On smoothness of connection of spatial curve cyclographic projection segments

The correlation between smoothness of connection of segments of a spatial curve and the respective cyclographic projection is considered. The sufficient condition for smoothness C α k − 1 at the point of connection of segments of cyclographic projection of a spatial curve described by a polynomial spline of order k +1 is formulated and verified. It has been demonstrated that despite the derivative of order k +1 of the spline function describing the spatial composite curve along with all the higher-order derivatives not being continuous, it is possible to connect segments of the cyclographic projection with smoothness C α r , where r = k + i , i = 0,1,2,…. Achieving smoothness C α k − 1 , C α r through the formulated condition of sufficiency is demonstrated on numeric examples. The obtained results of the study can serve as the basis of development of effective algorithms of cyclographic formation of geometric objects applied in geometric optics, road surface form design, and optimal cutting tool trajectory design in pocket machining.