[0101]

[0102] Therefore, the required fabric area can be calculated for its application in clothing:

[0103]

[0104] Where r is the radius of the arc, and R2 and R1 are the maximum and minimum values ​​of the arc radius, respectively;

[0105]

[0106] like Figure 8 As shown, the spatial state is intuitively displayed. Using the above calculation formula as the unfolding method can greatly reduce the distortion of the garment pattern structure angle and the area of ​​fabric used. When selecting the unfolded surface, the garment material can be used as the unfolded surface. This method provides a strong theoretical basis for future improvements in the application and unfolding of curved surfaces in the garment field. The overall message is that given the expression of a planar crease curve, its description in spatial state can be obtained using differential geometry.

[0107] As discussed above, the algorithms for calculating the curvature of the creases and the folding angles of the garment structure have been developed. This means that two known conditions are available for the developable surface required by the designer. Based on these conditions, a developable garment surface corresponding to the garment structure pieces can be constructed. In practical application, once the garment style is determined, the fashion designer uses the model's data to create a pattern, obtaining the prototype garment structure pattern.

[0108] This invention elucidates the characteristics of curved surface origami shapes and derives formulas for such origami based on differential geometry, providing a more rational approach to applying curved surface origami techniques in three-dimensional clothing design. It offers a practical method for transforming clothing from two-dimensional to three-dimensional forms, enabling true application and optimizing design thinking. This rational approach addresses challenges in clothing design, reducing time and material costs, and improving production efficiency and the presentation of three-dimensional effects.

[0109] In the several embodiments provided by this invention, it should be understood that the disclosed technical content can be implemented in other ways. The device embodiments described above are merely illustrative; for example, the division of units can be a logical functional division, and in actual implementation, there may be other division methods. For example, multiple units or components may be combined or integrated into another system, or some features may be ignored or not executed. Furthermore, the displayed or discussed mutual coupling, direct coupling, or communication connection can be through some interfaces; the indirect coupling or communication connection of units or modules can be electrical or other forms.

[0110] The units described as separate components may or may not be physically separate. The components shown as units may or may not be physical units; that is, they may be located in one place or distributed across multiple units. Some or all of the units can be selected to achieve the purpose of this embodiment according to actual needs.

[0111] Furthermore, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0112] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods described in the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, read-only memory (ROM), random access memory (RAM), portable hard drives, magnetic disks, or optical disks.

[0113] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for three-dimensional clothing design using the geometric principles of origami curves, characterized in that, The method includes: Fold a two-dimensional origami graphic to obtain a three-dimensional origami pattern. The origami pattern is scanned using a 3D scanner to obtain a 3D model of the pattern; Based on the Frenet frame in differential geometry, the three-dimensional model of the paper pattern is mathematically derived to obtain the mathematical mapping of the developable surface origami model. The mathematical mapping of the developable surface origami model represents the mapping relationship between the two-dimensional planar design drawing and the three-dimensional solid design drawing. Obtain two-dimensional clothing design drawings; the two-dimensional clothing design drawings include the model's body measurements and styling; The three-dimensional design drawing corresponding to the two-dimensional design drawing of the clothing is obtained by mathematical mapping of the developable origami model. Calculate the curvature of the surface curves and folding angles of the garment structure based on the three-dimensional design drawing; Based on the curvature of the crease and the folding angle of the garment structure, a developable surface is constructed to obtain the garment prototype piece structure pattern and the corresponding fabric usage.

2. The method for three-dimensional clothing design based on the geometric principles of origami curves according to claim 1, characterized in that, Obtain a 2D design drawing of the garment, including: selecting curve creases based on two basic developable surface types.

3. The method for three-dimensional clothing design based on the geometric principles of origami curves according to claim 1, characterized in that, Obtaining a 2D design drawing of the garment includes: selecting the basic developable surface types on both sides of the determined crease curve type.

4. The method for three-dimensional clothing design based on the geometric principles of origami curves according to claim 3, characterized in that, The crease curve is an arc. Based on the determined crease curve type, the basic developable surface types on both sides of the curve are selected, including: The basic developable surface type on both sides of the arc is a conical surface.

5. A method for three-dimensional clothing design based on the geometric principles of origami curves according to claim 1, characterized in that, Based on the Frenet frame in differential geometry, a mathematical derivation is performed on the three-dimensional model of the paper pattern to obtain the mathematical mapping of the developable surface origami model, including: The calculation of the folding angle of the developable surface in the three-dimensional model of the paper pattern based on the Frenet frame in differential geometry includes: Regarding point P, establish two basic developable surfaces ∑ L ,∑ R Unit frame: Where: e L =r L ×t L ;e R =r R ×t R ; Where, ∑ L ,∑ R There are two basic developable surfaces connected by a curved crease. On these surfaces are a pair of straight generatrices that intersect at point P. L r R These are the unit direction vectors of a pair of straight generatrices; t L t R Let e ​​be the unit vector of the tangent line to the crease curve drawn through point P; L e R These are the unit normal vectors of the tangent planes of the two expansion surfaces formed through the pair of straight generatrices; θ is e L e R The angle formed.

6. A method for three-dimensional clothing design based on the geometric principles of origami curves according to claim 5, characterized in that, Based on the Frenet frame in differential geometry, a mathematical derivation is performed on the three-dimensional model of the paper pattern to obtain the mathematical mapping of the developable surface origami model, which also includes: Establish a Frenet frame for the folded curve with point P as the base point. After folding, the curve forms a space curve s. Then, the unit tangent vector of the space curve l′0 at point p is represented by T, the principal normal vector by N, and the binormal vector by B. Draw two unit direction vectors r for the straight generatrices. L ,r R Through vector r L ,r R Construct the tangent planes of the two basic developable surfaces respectively, ∑ L ,∑ R The angles between the tangent plane and the closely spaced plane in the Frenet frame are β and β respectively. L ,β R Sectioning along the Frenet frame plane provides a clearer view of the included angle β. L ,β R When the positional relationship is in a two-dimensional planar state, the crease curve is l0, and at this time the curve only has curvature k. 2D Existing without torsion, when a two-dimensional developable origami model is folded to a certain extent, the curve will transform into a spatial curve s. This curve not only has curvature but also undergoes torsion, generating torsion. The curvature at point P is represented by k. 3D Let the torsion be denoted by τ(s), and let θ(s) be the folding angle function of the spatial crease curve s at point P. Then we have the following formula: |β L |＝|β R |； k 3D (s)cosθ(s)=k 2D (s); The formula for expressing curvature under general parameters is: The formula for expressing deflection under general parameters is: According to the above relationship, the unit direction vector on the straight generatrix is ​​expressed as: Construct tangent planes to the basic developable surfaces on both sides of the crease curve. The angle between the normal vectors of the two tangent planes is the fold angle. Let the normal vectors of the two tangent planes be f. L ,f R Then we have: Based on the above formula, the normal vectors of the two tangent planes are calculated respectively. Then, based on the two normal vectors f... L ,f R Further, the size of the folding angle of the model is determined by... Obtain the fold angle:

7. A method for three-dimensional clothing design using the geometric principles of origami curves according to claim 6, characterized in that, Obtain the garment prototype pattern and corresponding fabric usage, including: The required fabric area is calculated as follows: Where θ is the foldable angle of the surface; r is the radius of the arc, and R2 and R2 are the maximum and minimum values ​​of the arc radius, respectively.

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