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This work analyses the incidence of the parameter selection of B-spline curves, used in the design of displacement and motion functions , on its degree of freedom and shape. A complete design process based on the use of non-parametric B-spline curves and the convenience of selecting the curve parameters from the point of view of its practical application is shown. In order to make easy the design and use of the displacement function, the algorithms for derivation and integration of the B-splines used are presented. Three case studies validate the proposed design process and the selection of the adequate parameters. The first case presents the design of a displacement function of a roller follower driven by a disk cam; the corresponding cam profile and its prototype are shown. The second case presents the design of the motion function corresponding to the cutting unit of a manufacturing cardboard tube machine. The third case exposes the design of the displacement function of the bar feeding mechanism in a single-spindle automatic lathe, to produce a partial thread screw of hexagonal head. Response to Reviewers: The manuscript was properly revised as pointed out, so it is now acceptable for publication. The authors would thank to the reviewers and the editor their decision.

High-speed turning is an advanced and emerging machining technique that, in contrast to the conventional machining, allows the manufacture of the workpiece with high accuracy, efficiency and quality, with lower production costs and with a considerable reduction in the machining times. The cutting tools used for the conventional machining cannot be employed for high-speed machining due to a high temperature induced in machining and a lower tool life. Therefore, it is necessary to study the influence of high cutting speeds on the temperature distribution in different typologies of cutting tools, with the aim of evaluating their behavior. In this paper, a finite element method modeling approach with arbitrary Lagrangian-Eulerian fully coupled thermal-stress analysis is employed. The research presents the results of different cutting tools (two coated carbide tools and uncoated cermet) effects on average surface temperature fields on the cutting edge in the dry high-speed turning of AISI 1045 steel. The numerical experiments were designed based on different cutting tools like input parameters and different temperature field zones like dependent variables in the dry high-speed turning of AISI 1045 steel. The results indicate that the dry high-speed turning of AISI 1045 steel does not influence significantly the temperature field zones when P10, P15 or P25 inserts are used. Therefore, the use of a dry high-speed turning method, which reduces the amount of lubricant and increases productivity, may represent an alternative to turning to the extent here described.