Construction of a PH-curve of space by orthogonal projections of its hodograph

   In modern geometric modeling, a class of plane and spatial PH-curves (рythagorean hodograph curves) is known, which were theoretically substantiated by mathematician Rida T. Farouki in 2007. PH-curves have a unique property that is important in solving many different practical problems, namely: the "parametric speed" of these curves, i.e. the derivative of the arc length with respect to the curve parameter, is a polynomial (or rational) function of the parameter. This property is due to the fact that the coordinate components of the hodograph of the PH–curve are elements of the Pythagorean (n + 1)-tuple of coordinate polynomials. Due to this property, PH-curves are in demand in solving various practical problems: generating trajectories for UAVs (unmanned aerial vehicles), optimizing the path of mobile robots, calculating segments of the axis of a road that are optimal in shape and length, etc. etc. In the theory of spatial PH-curves, algorithms for their analytical construction have been developed. In this paper, an approach to constructing these curves is proposed, based on the theory of plane PH-curves and implemented by sequentially constructing images-projections of spatial PH-curves on two coordinate planes. That is, solving the spatial construction problem is reduced to solving two problems of sequential construction on coordinate planes. Numerical examples are given that demonstrate the efficiency of the proposed approach. At the same time, the calculation algorithms are simpler than in the case of the known spatial analytical approach. In the direction of developing the proposed approach, numerical examples of constructing a spatial composite PH-curve by smoothness C1 are given.

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