A Triple-tangency constraint formulation for ball-end mill positioning at tri-surface intersections

Abstract Accurate tool positioning at three-surface junctions remains a challenging problem in ball-end milling, particularly in regions where three surfaces intersect and residual material persists after roughing operations. Conventional offset-based or path-trimming strategies typically determine tool positions through sequential geometric constructions and do not directly enforce the coupled tangency conditions arising at multi-surface junctions, requiring explicit offset construction and additional numerical processing to obtain admissible tool positions. This work proposes a geometric formulation for computing admissible tool center positions based on a triple-tangency constraint system. The intersecting surfaces are represented implicitly, while the ball-end mill is modeled as a sphere of fixed radius whose center satisfies simultaneous signed-distance constraints with respect to all surfaces. A linear predictor derived from the surface normals at the intersection point is obtained by solving a linearized tangency system, providing a geometry-consistent initialization that enables rapid convergence of the subsequent Gauss–Newton refinement applied to the nonlinear constraint system. Local existence and conditioning are characterized through the rank properties of the associated normal matrix, enabling detection of degenerate or infeasible configurations for a given tool radius. The formulation provides a consistent mechanism for determining admissible tool penetration at tri-surface intersections and is applicable to residual material removal and corner finishing in CNC machining. The proposed framework establishes a unified geometric and numerical basis for robust ball-end mill positioning at complex surface junctions.