Accelerated Planar Development of Convex Free-form Mesh Patches Using a Variable Step-size Energy Dissipation Approach

Abstract Free-form complex surfaces are prevalent in modern graphic applications. With the increasing prevalence of complex 3D surfaces enabled by advances in range scanning and 3D printing technologies, minimising parameterization times for large meshes has become crucial. This paper proposes an efficient approach for the planar development of convex free-form mesh patches using an improved energy-based technique with a variable step-size algorithm. Building upon the energy model of Wang et al., our study addresses the limitations of conventional energy dissipation algorithms, which employ fixed step sizes. The proposed variable step-size method, particularly suitable for convex or disk-shaped mesh surfaces, dynamically adjusts steps, significantly reducing energy dissipation iterations. Leveraging our previous geometric flattening method, we further enhance planar surface development using an advanced mass-spring-based approach. Here, we show that our method accelerates the mechanical flattening process while maintaining high accuracy, achieving a shape error of 0.400 and an area error of 0.147 after 36 iterations for the Surf1 patch, reducing the required iterations by nearly half compared to the fixed step-size method. This study contributes to advancing the field of surface parameterization and flattening, with potential applications in various industries.