Garment pattern parameterization design system fusing anthropometry and multi-posture simulation

CN122287129APending Publication Date: 2026-06-26SHENZHEN ORIENTAL YISHANG CLOTHING CO LTD

Patent Information

Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
SHENZHEN ORIENTAL YISHANG CLOTHING CO LTD
Filing Date
2026-04-15
Publication Date
2026-06-26

AI Technical Summary

Technical Problem

Existing parametric design systems for garment patterns have shortcomings in controlling curve continuity and smoothness, especially in key areas such as the intersection of the sleeve cap curve and the undersleeve seam, the connection point between the neckline curve and the center front line, and the transition area between the waist and hip circumference. This results in small but obvious "sharp corners" or "creases" in the generated patterns, affecting the aesthetic quality of the garment and the stability of the simulation system.

Method used

The parametric design system for clothing patterns, which integrates human anthropometrics and multi-pose simulation, acquires three-dimensional scanning point cloud data of the human body and multi-pose joint angle sequence data through the data scanning module, constructs a topological map of the body surface curvature features, generates an initial pattern outline curve set, and constructs local curvature transition patches at splicing nodes through the continuity optimization module to achieve G2 curvature continuity. Combined with the deformation mapping module and the pattern adjustment module, the continuity of the pattern under various poses is optimized.

Benefits of technology

It improves the continuity and smoothness of the garment's curves, enhances the garment's aesthetic appeal and structural rationality, reduces mesh anomalies and stress singularities in virtual simulation, and improves the garment's wearing comfort and movement adaptability.

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Abstract

This invention belongs to the field of computer-aided clothing design technology. It discloses a parametric design system for clothing patterns that integrates anthropometrics and multi-pose simulation. The system extracts surface feature contours from 3D human body scan data and constructs a curvature feature topology map. Based on the topology map, an initial pattern outline is generated, and local curvature transition patches are optimized at splicing nodes to ensure G2 curvature continuity. Multi-pose joint angle data are mapped to a pattern deformation field to evaluate continuity deviations in dynamic states. Control point inverse compensation is implemented for nodes exceeding thresholds to generate a robust pattern that balances static aesthetics and dynamic adaptability. Global curvature energy assessment ensures overall pattern smoothness. This invention improves the aesthetic quality and functional performance of clothing in both static display and dynamic wearing.
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Description

Technical Field

[0001] This invention relates to the field of computer-aided clothing design technology, and more specifically, to a parametric design system for clothing patterns that integrates anthropometrics and multi-pose simulation. Background Technology

[0002] The current field of parametric garment pattern design faces challenges in controlling curve continuity and smoothness, particularly evident in the reconstruction process following parametric adjustments to patterns. When parametric design systems adjust key parameters (such as bust, armhole depth, and shoulder width) according to different body types or style requirements, the pattern outline needs to be regenerated. However, existing algorithms are severely inadequate in handling the continuity at curve connection points. Especially in complex structural areas, such as the intersection of the sleeve cap curve and the underarm seam, the connection point between the neckline curve and the center front line, and the transition area between the waist and hip, it is often impossible to guarantee the continuity of the G1 tangent or, more advancedly, the G2 curvature. This results in the generated pattern exhibiting subtle yet noticeable "sharp corners" or "creases" at the seams. These discontinuities appear abrupt and jarring even in static visual terms, compromising the aesthetic quality of the garment. When these patterns are used in virtual fitting and physical simulation, these continuity defects translate into mesh anomalies and computational singularities. This causes non-physical wrinkles, deformations, and even clipping in these areas of the simulated fabric, further leading to abnormal stress concentration and causing instability or even collapse of the simulation system. In actual production, these seemingly minor continuity issues transform into structural defects in physical garments, affecting wearing comfort and movement adaptability. These problems are particularly prominent in high-end sportswear, custom suits, and functional apparel.

[0003] In view of this, the present invention proposes a parametric design system for clothing patterns that integrates human body measurement and multi-posture simulation to solve the above problems. Summary of the Invention

[0004] To overcome the aforementioned shortcomings of the prior art and to achieve the above objectives, the present invention provides the following technical solution: a parametric design system for clothing patterns that integrates human anthropometrics and multi-pose simulation, comprising: The data scanning module is used to acquire human body 3D scanning point cloud data and multi-pose joint angle sequence data, and extract the body surface feature contour line set from the 3D scanning point cloud data along the preset cross-sectional direction; The body surface feature extraction module is used to calculate the curvature distribution of each contour line in the body surface feature contour line set, extract the curvature extreme points and curvature sign reversal points, and construct a body surface curvature feature topology map. The initial pattern generation module is used to generate an initial pattern outline curve group based on the surface curvature feature topology map, and to identify the set of splicing nodes of adjacent curve segments in the initial pattern outline curve group. The continuity optimization module is used to extract the end tangent vectors and curvature values ​​of the adjacent curve segments on both sides of each splicing node in the splicing node set, construct a splicing continuity state descriptor, and construct local curvature transition patches at each splicing node based on the splicing continuity state descriptor so that each splicing node achieves G2 curvature continuity. The deformation mapping module is used to convert multi-pose joint angle sequence data into deformation displacement fields on the initial pattern contour curve group through bone skin mapping. It recalculates the splicing continuity state descriptor for each splicing node after applying the deformation displacement field and obtains the continuity deviation after deformation. The pattern adjustment module is used to perform reverse compensation adjustment of the control points of the local curvature transition patch corresponding to the splicing node whose deviation exceeds the preset continuity threshold based on the continuity deviation after deformation, and generate a group of posture-robust pattern contour curves. The evaluation and output module is used to perform global curvature energy integral evaluation on the posture robust pattern contour curve group. When the global curvature energy value is lower than the preset smoothness threshold, the final parameterized pattern data is output. The modules are connected via wired and / or wireless means to enable data transmission between them.

[0005] The technical effects and advantages of this invention's parametric design system for clothing patterns, which integrates anthropometrics and multi-pose simulation, are as follows: This invention enhances the continuity and smoothness control of garment pattern curves, resolving the continuity breakage issue during curve reconstruction after parametric adjustment. By achieving high-order curvature continuity at splicing nodes, visual discontinuities and geometric singularities on the pattern outline are eliminated, resulting in smoother and more natural transitions in the generated garment pattern surface. This high-order continuity guarantee mechanism improves the quality of the pattern in complex structural areas, enhancing the aesthetic appeal and structural rationality of the garment. In the virtual simulation stage, the improved curvature continuity reduces mesh anomalies and stress singularities, avoiding common simulation instability and computational divergence problems, significantly improving the reliability and accuracy of virtual samples. The multi-pose adaptive optimization function of this invention enables the pattern to maintain ideal smoothness and continuity under various human dynamic states, improving the garment's wearing comfort and motion adaptability. Attached Figure Description

[0006] Figure 1 This is a schematic diagram of the parametric design system for clothing patterns that integrates human body measurement and multi-pose simulation according to the present invention. Detailed Implementation

[0007] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. 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 are within the scope of protection of the present invention.

[0008] This application provides a parametric design system for clothing patterns that integrates anthropometrics and multi-pose simulation. The system's execution entities include, but are not limited to, those running the system: clothing CAD platforms, digital clothing design workstations, virtual fitting systems, and intelligent garment manufacturing solutions, which can be considered as general computing nodes in this application. The parametric design system includes, but is not limited to, at least one of the following: a cloud-based human data analysis engine, a distributed pose simulation platform, and an intelligent pattern optimizer.

[0009] Please see Figure 1 In this embodiment of the invention, the parametric design system for clothing patterns that integrates anthropometrics and multi-pose simulation includes: The data scanning module acquires 3D human body point cloud data and multi-pose joint angle sequence data. It uses a high-precision 3D human body scanning device to obtain complete point cloud data of a static human body, while simultaneously recording joint angle sequences in different poses using a motion capture system. The point cloud data represents the geometric shape of the human body surface as a set of 3D coordinate points, containing hundreds of thousands of sampling points, accurately depicting subtle changes in the human body's curvature. The joint angle sequences record the skeletal movement state of the human body during typical movements such as walking, sitting, and stretching. The scanning process employs standardized poses and multi-angle fusion technology to ensure the integrity and accuracy of the point cloud data. Furthermore, this module extracts a set of surface feature contour lines from the 3D point cloud data along preset cross-sectional directions. Typically, three orthogonal directions are selected: horizontal, sagittal, and coronal planes. Multiple cross-sections are uniformly sampled in each direction (15-20 in each direction) to form a complete surface contour line network, providing basic data for subsequent feature extraction.

[0010] The body surface feature extraction module calculates the curvature distribution of each contour line in the set of body surface feature contour lines, extracts curvature extrema and curvature sign reversal points, and constructs a topological map of body surface curvature features. This module first provides a precise mathematical parameterization of each body surface contour line, then calculates the curvature distribution on the contour line, identifies key feature points, and finally constructs a topological structure reflecting the morphological characteristics of the human body surface. This process not only captures the geometric features of the human body surface but also establishes the topological relationships between these features, providing an ergonomic foundation for clothing pattern design.

[0011] The initial pattern generation module generates an initial pattern outline curve set based on the body surface curvature feature topology map, and identifies the set of splicing nodes for adjacent curve segments within the initial pattern outline curve set. This module maps 3D human body features onto a 2D pattern plane, constructs segmented curves according to garment structural design rules, and forms a complete pattern outline. Accurate identification of splicing nodes ensures the accuracy of subsequent continuous optimization, laying the foundation for achieving a smooth pattern.

[0012] The continuity optimization module extracts the end tangent vectors and curvature values ​​of the adjacent curve segments on both sides of each splicing node in the splicing node set, constructs a splicing continuity state descriptor, and builds local curvature transition patches at each splicing node based on the splicing continuity state descriptor, so that each splicing node achieves G2 curvature continuity. G2 curvature continuity is a key technology to ensure the smoothness of the garment pattern. By precisely controlling the connection relationship between curves, it eliminates visual discontinuities and stress concentrations that may occur during actual wear, improving the aesthetics and comfort of the garment.

[0013] The deformation mapping module transforms multi-pose joint angle sequence data into a deformation displacement field on the initial pattern contour curve group through skeletal skinning mapping. It then recalculates the splicing continuity state descriptor for each splicing node after applying the deformation displacement field, obtaining the continuity deviation after deformation. This module converts human motion into pattern deformation, assesses the impact of posture changes on the continuity of pattern curvature, and provides a quantitative basis for subsequent pattern adjustments.

[0014] The pattern adjustment module is used to perform reverse compensation adjustment of control points for local curvature transition patches corresponding to splicing nodes where the deviation exceeds a preset continuity threshold, based on the continuity deviation after deformation, to generate a set of posture-robust pattern contour curves. This module uses intelligent adjustment algorithms to ensure the pattern maintains good curvature continuity under various postures, improving the garment's adaptability and comfort in dynamic states.

[0015] The evaluation and output module performs a global curvature energy integral evaluation on the posture-robust pattern contour curve group. When the global curvature energy value is lower than a preset smoothness threshold, the final parametric pattern data is output. This module calculates and evaluates the overall smoothness of the pattern through mathematical integration, ensuring that the final output pattern satisfies both multi-posture adaptability and maintains good aesthetic characteristics.

[0016] The modules are connected via wired and / or wireless means to enable data transmission between them.

[0017] In this embodiment of the invention, the detailed implementation steps for constructing a topological map of body surface curvature features include: For each body surface feature contour line, cubic B-spline parametric fitting is performed to obtain a continuously differentiable parametric curve representation. Parametric fitting is a fundamental step in the mathematical representation of contour lines, transforming a discrete set of points into a continuous function form. The fitting process uses cubic B-spline basis functions, determines the control point positions using the least squares method, and generates parametric equations that accurately describe the contour line. Cubic B-splines have the characteristics of good local controllability and high computational efficiency, making them suitable for representing complex human body contours. For a cubic B-spline curve containing n control points, its parametric equation form is: ; in, For parametric curve representation, For the coordinates of the control points, For cubic B-spline basis functions, These are parameter variables. The basis functions are calculated using the Cox-de-Boor recursive formula, ensuring that the curves have... Continuity. The number of control points is dynamically adjusted according to the complexity of the contour, usually 1 / 5 to 1 / 10 of the size of the original point set, which reduces computational complexity while ensuring fitting accuracy.

[0018] The signed curvature value sequence at sampling points of equal arc length is calculated along a parametric curve. Curvature calculation is a core step in identifying contour features, quantifying the degree and direction of curve curvature. The calculation process first determines the sequence of equal arc length parameter values ​​through numerical integration with an adaptive step size, ensuring that the sampling points are uniformly distributed on the curve; then, the curvature value is calculated at each sampling point using a standard formula from differential geometry. For a planar curve, the signed curvature calculation formula is: ; in, For parameters The signed curvature at that point, , For the first derivative component of the curve, , This represents the second derivative component. The sign of curvature indicates the direction of curvature; a positive value indicates bending to the left, and a negative value indicates bending to the right. The larger the absolute value of curvature, the greater the degree of curvature. The number of sampling points is usually set as a function of the curve length to ensure sufficient morphological details are captured.

[0019] Extreme value detection is performed on a signed curvature value sequence to extract the coordinates of local curvature maxima and minima. Extreme value detection is a crucial step in identifying key feature points of the contour, locating inflection points where the curvature changes. The detection process employs an adaptive window local extremum detection algorithm, scanning the curvature sequence and marking points where the curvature value is greater than or less than that of all points in its neighborhood as extreme values. To improve the robustness of the detection, a significance threshold is introduced to filter out false extreme values ​​with minimal fluctuations. The significance is calculated based on the range of local curvature changes, using the following formula: ; in, For the first Significance score of each extreme point The curvature value at that point. , , These represent the average, maximum, and minimum curvature values ​​within the local window. Extreme points with a significance score greater than a preset threshold (usually 0.3-0.5) are retained as valid feature points, and their curve parameter values ​​and spatial coordinates are recorded to provide node data for subsequent topology graph construction.

[0020] Zero-crossing points are detected in a sequence of signed curvature values ​​where the curvature sign changes from positive to negative or vice versa, and these points are marked as curvature inflection points. Curvature inflection point identification is a crucial step in capturing the curvature transformation characteristics of curves, locating the points where the curve's bending direction changes. The detection process traverses the curvature sequence, and when adjacent sampling points have opposite curvature signs, linear interpolation is used to accurately locate the zero-crossing point. To improve recognition accuracy, local polynomial fitting is used to optimize the zero-crossing point parameter values ​​and eliminate noise interference. Curvature inflection points typically correspond to the boundary of surface curvature changes on the human body, such as the transition region from convex to concave surfaces, and are key locations that require special attention in clothing pattern design. Each curvature inflection point records its curve parameter values, spatial coordinates, and local rate of curvature change, providing supplementary node information for topology graph construction.

[0021] Using curvature extrema and inflection points as nodes, and the average curvature of curve segments between adjacent nodes as edge attributes, a topological graph of body surface curvature features is constructed. Topological graph construction is a crucial step in integrating discrete feature points into a structured representation, forming a network model reflecting the morphological features of the human body surface. The construction process first arranges all curvature extrema and inflection points in parameter order, establishing connections between adjacent feature points on the same contour line; then, lateral connections are established between adjacent feature points on different contour lines to form a complete feature network; finally, the attribute values ​​of each edge are calculated, including length, average curvature, and rate of change of curvature. The formula for calculating average curvature is: ; in, For nodes and The average curvature of the curve segments between them Let be the arc length of the curve segment, and the integration range be the nodes. and Corresponding parameter values and The integral is calculated numerically using the adaptive Simpson method to ensure accurate capture of curvature distribution. The final topological map of body surface curvature features is stored in a graph data structure, including a node table and an edge table. The node table records the type, location, and local curvature characteristics of feature points, while the edge table records the connection relationships and edge attributes between nodes, providing a comprehensive representation of human morphological features for subsequent pattern generation.

[0022] In this embodiment of the invention, the detailed implementation steps for generating an initial pattern outline curve group based on the surface curvature feature topology map and identifying the splicing node set of adjacent curve segments in the initial pattern outline curve group include: Node mapping in the topological map of the body's surface curvature features is applied to corresponding positions on the unfolded 2D pattern surface, generating a set of pattern feature control points. Node mapping is a crucial transformation step from 3D human body to 2D pattern, preserving morphological fidelity by maintaining the topological relationships of key feature points. The mapping process employs a geometric flattening algorithm based on minimum deformation energy. First, the human body surface is divided into unfoldable regions. Then, conformal mapping technology is applied to each region to transform the coordinates of 3D feature points to the 2D plane, while minimizing the seam stress at the region boundaries. The objective function of the mapping transformation is: ; in, Total deformation energy For area distortion energy, For angular distortion energy, The energy for feature point distance distortion. , , These are weighting coefficients, dynamically adjusted according to clothing type. The feature point mapping process pays special attention to maintaining the dimensional proportions and curvature characteristics of key body parts (such as bust, waist, and hip circumference) to ensure the generated pattern conforms to ergonomic principles.

[0023] Based on garment structural design rules, piecewise cubic B-spline curves are constructed between the set of pattern feature control points to form an initial pattern outline curve set. Curve construction is the core step in defining the pattern shape, precisely expressing the pattern outline through mathematical curves. The construction process first determines the pattern structural rules according to the garment type and style, such as straight, fitted, or loose; then, it divides the pattern into sections according to structural lines, such as the front, back, and sleeve pieces; finally, within each section, the corresponding feature control points are connected to form a closed outline. Piecewise cubic B-spline curves are used for connection, maintaining the smoothness of the shape while providing sufficient flexibility for local control. To ensure the curve shape is reasonable, curvature constraints are set at key feature points to ensure that the curvature of the curve at these locations meets the garment design requirements. At the same time, garment manufacturing experience parameters are introduced into the functional areas of the pattern (such as the neckline, armhole, and shoulder line) to adjust the curve shape to optimize wearing comfort and appearance.

[0024] The process iterates through all curve segment endpoints in the initial pattern outline curve group, identifying endpoints shared by two or more curve segments and marking them as splicing nodes. Splicing node identification is fundamental for subsequent continuous optimization, precisely locating key positions requiring special handling. The identification process employs an efficient endpoint matching algorithm, constructing an endpoint hash table to group endpoints with the same or similar coordinates into the same splicing node. To handle numerical errors, a spatial tolerance threshold is set; endpoints with a distance less than the threshold are considered to be at the same location. Each splicing node is assigned a unique identifier, recording its spatial coordinates and information about all associated curve segments, forming a splicing node set. Particularly important key splicing locations include pattern transitions, seam intersections, and functional structural line intersections. These locations significantly impact the aesthetics and comfort of the garment, requiring high-precision continuous control.

[0025] For each splicing node, the identifiers of all associated curve segments and the parameter directions of each curve segment at that splicing node are recorded, forming a splicing node set. Recording parameter directions is a crucial step in ensuring the correctness of curve connections, defining the entry or exit direction of curve segments at the splicing point. The recording process first determines the parameterized direction of each curve segment, typically using a standard of increasing parameter values ​​from the start to the end point. Then, for each splicing node, it is determined whether it serves as the start or end point of an associated curve segment, and the corresponding parameter direction flag is recorded (1 indicates exit, -1 indicates entry). Parameter direction information is essential for subsequent continuity analysis and optimization, ensuring the correct calculation of the curve's tangent and curvature at the splicing point and avoiding continuity errors caused by direction confusion. The complete information of the splicing node set includes node coordinates, a list of associated curve identifiers, and a corresponding list of parameter directions, providing comprehensive data support for the construction of the continuity state descriptor.

[0026] In this embodiment of the invention, the detailed implementation steps for constructing local curvature transition patches at each splicing node based on the splicing continuity state descriptor to achieve G2 curvature continuity at each splicing node include: For each splicing node, the end tangent vectors and curvature values ​​of the curve segments on both sides are read from the splicing continuity state descriptor. Continuity state reading is the starting point for optimization, obtaining the current connection state information at the splicing point. The reading process first determines the entry or exit state of the associated curve segment based on the parameter direction information of the splicing node; then, it calculates the tangent vector and curvature value of each curve segment at the splicing point. The tangent vector is obtained using the first derivative of the curve, and the curvature value is calculated using the second derivative. For cubic B-spline curves, the derivative calculation utilizes the derivative basis function expression to ensure high-precision differential results. The tangent vector is normalized, retaining only the direction information for subsequent angle analysis; the curvature value retains its sign, reflecting the bending direction and degree of the curve. The continuity state descriptor fully records the geometric characteristics at the splicing point, providing a quantitative basis for judging the continuity level and subsequent optimization.

[0027] The algorithm determines whether the angle between the tangent vectors on both sides is less than a preset tangent deviation threshold. If not, it fine-tunes the position of the nearest control point on both sides of the curve segment to align the tangent directions, achieving G1 tangent continuity. Ensuring G1 continuity is the first stage of optimization, eliminating visually sharp angles at the splicing points. The determination process calculates the angle between the tangent vectors using the vector dot product formula: ; in, Angle and Let be the tangent vectors of the curves on both sides. When When the angle exceeds a preset threshold (typically 5°-10°), tangent alignment adjustment is required. This adjustment uses a control point fine-tuning method, changing the tangent direction at the splice point by moving control points closer to the splice point. For cubic B-spline curves, the tangent direction at splice point P is determined by the two nearest control points, and the adjustment formula is: ; in, For splicing control points, This is the control point preceding the splicing point. The adjusted position. The average direction (normalized) of the tangent vectors on both sides. To adjust the strength coefficient (typically 0.8-1.2), the tangent angle is gradually reduced through iterative adjustments until the G1 continuity standard is achieved.

[0028] Based on the G1 tangent continuity, the arithmetic mean of the curvature values ​​on both sides is used as the target curvature, and an auxiliary control point is introduced on each side of the splicing node. G2 continuity optimization is the core step in improving smoothness, eliminating visual and physical problems caused by curvature discontinuities at the splicing point. The process of introducing auxiliary control points first calculates the target curvature, i.e., the arithmetic mean of the original curvatures on both sides; then, auxiliary control points are placed at appropriate distances along the tangent direction on both sides of the splicing point, with the initial positions determined by empirical formulas. ; in, To determine the initial distance from the auxiliary control point to the splicing point, The average length of the associated curve segment. The target curvature value is used. When the target curvature is close to zero, a fixed distance is taken to avoid numerical instability. The introduction of auxiliary control points provides degrees of freedom for subsequent curvature adjustments. By precisely controlling the position of these points, a smooth transition of curvature at the splicing point can be achieved.

[0029] The position coordinates of the auxiliary control points are solved by simultaneously solving the target curvature constraint and the minimum bending energy condition, thus constructing a local curvature transition patch. Control point position optimization is a key step in the continuous implementation of G2, and the optimal layout is solved through mathematical optimization. The optimization problem is set as constrained energy minimization, the objective function is the bending energy of the local curvature transition patch, and the constraint condition is that the curvature at the splicing point equals the target curvature. The integral expression for bending energy is: ; in, For bending energy, For parameters curvature at that point Let be an arc-length infinitesimal element. To solve this optimization problem, the constrained optimization is transformed into an unconstrained problem using the Lagrange multiplier method, and the Lagrange function is constructed as follows: ; in, For the auxiliary control point position vector, It is a Lagrange multiplier. These are the splicing point parameter values. Solved... and The auxiliary control point positions that satisfy G2 continuity and minimize bending energy are obtained. The specific solution employs Newton's iteration method, utilizing analytical gradients and the Hessian matrix to accelerate convergence. The resulting local curvature transition patch seamlessly connects to the original curve, achieving G2 curvature continuity at the splicing point, significantly improving the smoothness and aesthetics of the pattern.

[0030] Verify the continuity of the second derivative of the local curvature transition patch at the splicing node to confirm G2 curvature continuity. Verification is the final step in quality assurance, ensuring that the optimization results meet design requirements. The verification process calculates the curvature values ​​of the curves on both sides of the splicing point and compares whether their differences are within the allowable error range. For numerical calculations, a relative error tolerance of 1e-4 is set; when the relative difference in curvature on both sides is less than this value, the G2 continuity standard is considered to be met. In addition, a visual inspection is performed, using curvature gradient visualization and reflective ray rendering to intuitively assess the smoothness of the splicing point. For splicing points that pass verification, they are marked as G2 optimization completed; for those that fail verification, the reasons for failure are recorded, optimization parameters are adjusted, and the optimization process is re-executed. Complete verification results are recorded in the system log, providing detailed evidence for pattern quality assessment.

[0031] In this embodiment of the invention, the detailed implementation steps for converting multi-pose joint angle sequence data into a deformation displacement field on an initial pattern contour curve group through skeleton skinning mapping include: For each pose frame in the multi-pose joint angle sequence data, the spatial transformation matrix of each bone segment is calculated based on the skeletal kinematic chain. Skeletal transformation calculation is fundamental to pose mapping, converting joint angles into spatial displacements. The calculation process employs forward kinematics, starting from the root bone (usually the pelvis) and calculating the position and orientation of each bone segment hierarchically. For joint angle representation, Euler angles or quaternions are used, converted to an expression in the same coordinate system via rotation matrices. The spatial transformation matrix for each bone segment is a 4×4 homogeneous transformation matrix, containing both rotation and translation components, fully describing the spatial changes of the skeleton from the standard pose to the current pose. The calculation of the transformation matrix considers joint motion constraints to ensure that the generated pose conforms to human biomechanical characteristics. For complex poses, inter-joint cooperative constraints are also introduced to simulate the muscle linkage effects in real human movement, improving the naturalness and accuracy of the pose.

[0032] The control points of the initial pattern outline curve group are weighted and transformed using skinning weights and a spatial transformation matrix to obtain the deformation position of each control point in the pose frame. Skinning mapping is a key step in converting skeletal motion into surface deformation, enabling the pattern to change naturally with the pose. The mapping process first assigns skinning weights to each control point of the pattern, reflecting the degree to which it is affected by each skeleton. The weight allocation is based on the relative position of the control point to the skeleton and the structural characteristics of the garment. Then, a linear hybrid skinning algorithm is used to calculate the deformation position of the control points. The deformation calculation formula is: ; in, Control points The position after deformation, The initial position, Control points Bone The weight of influence For bones The spatial transformation matrix, To influence the number of bones at this control point. The weights satisfy... and To ensure the physical plausibility of the deformation, an improved dual quaternion skinning algorithm is used in key areas (such as near joints) to reduce the volume loss problem of traditional linear hybrid skinning when rotating at large angles, thus maintaining the natural folds and tension distribution of the garment.

[0033] The displacement difference of each control point before and after deformation is calculated to construct the deformation displacement field for that attitude frame. The displacement field construction is the foundation of deformation analysis, quantifying the impact of attitude changes on the pattern. The construction process calculates the position difference vector of each control point before and after deformation, recording its direction and magnitude. The formula for calculating the displacement difference is: ; in, Control points The displacement vector is used. For ease of subsequent analysis, the displacement field is decomposed into normal and tangential components, reflecting the tensile / compression and shear deformation of the garment, respectively. The displacement field visually demonstrates the degree of influence of posture changes on different areas of the pattern, identifying areas of concentrated deformation and stable areas, providing a spatial distribution basis for subsequent continuity assessment and pattern adjustment. High-risk areas of particular concern include parts with large joint range of motion (such as elbows and knees) and structural lines where garment stress is concentrated (such as armholes and shoulder lines). These areas are usually prone to continuity failure and require focused analysis and optimization.

[0034] The deformation displacement fields are arranged according to the attitude frame sequence to form a control point displacement time series, which is used to re-evaluate the continuity state of each splicing node. Time series organization is the foundation of dynamic analysis, revealing the temporal characteristics of pattern changes. The organization process arranges the displacement fields under each attitude frame in chronological order, forming a four-dimensional dataset (x-coordinate, y-coordinate, z-coordinate, and time), completely recording the deformation history of the pattern during the action. For ease of analysis, the displacement time series of the relevant control points of each splicing node are extracted to form the dynamic behavior characteristics of the node. Based on the displacement time series, the splicing continuity state descriptor of each splicing node under different attitudes is recalculated to evaluate the degree of G2 continuity preservation. The continuity deviation is calculated by comparing the difference between the current curvature and the target curvature. Nodes with excessive deviations are marked as targets requiring optimization, providing precise positioning for subsequent pattern adjustments. The dynamic evaluation not only examines continuity under extreme attitudes but also analyzes the continuity change trend throughout the entire action process, comprehensively evaluating the attitude adaptability performance of the pattern.

[0035] In this embodiment of the invention, the detailed implementation steps for performing reverse compensation adjustment of control points on the local curvature transition patch corresponding to the splicing node whose deviation exceeds a preset continuity threshold, based on the continuity deviation after deformation, include: Nodes whose continuity deviation exceeds a preset continuity threshold after deformation are marked as nodes requiring compensation. Node selection is a crucial step in optimizing resource concentration, ensuring that adjustments focus on problem areas. The selection process first sets a continuity deviation threshold, typically 15%-25% of the target curvature, dynamically adjusted according to clothing type and requirements. Then, it iterates through all splicing nodes, calculates the maximum continuity deviation under each posture, and compares it with the threshold for selection. To comprehensively evaluate node status, a multi-posture integrated scoring mechanism is used, comprehensively considering the maximum value, average value, and frequency of exceeding the threshold to calculate a risk score. ; in, For nodes Risk score, For nodes In posture The curvature value below, For the target curvature value, For the total number of postures, This represents the number of poses exceeding the threshold. , , These are the weighting coefficients, and Nodes with higher risk scores are marked as nodes requiring compensation and proceed to the next stage of the optimization process. This risk-score-based selection method ensures the effective allocation of optimization resources, focusing on addressing the most significant problem areas.

[0036] For each node to be compensated, a sequence of continuous deviations across all attitude frames is extracted, and the maximum deviation value and corresponding extreme attitude frame are identified. Extreme attitude identification is a crucial step in determining the optimization target, pinpointing the most critical operating conditions requiring improvement. The identification process analyzes the time series of continuous deviations, uses a peak detection algorithm to identify the time point with the largest deviation, and extracts the corresponding attitude parameters. For multiple attitudes with similar deviation peaks, cluster analysis is used to identify representative extreme attitude types, preventing optimization from being overly focused on specific attitudes and neglecting overall performance. Extreme attitude identification considers not only the absolute deviation value but also the duration and rate of change of the deviation, ensuring that the truly critical challenging attitudes are captured. For each identified extreme attitude, its joint angle configuration, resulting pattern deformation, and continuity failure mode are recorded, providing detailed reference for subsequent inverse compensation.

[0037] Using extreme attitude frames as extreme conditions, the required position offset for the auxiliary control points of the local curvature transition patch is calculated in reverse, reducing the curvature deviation at the stitching node to within a preset continuity threshold under this extreme condition. Reverse compensation is the core optimization strategy, adjusting the original design based on predicted deformation to achieve dynamic adaptability. The calculation process establishes a sensitivity analysis model of the control point position and curvature changes, using Taylor expansion to approximate the local relationships. ; in, The change in curvature Let be the Jacobian matrix of curvature with respect to the control point positions. Let be the change in the position of the control point. The goal is to solve for... This ensures that the curvature deviation after deformation remains within acceptable limits. Since the curvature's dependence on position is non-linear, an iterative solution strategy is employed, updating the Jacobian matrix in each iteration to gradually approach the optimal solution. To ensure the stability of the solution, a regularization term is introduced to prevent excessive position adjustments, and the optimization objective is modified as follows: ; in, is the regularization coefficient, determined through cross-validation. The analytical solution is: ; The positional offset calculated using this method can effectively compensate for curvature deviations under extreme postures, thereby improving the dynamic adaptability of the pattern.

[0038] The position offset is superimposed onto the initial position coordinates of the auxiliary control points. The local curvature transition patch is then reconstructed using the updated auxiliary control points, and the continuity deviation is verified to not exceed a preset continuity threshold across all attitude frames, forming a set of attitude-robust pattern contour curves. Patch reconstruction is the final step in the optimization process, translating the calculation results into actual design modifications. The reconstruction process first updates the auxiliary control point positions, applying the calculated offsets; then, it regenerates the local curvature transition patch, ensuring a seamless connection with the original curve; finally, it verifies the continuity performance across all attitudes to confirm the optimization effect. Verification uses the Monte Carlo method, randomly sampling multiple configurations in the attitude space to comprehensively evaluate the dynamic performance of the optimized pattern. For patches that pass verification, the optimization results are retained; for patches with remaining issues, deficiencies are recorded, optimization parameters are adjusted, and the calculation is re-executed. The final generated set of attitude-robust pattern contour curves maintains good smoothness in both static and dynamic states, achieving the design goal of multi-attitude adaptation. The entire optimization process records detailed adjustment history and verification data, providing complete technical support and decision-making basis for pattern design.

[0039] In this embodiment of the invention, the detailed implementation steps for performing global curvature energy integral evaluation on the attitude-robust profile curve group include: The arc-length integral of the square of curvature is calculated for each curve segment in the attitude-robust profile curve set, and denoted as the bending energy of that curve segment. Bending energy calculation is fundamental to smoothness assessment, quantifying the overall degree of curvature of the curve. The calculation process involves uniformly sampling along the curve parameters, calculating the square of curvature at each sampling point, and then accumulating the total energy using a numerical integration method. For parametric curves, the bending energy integral formula is: ; in, For curve segments The bending energy, For parameters curvature at that point The velocity vector magnitude is represented by the arc length infinitesimal element. The integration employs an adaptive Simpson method, dynamically adjusting the sampling density based on the drastic change in curvature to ensure computational accuracy. Bending energy visually reflects the "unsmoothness" of the curve; higher energy indicates a more tortuous curve, potentially leading to visual discomfort and wearing discomfort in clothing. The bending energy of each curve segment is recorded along with its length and functional location, providing detailed information for subsequent global evaluation and optimization.

[0040] The global curvature energy value is obtained by summing the bending energies of all curve segments. Global energy calculation is a crucial step in the overall evaluation, providing a comprehensive measure of pattern smoothness. The calculation process is simple and direct: the bending energies of all curve segments are linearly added to obtain a single index reflecting the overall smoothness of the pattern. To account for the impact of curve length differences, a length-weighted average is sometimes used, assigning higher weights to longer curves to more accurately reflect visual saliency. The global curvature energy value is the primary objective function for pattern optimization, directly influencing the final output decision. Generally, a lower energy value indicates a smoother pattern, but excessively low energy may lead to an overly flat pattern lacking necessary design features; therefore, a proper balance needs to be found between smoothness and shape.

[0041] The ratio of the global curvature energy value to the global curvature energy value of the initial pattern profile curve set is calculated and denoted as the smoothness degradation coefficient. Calculating the degradation coefficient is a crucial step in evaluating the optimization effect, quantifying the impact of multi-pose adaptive optimization on static smoothness. The calculation formula is a simple ratio division: ; in, The coefficient for deterioration of smoothness. The global curvature energy value for the attitude-robust pattern. This represents the global curvature energy value of the initial pattern. Ideally, the degradation coefficient should be close to 1, indicating that the optimization process has maintained the original smoothness; a coefficient greater than 1 indicates that some static smoothness has been sacrificed to achieve multi-pose adaptability, with a larger coefficient indicating greater sacrifice. Setting an appropriate preset degradation threshold (usually 1.2-1.5) is a key decision in balancing multi-pose adaptability and static aesthetics, and different threshold standards can be used for different types of clothing.

[0042] When the smoothness degradation coefficient exceeds a preset degradation threshold, control point optimization is re-executed on the curve segment with the highest bending energy to minimize the bending energy of that segment without breaking the G2 continuity constraint of its end splice nodes. Control point optimization is a fine-tuning step to improve smoothness, enhancing aesthetic quality while maintaining key performance characteristics. The optimization process first identifies the curve segments with the highest bending energy, which are typically the main contributors to smoothness degradation. Then, an optimization problem is constructed, with the bending energy of the curve as the objective function and the position of the endpoints, tangents, and curvature as constraints. Finally, the optimal control point configuration is solved using numerical optimization methods. The optimization employs interior-point methods or sequential quadratic programming algorithms, which can efficiently handle constrained nonlinear optimization problems. To avoid over-flattening, shape preservation terms can be added to ensure that the optimized curve retains the original design intent and characteristics. After each optimization, the global curvature energy and degradation coefficient are recalculated, and the improvement effect is evaluated until a satisfactory result is achieved or the maximum number of iterations is reached.

[0043] The optimization process iteratively executes until the smoothness degradation coefficient is no greater than a preset degradation threshold, outputting the final parametric pattern data. Iterative optimization is an adaptive process that achieves balance by repeatedly fine-tuning to reach the predetermined goal. The iterative process is designed with a greedy strategy, processing the curve segment with the highest energy at each iteration to gradually improve global smoothness. To avoid infinite loops, a maximum number of iterations (usually 10-20) and a minimum improvement threshold are set, and optimization ends when these termination conditions are met. The final output parametric pattern data includes complete curve definitions (control point coordinates and parameter settings), splicing node information, multi-pose verification results, and smoothness evaluation data, providing comprehensive technical specifications for garment production. The output format supports mainstream garment CAD system standards, ensuring seamless integration with existing workflows. The final pattern maintains good aesthetic characteristics in a static state and maintains necessary smoothness and comfort in dynamic poses, achieving the comprehensive optimization goal of integrating anthropometrics and multi-pose simulation.

[0044] In this embodiment of the invention, the detailed implementation steps for solving the position coordinates of the auxiliary control point by simultaneously solving the target curvature constraint and the minimum bending energy condition include: Using the coordinates of the auxiliary control points as optimization variables, an equality constraint is established with the curvature at the splicing node equal to the target curvature. The variable and constraint settings form the foundation of the optimization problem, clarifying the solution objective and boundary conditions. The setting process first defines the optimization variables, namely the four coordinate components of the two auxiliary control points (two coordinates per point in the two-dimensional case); then, an equality constraint is constructed, requiring the curvature at the splicing node to be precisely equal to the preset target curvature value. For cubic B-spline curves, the curvature at the node can be calculated using the analytical formula for the control point positions, forming the constraint equation. To simplify the problem, the positions of the auxiliary control points can be parameterized as distances along the tangent direction and offsets along the normal direction, reducing the degrees of freedom and making the optimization more stable. The constraint equation ensures the core requirement of G2 continuity, a condition that must be strictly satisfied during the optimization process.

[0045] The objective function is the integral of the squared arc length of the curvature of the local curvature transition patch within the neighborhood of the auxiliary control point. The design of the objective function is crucial for ensuring optimization effectiveness, defining the criteria for evaluating the optimal solution. The design process uses the bending energy integral as the primary objective; this integral has a clear physical meaning, corresponding to the overall curvature of the curve. Minimizing the bending energy usually yields the visually smoothest curve. The integration range of the objective function is limited to the local region influenced by the auxiliary control point, typically the span of a control point on each side of the splicing point. This allows for focused optimization of the smoothness of the transition region without affecting the curve shape far from the splicing point. The objective function is mathematically expressed as the standard bending energy integral, calculated using high-precision numerical methods, providing accurate evaluation criteria for subsequent optimization.

[0046] The Lagrange multiplier method is used to merge equality constraints with the objective function, constructing an unconstrained optimization problem. Problem transformation is a key technical aspect of optimization computation, converting constrained problems into more solvable forms. This transformation process introduces Lagrange multipliers to construct a Lagrange function, integrating the constraints into the objective function. The expression for the Lagrange function is: ; in, and Two auxiliary control points and Location coordinates, It is a Lagrange multiplier. Let the curvature be the curvature at the splicing point. Let be the objective curvature value. By incorporating constraints into the objective function, the original problem is transformed into finding the stationary points of the Lagrange function, i.e., solving for the points where its first derivative is zero. This transformation allows the problem to be solved using standard unconstrained optimization methods, greatly simplifying the computational process.

[0047] For an unconstrained optimization problem, the stationary points are determined to obtain the coordinates of auxiliary control points that satisfy G2 continuity and minimize bending energy. Stationary point determination is the final step in obtaining the optimal solution, using numerical methods to find the extreme points of the objective function. The solution process first calculates the partial derivatives of the Lagrangian function with respect to each variable, constructing a gradient vector; then, it uses Newton's method or a quasi-Newton method to find the points where the gradient is zero. For this problem, due to the complexity of the bending energy integral, partial derivatives usually need to be calculated using numerical differentiation methods. The solution algorithm adopts the L-BFGS method, which is both efficient and stable in handling medium-sized nonlinear optimization problems. To improve the success rate, a multi-starting-point strategy is used, optimizing from different initial points and selecting the best result to avoid getting trapped in local optima. After optimization, the results are verified to ensure that the original constraints and optimality conditions are met; if so, the solution is accepted; otherwise, the parameters are adjusted and the solution is re-solved. The final coordinates of the auxiliary control points ensure both G2 curvature continuity at the splicing point and minimizes the bending energy in the transition region, achieving a smooth connection with local optima.

[0048] This invention achieves a parametric design system for clothing patterns by integrating anthropometry and multi-pose simulation. The curvature analysis and G2 continuous optimization method of this invention can precisely control the smoothness of the pattern, while multi-pose robust adjustment ensures the comfort and aesthetics of the clothing in dynamic states.

[0049] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

[0050] It should be noted that all formulas in this manual are calculated by removing dimensions and taking their numerical values. The formulas are derived from software simulations based on a large amount of collected data to obtain the most recent real-world results. The preset parameters and thresholds in the formulas are set by those skilled in the art according to the actual situation.

[0051] Although embodiments of the invention have been shown and described, those skilled in the art will understand that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the claims and their equivalents.

Claims

1. A parametric design system for clothing patterns that integrates anthropometrics and multi-pose simulation, characterized in that: include: The data scanning module is used to acquire human body three-dimensional scanning point cloud data and multi-pose joint angle sequence data, and extract a set of body surface feature contour lines from the three-dimensional scanning point cloud data along a preset cross-sectional direction; The body surface feature extraction module is used to calculate the curvature distribution of each contour line in the body surface feature contour line set, extract the curvature extreme points and curvature sign reversal points, and construct a body surface curvature feature topology map. The initial pattern generation module is used to generate an initial pattern outline curve group based on the surface curvature feature topology map, and to identify the set of splicing nodes of adjacent curve segments in the initial pattern outline curve group. The continuity optimization module is used to extract the end tangent vectors and curvature values ​​of the adjacent curve segments on both sides of each splicing node in the splicing node set, construct a splicing continuity state descriptor, and construct local curvature transition patches at each splicing node based on the splicing continuity state descriptor so that each splicing node achieves G2 curvature continuity. The deformation mapping module is used to convert the multi-pose joint angle sequence data into a deformation displacement field on the initial pattern contour curve group through bone skin mapping, and to recalculate the splicing continuity state descriptor for each splicing node after applying the deformation displacement field to obtain the continuity deviation after deformation. The pattern adjustment module is used to perform reverse compensation adjustment of the control points of the local curvature transition patch corresponding to the splicing node whose deviation exceeds the preset continuity threshold according to the continuity deviation after deformation, and generate a group of posture robust pattern contour curves. The evaluation and output module is used to perform global curvature energy integral evaluation on the posture robust pattern contour curve group. When the global curvature energy value is lower than the preset smoothness threshold, the final parameterized pattern data is output.

2. The parametric design system for garment patterns according to claim 1, characterized in that, The step of calculating the curvature distribution of each contour line in the set of body surface feature contour lines, extracting curvature extrema points and curvature sign reversal points, and constructing a body surface curvature feature topology map includes: For each of the aforementioned body surface feature contour lines, a cubic B-spline parameterized fit is performed to obtain a continuously differentiable parameterized curve representation. The sequence of signed curvature values ​​at equal arc length sampling points is calculated along the parameterized curve. Extremum detection is performed on the signed curvature value sequence to extract the coordinates of local curvature maxima and minima; The zero-crossing positions in the signed curvature value sequence where the curvature sign changes from positive to negative or from negative to positive are detected and marked as curvature inflection points. Using the curvature extrema and curvature inflection points as nodes, and the average curvature of the curve segments between adjacent nodes as the edge attribute, a topological graph of the surface curvature features is constructed.

3. The parametric design system for garment patterns according to claim 1, characterized in that, The process of generating an initial pattern outline curve group based on the surface curvature feature topology map, and identifying the set of splicing nodes of adjacent curve segments in the initial pattern outline curve group, includes: Map the nodes in the surface curvature feature topology map to the corresponding positions on the two-dimensional pattern unfolding surface to generate a pattern feature control point set; Based on the rules of garment structure design, a piecewise cubic B-spline curve is constructed between the set of pattern feature control points to form the initial pattern outline curve set. Traverse all curve segment endpoints in the initial template outline curve group, identify endpoints shared by two or more curve segments, and mark them as splicing nodes; For each splicing node, record the identifiers of all curve segments associated with it and the parameter direction of each curve segment at that splicing node, forming the splicing node set.

4. The parametric design system for garment patterns according to claim 1, characterized in that, The construction of local curvature transition patches at each splicing node based on the splicing continuity state descriptor, so that each splicing node achieves G2 curvature continuity, includes: For each splicing node, the end tangent vectors and curvature values ​​of the curve segments on both sides are read from the splicing continuity state descriptor; Determine whether the angle between the tangent vectors on both sides is less than the preset tangent deviation threshold. If not, make fine adjustments to the position of the nearest control point of the curve segment on both sides to align the tangent direction and achieve G1 tangent continuity. Based on the continuity of the G1 tangent, the arithmetic mean of the curvature values ​​on both sides is used as the target curvature, and an auxiliary control point is introduced on each side of the splicing node. The position coordinates of the auxiliary control point are solved by simultaneously solving the target curvature constraint and the minimum bending energy condition, and the local curvature transition patch is constructed. The continuity of the second derivative of the local curvature transition patch at the splicing node was verified, confirming that G2 curvature continuity was achieved.

5. The parametric design system for garment patterns according to claim 1, characterized in that, The step of converting the multi-pose joint angle sequence data into a deformation displacement field on the initial pattern contour curve group through skeleton skinning mapping includes: For each pose frame in the multi-pose joint angle sequence data, the spatial transformation matrix of each bone segment is calculated according to the skeletal kinematic chain; The control points of the initial profile curve group are weighted by skin weights and the spatial transformation matrix to obtain the deformation position of each control point in the pose frame. Calculate the displacement difference before and after deformation at each control point to form the deformation displacement field under this attitude frame; The deformation displacement field is arranged according to the attitude frame sequence to form a control point displacement time series, which is used to re-evaluate the continuity state of each splicing node.

6. The parametric design system for garment patterns according to claim 1, characterized in that, The step of performing reverse compensation adjustment of control points for the local curvature transition patch corresponding to the splicing node whose deviation exceeds a preset continuity threshold based on the continuity deviation after deformation includes: Filter the splicing nodes whose continuity deviation after deformation exceeds the preset continuity threshold and mark them as nodes to be compensated; For each node to be compensated, extract the continuous deviation sequence across all attitude frames, and identify the maximum deviation value in the sequence and the corresponding extreme attitude frame. Using the extreme attitude frame as an extreme working condition, the position offset to be applied to the auxiliary control point of the local curvature transition patch is calculated in reverse, so that the curvature deviation at the splicing node under the extreme working condition is reduced to within the preset continuity threshold. The position offset is superimposed on the initial position coordinates of the auxiliary control point, and the local curvature transition patch is reconstructed with the updated auxiliary control point. The continuity deviation in all attitude frames is verified to be no more than the preset continuity threshold, thus forming the attitude robust profile curve set.

7. The parametric design system for garment patterns according to claim 1, characterized in that, The global curvature energy integral evaluation of the attitude-robust profile curve set includes: The arc length integral of the square of curvature is calculated for each curve segment in the posture robust profile curve group, and is denoted as the bending energy of that curve segment. The global curvature energy value is obtained by summing the bending energies of all curve segments. The ratio of the global curvature energy value to the global curvature energy value of the initial pattern contour curve group is calculated and denoted as the smoothness deterioration coefficient. When the smoothness deterioration coefficient is greater than the preset deterioration threshold, the control point optimization is re-executed on the curve segment with the largest bending energy in order to minimize the bending energy of the curve segment without breaking the G2 continuity constraint of its end splicing node. The above optimization is performed iteratively until the smoothness degradation coefficient is no greater than the preset degradation threshold, and then the final parameterized pattern data is output.

8. The parametric design system for garment patterns according to claim 4, characterized in that, The process of solving for the position coordinates of the auxiliary control point by simultaneously solving the target curvature constraint and the minimum bending energy condition includes: Using the position coordinates of the auxiliary control points as optimization variables, an equality constraint is established with the curvature at the splicing node equal to the target curvature. The integral of the square arc length of the curvature of the local curvature transition patch within the neighborhood of the auxiliary control point is used as the objective function. The Lagrange multiplier method is used to combine the equality constraints with the objective function to construct an unconstrained optimization problem. Solve the stationary point for the unconstrained optimization problem to obtain the position coordinates of the auxiliary control point that satisfies G2 continuity and minimizes bending energy.