Generating spiral tool path to machine free-form surface with complex topology based on fusing constraint mapping and enriched Voronoi diagram

  • Zhiping Liu
  • Xiongbing Li
  • Bing YiEmail author


Although there are methods to machine free-form surfaces, serious distortion in the concave–convex characteristic of the flattened-plane boundary, high deformation of the surface geometry, and single limitation of the surface topology are usually produced. Thus, a novel surface flattening method is proposed in this paper to retain the concave–convex characteristic and reduce the deformation, and a spiral path is generated to machine the free-form surface with various topologies. The machined surface is mapped to a planar region with a free boundary using a fusing constraint mapping method. First, the spring-mass-based stretching constraint is used to minimize the length differences of the triangular sides, which are caused by surface flattening. Subsequently, in order to flatten surfaces with an isometric deformation, we perform this operation under the constraint of hinge-based bending. Eventually, the global constraint, the minimization of the global energy, is employed to acquire a less distorted plane. Then, to generate a planar spiral path fit for machining various planes, which are concave, convex, simply-connected, or multiply connected, enrichment of the conventional Voronoi diagram, interpolation between the wave fronts, and rounding of the spiral polyline are implemented. For machining a free-form surface, by inversely mapping the planar path, a spiral tool path is planned. Experimental results are given to illustrate the effectiveness of the presented methods.


Fusing constraint mapping Enriched Voronoi diagram Free-form surface machining Error analysis 


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The authors would like to express their gratitude to Travis D. Ashworth for many useful comments on English writing.

Funding information

This work is supported by the National Natural Science Foundation of China under Grant No.51605495, the Fundamental Research Funds for the Central Universities of Central South University under Grant NO.2017zzts195, the International Postdoctoral Exchange Fellowship Program Grant NO.2017[59], and Research on Laboratory Construction and Management of Central South University Program NO. 201719.


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© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Traffic and Transportation EngineeringCentral South UniversityChangshaChina
  2. 2.University of MichiganAnn ArborUSA

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