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<p>In the realm of computational geometry and CAD design, accurately modelling fundamental transcendental curves such as cylindrical helices and circle involutes presents significant challenges. This paper addresses these challenges by introducing a novel method for constructing CAD models of these curves, which is particularly relevant in the design, quality control and manufacturing of gears and transmission systems. The proposed method employs a mapping of the diagonal from <i>uv</i>-domain onto the cylindrical surface to create the CAD model of the helix — the UV helix. Similarly, the UV involute is generated by mapping the diagonal from <i>uv</i>-domain onto the tangent developable surface of the UV helix. It is noteworthy that commercial CAD systems do not provide built-in support for generating involute models other than using a curve given by parametric equations, which underlines the importance of our proposed method. We rigorously evaluate the geometric accuracy and curvature of these models using quantitative metrics and comparative analyses with existing methods, demonstrating significant improvements in precision and efficiency. With a preset software tolerance of 0.001 mm, the accuracy of the cylindrical UV helix was achieved with a maximum deviation of 0.000616 mm from the theoretical helix and a maximum curvature deviation of 0.000900&#xa0;mm. For a generalised UV helix located on a general surface of revolution, the maximum deviation from the theoretical helix was 0.000729&#xa0;mm and the maximum curvature deviation 0.003150&#xa0;mm. The accuracy of the circular UV involute was achieved with a&#xa0;maximum deviation of 0.00378 mm. The implications of this work are significant, as the newly developed methods facilitate the efficient creation and analysis of geometrically accurate CAD models, ultimately enhancing the design process for engineers. Future research may explore the application of these techniques in more complex geometrical configurations and their integration into existing CAD systems.</p>

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A novel approach to modelling helix and involute curves through UV mapping

  • 期刊论文

  • 发表日期：

    2025年10月7日

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  • Marta Hlavová,
  • Ivana Linkeová

摘要

In the realm of computational geometry and CAD design, accurately modelling fundamental transcendental curves such as cylindrical helices and circle involutes presents significant challenges. This paper addresses these challenges by introducing a novel method for constructing CAD models of these curves, which is particularly relevant in the design, quality control and manufacturing of gears and transmission systems. The proposed method employs a mapping of the diagonal from uv-domain onto the cylindrical surface to create the CAD model of the helix — the UV helix. Similarly, the UV involute is generated by mapping the diagonal from uv-domain onto the tangent developable surface of the UV helix. It is noteworthy that commercial CAD systems do not provide built-in support for generating involute models other than using a curve given by parametric equations, which underlines the importance of our proposed method. We rigorously evaluate the geometric accuracy and curvature of these models using quantitative metrics and comparative analyses with existing methods, demonstrating significant improvements in precision and efficiency. With a preset software tolerance of 0.001 mm, the accuracy of the cylindrical UV helix was achieved with a maximum deviation of 0.000616 mm from the theoretical helix and a maximum curvature deviation of 0.000900 mm. For a generalised UV helix located on a general surface of revolution, the maximum deviation from the theoretical helix was 0.000729 mm and the maximum curvature deviation 0.003150 mm. The accuracy of the circular UV involute was achieved with a maximum deviation of 0.00378 mm. The implications of this work are significant, as the newly developed methods facilitate the efficient creation and analysis of geometrically accurate CAD models, ultimately enhancing the design process for engineers. Future research may explore the application of these techniques in more complex geometrical configurations and their integration into existing CAD systems.