Riemannian Geometry–Based Thermal Metrics for Optimal Toolpath Planning in Titanium Machining

Machining of titanium alloys is characterized by severe thermal loads, localized plastic deformation,and accelerated tool wear. Conventional toolpath planning relies on Euclidean geometry and kinematicconstraints, largely ignoring the spatial variability of thermal fields induced during cutting. In this work, wepropose a novel framework in which the machining domain is endowed with a temperature-dependentRiemannian metric. Under this formulation, toolpaths correspond to geodesics minimizing a thermally weightedlength functional rather than purely geometric distance. This paper develops the full mathematical formulation ofthe Riemannian metric, derive the associated geodesic equations, and present a numerical example involving theexternal turning of a titanium tube. The results demonstrate how thermally unfavorable regions act as zones ofhigh curvature, naturally steering optimal paths away from excessive heat accumulation.