Spline Curves Formation Given Extreme Derivatives

This paper is dedicated to development of mathematical models for polynomial spline curve formation given extreme vector derivatives. This theoretical problem is raised in the view of a wide variety of theoretical and practical problems considering motion of physical objects along certain trajectories with predetermined laws of variation of speed, acceleration, jerk, etc. The analysis of the existing body of work on computational geometry performed by the authors did not reveal any systematic research in mathematical model development dedicated to solution of similar tasks. The established purpose of the research is therefore to develop mathematical models of formation of spline curves based on polynomials of various orders modeling the determined trajectories. The paper presents mathematical models of spline curve formation given extreme derivatives of the initial orders. The paper considers construction of Hermite and Bézier spline curves of various orders consisting of various segments. The acquired mathematical models are generalized for the cases of vector derivatives of higher orders. The presented models are of systematic nature and are universal, i.e., they can be applied in formation of any polynomial spline curves given extreme vector derivatives. The paper provides a number of examples validating the presented models.