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August 17, 2025Open Access

Triangulation of helical surfaces in forming processes

Key Points

The proposed method offers a continuous representation of differential geometry for helical surfaces, enhancing accuracy.
Key findings indicate that previous flat triangulation methods introduce significant errors in tool shape analysis.
The developed approach utilizes helical trajectories to accurately triangulate surfaces, improving geometric fidelity.
Implementation of this technique supports better understanding of forming processes and initial tool design requirements.

Abstract

Using modern triangulation methods allows simplifying problems of representing complex shaped surfaces when study-ing processes of forming parts and designing shaped cutting tools.The main problem in solving shaping problems is the problem of adequate representation of local differential character-istics of the surface, namely: tangent, normal, curvature and torsion. The surface represented by a set of flat triangulation elements has only discrete values of the specified parameters of the local geometry of the surface, while for the nominal surface of the part they are continuous.Analysis of research and publications on triangulation of shaped helical surfaces by flat finite elements showed that the use of flat elements introduces significant errors into the study of shaping processes and the search for a theoretically accurate profile of the cutting tool.The aim of the article is to develop the method for triangulating a helical surface using helical trajectories, which makes it possible to move from discrete representation of differential characteristics of helical surface to continuous one and to for-mulate mathematical dependencies that describe characteristics of the surface.Based on the differential theory of surfaces, the method for triangulating helical surface with spatial cylindrical elements whose boundaries are helical trajectories of the points of the end section of the part has been developed. Criteria for reliable representation of the triangulated surface have been formulated. Mathematical dependencies have been derived for determin-ing differential characteristics of the local geometry of the surface after its triangulation by helical trajectories.In the article, the method for triangulating helical surface using a set of helical trajectories was proposed for the first time, which allows moving from discrete determination of differential characteristics of the local geometry of the triangulated surface to continuous one. The presented materials can be used to study processes of forming and search for the profile of the initial tool surfaces in conventional processes of processing parts.

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