Mathematical Modeling for Costume Design & Fashion Technology: A Parametric Computational Framework for Contemporary Design

Abstract

The fashion and costume design industry is rapidly evolving through the integration of computational tools, requiring systematic models that balance creativity with technical accuracy. This paper presents a compact mathematical framework that unifies three critical components parametric pattern generation, physics-based cloth simulation, and generative design to support modern fashion product development. The proposed model begins with a parametric geometric representation, which enables accurate and flexible pattern construction adaptable to diverse body measurements and design specifications. This mathematical layer allows for automated size grading and precise adjustment of silhouettes, thus minimizing manual intervention. The second component introduces a physics-informed simulation, where fabric behavior is modeled using measurable parameters such as elasticity, drape, and tensile strength. This ensures that virtual garments reflect real-world performance, improving both fitting accuracy and material optimization. The third component incorporates a generative aesthetic design layer, which draws upon stylistic priors and computational creativity to accelerate the exploration of new forms, textures, and patterns. Together, these three modules establish an end-to-end digital pipeline that integrates rapid ideation, virtual fitting, manufacturability assessment, and sustainability checks. Implementation strategies are briefly outlined, focusing on algorithmic adaptability and integration with existing computer-aided design (CAD) environments. Validation strategies suggest experimental prototyping using digital twins and virtual fashion platforms to evaluate model efficiency. Key advantages of the framework include reduction in material wastage during sampling, accelerated iteration cycles from concept to prototype, and enhanced systematic control over both aesthetic and functional constraints. By embedding mathematics into fashion workflows, the framework provides a scalable pathway for researchers, technologists, and designers seeking to align artistic innovation with efficiency, sustainability, and precision in contemporary costume and fashion technology.

Introduction

Fashion and costume design are rapidly evolving through computational methods. Tools like parametric modeling, 3D simulation, and generative AI are transforming how garments are designed, prototyped, and evaluated. However, these technologies often function in silos, lacking integration.

This paper proposes a unified mathematical framework that combines:

• Parametric geometry for automated pattern generation,

• Fabric physics for realistic draping,

• Generative AI for stylistic variation.

Key Contributions:

1. Unified Design Framework that integrates:

   • Parametric pattern generation,

   • Physics-based cloth simulation,

   • Generative design and aesthetic modeling.

2. Closed-Loop System Architecture:

   • With modules for Parametric, Physics, and Generative layers.

   • Feedback loops for manufacturability and design optimization.

3. Compact Mathematical Model:

   • Patterns are modeled using B-splines/Bézier curves defined by parameters like length (l), angle (α), and curvature (r).

   • Fabric behavior is modeled via physical properties (e.g., stiffness, mass) and simulated using finite element analysis (FEA).

   • Generative models (e.g., GANs, diffusion transformers) explore stylistic diversity from latent space.

Motivation:

Traditional design workflows are time-consuming and produce material waste. Emerging tools improve parts of the process but are fragmented. A unified, mathematically grounded approach can:

• Speed up design cycles,

• Improve garment realism and fit,

• Reduce waste and cost,

• Enable creative flexibility.

Review of Prior Work:

• Physics Simulations (e.g., DiffCloth, 2024) introduced differentiable simulations for inverse design.

• Parametric Pattern Drafting (e.g., Kim & Park, 2025) enabled automated pattern generation from formulas.

• Generative AI (e.g., GarmentDiffusion, 2025) leveraged multimodal models to generate vector patterns from image/text inputs.

• Limitations: Lack of integrated systems that combine geometry, physics, and generative design.

Framework Components:

1. Parametric Pattern Module:

• Encodes garment panels as 2D parametric curves (e.g., B-splines).

• Parameters (θ) control length, angles, and curvature.

• Supports automated size grading using anthropometric data.

2. Physics-Based Cloth Layer:

• Uses finite element simulation to model fabric behavior.

• Captures stretching, bending, and collision with the body/environment.

• Supports differentiable simulations for optimization (e.g., material tuning, fit prediction).

3. Generative Aesthetic Layer:

• Style is encoded as latent variable z from a generative model (GAN, VAE, Diffusion).

• Generates silhouettes, textures, trims.

• Aesthetic loss functions guide outputs to match style targets.

• Feasibility is ensured through filtering with manufacturability constraints.

Mathematical Summary:

• P: Parametric pattern domain, governed by θ (e.g., length, curvature).

• F: Fabric properties (e.g., stiffness k, bending b, density ρ, friction μ).

• z: Latent style variables from generative model G.

Pipeline:

1. θ → 2D pattern → simulated 3D mesh (Ψ),

2. Mesh + z → stylized render (Φ),

3. Optimize (θ, F, z) for fit, comfort, aesthetics, manufacturability.

Objective Function:

min?θ,F,z(λ1Lfit+λ2Lcomf+λ3Laes+λ4Lmanuf)\min_{\theta, F, z} \left( \lambda_1 L_{\text{fit}} + \lambda_2 L_{\text{comf}} + \lambda_3 L_{\text{aes}} + \lambda_4 L_{\text{manuf}} \right)θ,F,zmin?(λ1?Lfit?+λ2?Lcomf?+λ3?Laes?+λ4?Lmanuf?)

Optimization and Implementation:

Optimization Strategy:

• Step 1: Coarse search using surrogate models (e.g., Gaussian Processes).

• Step 2: Fine-tuning with gradient descent via differentiable simulation.

Practical Tips:

• Use low-res meshes during iteration; upscale for final output.

• Precompute fabric parameter tables for speed.

• Incorporate PM4Fashion-style scripting for designer-friendly workflows.

Applications:

• Fashion CAD systems, digital garment sampling,

• Virtual try-ons with personalized fit prediction,

• Sustainable fashion via reduced prototyping cycles,

• Creative AI tools for designers to explore styles, materials, and fits.

Conclusion

The proposed work compact mathematical framework that unifies parametric pattern modeling, physics-based cloth simulation, and generative design into a single cohesive pipeline for costume and fashion design. The aim is to create a system that not only accelerates design iteration but also ensures manufacturability and creative flexibility. At the foundation, parametric pattern models encode garment panels using geometric descriptors such as lengths, angles, and curvature radii. This constructive representation makes it possible to define seams, grading rules, and size adjustments through linear or low-order nonlinear constraints. Designers can interactively manipulate these parameters, while the system guarantees geometric consistency and scalability across different body landmarks. On top of this, physics-based cloth simulation provides realistic virtual fitting. Using triangle mesh models with stretching, bending, and contact energies, the simulator captures how fabric responds to drape, body movement, and styling. Differentiable simulation techniques further allow inverse design: material or structural parameters can be optimized to achieve a target fit or drape. This closes the loop between abstract parameters and physically grounded garment behavior. Finally, generative design modules introduce creativity by sampling from latent style distributions. A learned decoder maps latent codes into texture maps, trim placements, or silhouette variations, while filtering mechanisms enforce manufacturability thresholds. This ensures that novel aesthetic variants remain feasible for production. Together, these three layers form an integrated framework that supports rapid ideation, virtual prototyping, and early manufacturability checks. By reducing reliance on physical sampling, the pipeline accelerates workflows and broadens the creative space available to designers. The next crucial step is empirical validation: testing with measured fabric properties and conducting designer-in-the-loop studies. Such experiments will quantify improvements in efficiency, creative exploration, and production readiness, offering evidence of the framework’s practical impact on contemporary fashion technology.

References

[1] Survey — Mathematics meets the fashion industry on path to product (Polimi survey). [2] PM4Fashion — A Scriptable Parametric Modeling Interface for Conceptual Fashion (2024). [3] Park, et al. — Developing parametric design fashion products using 3D printing technology (2021). [4] Garment3DGen — 3D Garment Stylization and Texture Generation (2024/2025 preprints). [5] DiffCloth / Differentiable cloth simulation resources (Projective Dynamics extensions). [6] Physics-based graph network model for cloth animation (PGN-Cloth). [7] AI-driven computational creativity in fashion design: a review (2024–2025 review). [8] Industry blog / application — How Does 3D Clothing Simulation Transform Fashion Design? (2025). [9] Li, et al. (2024). DiffCloth: Differentiable Cloth Simulation with Dry Frictional Contact. [10] Yu, Cordier, & Seo. (2024). Inverse Garment and Pattern Modeling with a Differentiable Simulator. [11] Kim, & Park. (2025). Development of a Parametric Production Jacket Pattern for an Automated Pattern-Making System. Fashion and Textiles, Springer Open. [12] Oh, & Kim. (2023). Automatic Generation of Parametric Patterns from Grading Patterns using Artificial Intelligence. International Journal of Clothing Science and Technology, Emerald. [13] Li, Yao, & Wang. (2025). GarmentDiffusion: 3D Garment Sewing Pattern Generation with Multimodal Diffusion Transformers. Garment3DGen Team. (2024). Garment3DGen: 3D Garment Stylization and Texture Generation.

Copyright

Copyright © 2025 P. Selvi, M. Santhiya, M. Madhumitha, V. A. Shrimathi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.